Integrand size = 30, antiderivative size = 532 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=-\frac {f x \sqrt {a+b x^2}}{6 e (d e-c f) \left (e+f x^2\right )^3}+\frac {f (a f (11 d e-5 c f)-2 b e (5 d e-2 c f)) x \sqrt {a+b x^2}}{24 e^2 (b e-a f) (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f \left (4 b^2 e^2 \left (11 d^2 e^2-7 c d e f+2 c^2 f^2\right )+3 a^2 f^2 \left (19 d^2 e^2-16 c d e f+5 c^2 f^2\right )-2 a b e f \left (52 d^2 e^2-41 c d e f+13 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{48 e^3 (b e-a f)^2 (d e-c f)^3 \left (e+f x^2\right )}-\frac {d^3 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (d e-c f)^4}+\frac {\left (16 b^3 d^3 e^6+2 a^2 b e f^2 \left (45 d^3 e^3-40 c d^2 e^2 f+25 c^2 d e f^2-6 c^3 f^3\right )-a^3 f^3 \left (35 d^3 e^3-35 c d^2 e^2 f+21 c^2 d e f^2-5 c^3 f^3\right )-8 a b^2 e^2 f \left (9 d^3 e^3-6 c d^2 e^2 f+4 c^2 d e f^2-c^3 f^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{16 e^{7/2} (b e-a f)^{5/2} (d e-c f)^4} \] Output:
-1/6*f*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)/(f*x^2+e)^3+1/24*f*(a*f*(-5*c*f+11*d *e)-2*b*e*(-2*c*f+5*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-a*f+b*e)/(-c*f+d*e)^2/(f *x^2+e)^2-1/48*f*(4*b^2*e^2*(2*c^2*f^2-7*c*d*e*f+11*d^2*e^2)+3*a^2*f^2*(5* c^2*f^2-16*c*d*e*f+19*d^2*e^2)-2*a*b*e*f*(13*c^2*f^2-41*c*d*e*f+52*d^2*e^2 ))*x*(b*x^2+a)^(1/2)/e^3/(-a*f+b*e)^2/(-c*f+d*e)^3/(f*x^2+e)-d^3*(-a*d+b*c )^(1/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(1/2)/(-c*f+ d*e)^4+1/16*(16*b^3*d^3*e^6+2*a^2*b*e*f^2*(-6*c^3*f^3+25*c^2*d*e*f^2-40*c* d^2*e^2*f+45*d^3*e^3)-a^3*f^3*(-5*c^3*f^3+21*c^2*d*e*f^2-35*c*d^2*e^2*f+35 *d^3*e^3)-8*a*b^2*e^2*f*(-c^3*f^3+4*c^2*d*e*f^2-6*c*d^2*e^2*f+9*d^3*e^3))* arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(7/2)/(-a*f+b*e)^(5/ 2)/(-c*f+d*e)^4
Time = 12.20 (sec) , antiderivative size = 485, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\frac {\frac {f (d e-c f) x \sqrt {a+b x^2} \left (-8 e^2 (d e-c f)^2-\frac {2 e (d e-c f) (2 b e (5 d e-2 c f)+a f (-11 d e+5 c f)) \left (e+f x^2\right )}{b e-a f}-\frac {\left (4 b^2 e^2 \left (11 d^2 e^2-7 c d e f+2 c^2 f^2\right )+3 a^2 f^2 \left (19 d^2 e^2-16 c d e f+5 c^2 f^2\right )-2 a b e f \left (52 d^2 e^2-41 c d e f+13 c^2 f^2\right )\right ) \left (e+f x^2\right )^2}{(b e-a f)^2}\right )}{e^3 \left (e+f x^2\right )^3}+\frac {48 d^3 \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}+\frac {3 \left (16 b^3 d^3 e^6+2 a^2 b e f^2 \left (45 d^3 e^3-40 c d^2 e^2 f+25 c^2 d e f^2-6 c^3 f^3\right )-8 a b^2 e^2 f \left (9 d^3 e^3-6 c d^2 e^2 f+4 c^2 d e f^2-c^3 f^3\right )+a^3 f^3 \left (-35 d^3 e^3+35 c d^2 e^2 f-21 c^2 d e f^2+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{7/2} (-b e+a f)^{5/2}}}{48 (d e-c f)^4} \] Input:
Integrate[Sqrt[a + b*x^2]/((c + d*x^2)*(e + f*x^2)^4),x]
Output:
((f*(d*e - c*f)*x*Sqrt[a + b*x^2]*(-8*e^2*(d*e - c*f)^2 - (2*e*(d*e - c*f) *(2*b*e*(5*d*e - 2*c*f) + a*f*(-11*d*e + 5*c*f))*(e + f*x^2))/(b*e - a*f) - ((4*b^2*e^2*(11*d^2*e^2 - 7*c*d*e*f + 2*c^2*f^2) + 3*a^2*f^2*(19*d^2*e^2 - 16*c*d*e*f + 5*c^2*f^2) - 2*a*b*e*f*(52*d^2*e^2 - 41*c*d*e*f + 13*c^2*f ^2))*(e + f*x^2)^2)/(b*e - a*f)^2))/(e^3*(e + f*x^2)^3) + (48*d^3*Sqrt[-(b *c) + a*d]*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/Sqrt[ c] + (3*(16*b^3*d^3*e^6 + 2*a^2*b*e*f^2*(45*d^3*e^3 - 40*c*d^2*e^2*f + 25* c^2*d*e*f^2 - 6*c^3*f^3) - 8*a*b^2*e^2*f*(9*d^3*e^3 - 6*c*d^2*e^2*f + 4*c^ 2*d*e*f^2 - c^3*f^3) + a^3*f^3*(-35*d^3*e^3 + 35*c*d^2*e^2*f - 21*c^2*d*e* f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])] )/(e^(7/2)*(-(b*e) + a*f)^(5/2)))/(48*(d*e - c*f)^4)
