Integrand size = 30, antiderivative size = 376 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {d^2 (b c-a d) x \sqrt {a+b x^2}}{2 c (d e-c f)^3 \left (c+d x^2\right )}-\frac {f (b e-a f) x \sqrt {a+b x^2}}{4 e (d e-c f)^2 \left (e+f x^2\right )^2}+\frac {f (a f (11 d e-3 c f)-2 b e (3 d e+c f)) x \sqrt {a+b x^2}}{8 e^2 (d e-c f)^3 \left (e+f x^2\right )}-\frac {d \sqrt {b c-a d} (a d (d e-7 c f)+2 b c (d e+2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (d e-c f)^4}-\frac {\left (8 a b d e^2 f (5 d e+c f)-8 b^2 d e^3 (d e+2 c f)-a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} \sqrt {b e-a f} (d e-c f)^4} \] Output:
-1/2*d^2*(-a*d+b*c)*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)^3/(d*x^2+c)-1/4*f*(-a*f +b*e)*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)^2/(f*x^2+e)^2+1/8*f*(a*f*(-3*c*f+11*d *e)-2*b*e*(c*f+3*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-c*f+d*e)^3/(f*x^2+e)-1/2*d* (-a*d+b*c)^(1/2)*(a*d*(-7*c*f+d*e)+2*b*c*(2*c*f+d*e))*arctanh((-a*d+b*c)^( 1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(3/2)/(-c*f+d*e)^4-1/8*(8*a*b*d*e^2*f*(c *f+5*d*e)-8*b^2*d*e^3*(2*c*f+d*e)-a^2*f^2*(3*c^2*f^2-14*c*d*e*f+35*d^2*e^2 ))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/(-a*f+b*e)^ (1/2)/(-c*f+d*e)^4
Time = 11.18 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (x \sqrt {a+b x^2} \left (\frac {4 d^2 (b c-a d)}{c (-d e+c f)^3 \left (c+d x^2\right )}+\frac {2 f (-b e+a f)}{e (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f (2 b e (3 d e+c f)+a f (-11 d e+3 c f))}{e^2 (d e-c f)^3 \left (e+f x^2\right )}\right )+\frac {4 d \sqrt {-b c+a d} (a d (d e-7 c f)+2 b c (d e+2 c f)) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2} (d e-c f)^4}+\frac {\left (-8 a b d e^2 f (5 d e+c f)+8 b^2 d e^3 (d e+2 c f)+a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} \sqrt {-b e+a f} (d e-c f)^4}\right ) \] Input:
Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^2*(e + f*x^2)^3),x]
Output:
(x*Sqrt[a + b*x^2]*((4*d^2*(b*c - a*d))/(c*(-(d*e) + c*f)^3*(c + d*x^2)) + (2*f*(-(b*e) + a*f))/(e*(d*e - c*f)^2*(e + f*x^2)^2) - (f*(2*b*e*(3*d*e + c*f) + a*f*(-11*d*e + 3*c*f)))/(e^2*(d*e - c*f)^3*(e + f*x^2))) + (4*d*Sq rt[-(b*c) + a*d]*(a*d*(d*e - 7*c*f) + 2*b*c*(d*e + 2*c*f))*ArcTan[(Sqrt[-( b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(3/2)*(d*e - c*f)^4) + ((-8* a*b*d*e^2*f*(5*d*e + c*f) + 8*b^2*d*e^3*(d*e + 2*c*f) + a^2*f^2*(35*d^2*e^ 2 - 14*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^(5/2)*Sqrt[-(b*e) + a*f]*(d*e - c*f)^4))/8
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\int -\frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b (3 d e-c f) x^2+a (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\int -\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {\int \frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 422 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}\right )}{d}\) |
Input:
Int[(a + b*x^2)^(3/2)/((c + d*x^2)^2*(e + f*x^2)^3),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*a*d, 0]))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2 ]), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/((a + b*x^2)*Sqrt[e + f*x^2]), x], x] - Simp[d/(b*c - a*d) Int[1/((c + d*x^2)*Sqrt[e + f*x^2]), x], x] / ; FreeQ[{a, b, c, d, e, f}, x]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[-d/(b*c - a*d) Int[(c + d*x^2)^q*(e + f*x^2)^r, x], x] + Simp[b/(b*c - a*d) Int[(c + d*x^2)^(q + 1)*((e + f*x^2)^r/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LeQ[q, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d) Int[(a + b*x^2)^p*(c + d*x^2)^ (q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d) Int[(a + b*x^2)^(p + 1 )*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
Time = 1.88 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.21
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (c \left (x^{2} d +c \right ) \left (\frac {8 b^{2} d^{2} e^{4}}{3}-\frac {40 \left (a d -\frac {2 b c}{5}\right ) d b f \,e^{3}}{3}+\frac {35 a d \left (a d -\frac {8 b c}{35}\right ) f^{2} e^{2}}{3}-\frac {14 a^{2} c d e \,f^{3}}{3}+a^{2} c^{2} f^{4}\right ) \sqrt {\left (a d -b c \right ) c}\, \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {28 \sqrt {\left (a f -b e \right ) e}\, \left (\left (a d -b c \right ) d \left (x^{2} d +c \right ) \left (-\frac {d \left (a d +2 b c \right ) e}{7}+c f \left (a d -\frac {4 b c}{7}\right )\right ) \left (f \,x^{2}+e \right )^{2} e^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\frac {5 \left (c f -d e \right ) \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, \left (-\frac {4 d^{2} \left (a d -b c \right ) e^{4}}{5}-\frac {8 d f \left (a \,d^{2} x^{2}-2 b c d \,x^{2}-b \,c^{2}\right ) e^{3}}{5}-\frac {13 d \,f^{2} \left (\left (-\frac {6 b \,x^{2}}{13}+a \right ) c^{2}+d \,x^{2} \left (-\frac {10 b \,x^{2}}{13}+a \right ) c +\frac {4 a \,d^{2} x^{4}}{13}\right ) e^{2}}{5}+c \left (c \left (\frac {2 b \,x^{2}}{5}+a \right )-\frac {11 a d \,x^{2}}{5}\right ) \left (x^{2} d +c \right ) f^{3} e +\frac {3 x^{2} f^{4} c^{2} a \left (x^{2} d +c \right )}{5}\right ) x}{28}\right )}{3}\right )}{8 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{4} \left (f \,x^{2}+e \right )^{2} e^{2} c \left (x^{2} d +c \right )}\) | \(455\) |
| default | \(\text {Expression too large to display}\) | \(10782\) |
Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
-3/8/((a*d-b*c)*c)^(1/2)*(c*(d*x^2+c)*(8/3*b^2*d^2*e^4-40/3*(a*d-2/5*b*c)* d*b*f*e^3+35/3*a*d*(a*d-8/35*b*c)*f^2*e^2-14/3*a^2*c*d*e*f^3+a^2*c^2*f^4)* ((a*d-b*c)*c)^(1/2)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^( 1/2))-28/3*((a*f-b*e)*e)^(1/2)*((a*d-b*c)*d*(d*x^2+c)*(-1/7*d*(a*d+2*b*c)* e+c*f*(a*d-4/7*b*c))*(f*x^2+e)^2*e^2*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c) *c)^(1/2))+5/28*(c*f-d*e)*((a*d-b*c)*c)^(1/2)*(b*x^2+a)^(1/2)*(-4/5*d^2*(a *d-b*c)*e^4-8/5*d*f*(a*d^2*x^2-2*b*c*d*x^2-b*c^2)*e^3-13/5*d*f^2*((-6/13*b *x^2+a)*c^2+d*x^2*(-10/13*b*x^2+a)*c+4/13*a*d^2*x^4)*e^2+c*(c*(2/5*b*x^2+a )-11/5*a*d*x^2)*(d*x^2+c)*f^3*e+3/5*x^2*f^4*c^2*a*(d*x^2+c))*x))/((a*f-b*e )*e)^(1/2)/(c*f-d*e)^4/(f*x^2+e)^2/e^2/c/(d*x^2+c)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/2)/(d*x**2+c)**2/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{2} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^2*(f*x^2 + e)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (340) = 680\).
Time = 3.92 (sec) , antiderivative size = 1411, normalized size of antiderivative = 3.75 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="giac")
Output:
1/2*(2*b^(5/2)*c^2*d^2*e - a*b^(3/2)*c*d^3*e - a^2*sqrt(b)*d^4*e + 4*b^(5/ 2)*c^3*d*f - 11*a*b^(3/2)*c^2*d^2*f + 7*a^2*sqrt(b)*c*d^3*f)*arctan(1/2*(( sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/ ((c*d^4*e^4 - 4*c^2*d^3*e^3*f + 6*c^3*d^2*e^2*f^2 - 4*c^4*d*e*f^3 + c^5*f^ 4)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/8*(8*b^(5/2)*d^2*e^4 + 16*b^(5/2)*c*d*e^3 *f - 40*a*b^(3/2)*d^2*e^3*f - 8*a*b^(3/2)*c*d*e^2*f^2 + 35*a^2*sqrt(b)*d^2 *e^2*f^2 - 14*a^2*sqrt(b)*c*d*e*f^3 + 3*a^2*sqrt(b)*c^2*f^4)*arctan(1/2*(( sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/ ((d^4*e^6 - 4*c*d^3*e^5*f + 6*c^2*d^2*e^4*f^2 - 4*c^3*d*e^3*f^3 + c^4*e^2* f^4)*sqrt(-b^2*e^2 + a*b*e*f)) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2 )*c^2*d - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c*d^2 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*d^3 + a^2*b^(3/2)*c*d^2 - a^3*sqrt(b)*d^3) /((c*d^3*e^3 - 3*c^2*d^2*e^2*f + 3*c^3*d*e*f^2 - c^4*f^3)*((sqrt(b)*x - sq rt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)) - 1/4*(8*(sqrt(b)*x - sqrt(b*x^2 + a))^ 6*b^(5/2)*d*e^3*f + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c*e^2*f^2 - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d*e^2*f^2 + 11*(sqrt(b)*x - s qrt(b*x^2 + a))^6*a^2*sqrt(b)*d*e*f^3 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6* a^2*sqrt(b)*c*f^4 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d*e^4 + 16* (sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*e^3*f - 136*(sqrt(b)*x - sqrt...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^2\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((a + b*x^2)^(3/2)/((c + d*x^2)^2*(e + f*x^2)^3),x)
Output:
int((a + b*x^2)^(3/2)/((c + d*x^2)^2*(e + f*x^2)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{2} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x)
Output:
int((b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x)