Time = 1.04 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.14, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.767, Rules used = {421, 401, 25, 27, 402, 402, 27, 291, 221, 421, 301, 224, 219, 291, 221, 401, 25, 27, 398, 224, 219, 291, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^4}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}-\frac {\int -\frac {f \left (2 b (5 d e-2 c f) x^2+a (11 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e f}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b (5 d e-2 c f) x^2+a (11 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b (5 d e-2 c f) x^2+a (11 d e-5 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\int \frac {a (2 b e (17 d e-8 c f)-3 a f (11 d e-5 c f))-2 b (a f (11 d e-5 c f)-2 b e (5 d e-2 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\frac {\int -\frac {3 a \left (-8 b^2 (2 d e-c f) e^2+2 a b f (13 d e-6 c f) e-a^2 f^2 (11 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}-\frac {\int -\frac {f \left (2 b d e x^2+a (3 d e-c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e f}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b d e x^2+a (3 d e-c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b d e x^2+a (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 b d e \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 b d e \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 \sqrt {b} d e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 \sqrt {b} d e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 \sqrt {b} d e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 b d e^2-a f (3 d e-c f)\right )}{\sqrt {e} f \sqrt {b e-a f}}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {-\frac {3 a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (11 d e-5 c f)+2 a b e f (13 d e-6 c f)-8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-3 a^2 f^2 (11 d e-5 c f)+2 a b e f (28 d e-13 c f)-4 b^2 e^2 (5 d e-2 c f)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (11 d e-5 c f)-2 b e (5 d e-2 c f))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3}\right )}{(d e-c f)^2}\) |
Input:
Int[Sqrt[a + b*x^2]/((c + d*x^2)*(e + f*x^2)^4),x]
Output:
-((f*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(6*e*(e + f*x^2)^3) + (-1/4*((a*f*(1 1*d*e - 5*c*f) - 2*b*e*(5*d*e - 2*c*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)* (e + f*x^2)^2) + (-1/2*((2*a*b*e*f*(28*d*e - 13*c*f) - 3*a^2*f^2*(11*d*e - 5*c*f) - 4*b^2*e^2*(5*d*e - 2*c*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) - (3*a*(2*a*b*e*f*(13*d*e - 6*c*f) - a^2*f^2*(11*d*e - 5*c*f) - 8*b^2*e^2*(2*d*e - c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x ^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f)))/(6*e)))/(d*e - c* f)^2) + (d^2*((d^2*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sq rt[b*c - a*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqr t[c]*d)))/(d*e - c*f)^2 - (f*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(e + f* x^2)) + ((2*Sqrt[b]*d*e*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - ((2*b*d* e^2 - a*f*(3*d*e - c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x ^2])])/(Sqrt[e]*f*Sqrt[b*e - a*f]))/(2*e)))/(d*e - c*f)^2))/(d*e - c*f)^2
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[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
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[(((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]
Time = 2.16 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (\left (\frac {16 b^{3} d^{3} e^{6}}{5}-\frac {72 a \,b^{2} d^{3} e^{5} f}{5}+18 a \,d^{2} \left (a d +\frac {8 b c}{15}\right ) b \,f^{2} e^{4}-7 a d \left (a^{2} d^{2}+\frac {16}{7} a b c d +\frac {32}{35} b^{2} c^{2}\right ) f^{3} e^{3}+7 a c \left (a^{2} d^{2}+\frac {10}{7} a b c d +\frac {8}{35} b^{2} c^{2}\right ) f^{4} e^{2}-\frac {21 \left (a d +\frac {4 b c}{7}\right ) a^{2} c^{2} f^{5} e}{5}+a^{3} c^{3} f^{6}\right ) \sqrt {\left (a d -b c \right ) c}\, \left (f \,x^{2}+e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {16 \sqrt {\left (a f -b e \right ) e}\, \left (d^{3} e^{3} \left (f \,x^{2}+e \right )^{3} \left (a f -b e \right )^{2} \left (a d -b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )-\frac {11 \left (\frac {24 b^{2} d^{2} e^{6}}{11}-\frac {54 d b f \left (-\frac {2}{3} b d \,x^{2}+a d +\frac {4}{9} b c \right ) e^{5}}{11}+\frac {29 \left (\frac {44 b^{2} d^{2} x^{4}}{87}+\left (-\frac {250}{87} a b \,d^{2}-\frac {28}{29} b^{2} c d \right ) x^{2}+a^{2} d^{2}+2 a b c d +\frac {8 b^{2} c^{2}}{29}\right ) f^{2} e^{4}}{11}-\frac {32 \left (\left (\frac {13}{12} a b \,d^{2}+\frac {7}{24} b^{2} c d \right ) x^{4}+\left (-\frac {17}{12} a^{2} d^{2}-\frac {7}{3} a b c d -\frac {1}{4} b^{2} c^{2}\right ) x^{2}+a c \left (a d +\frac {5 b c}{8}\right )\right ) f^{3} e^{3}}{11}+\left (\left (\frac {19}{11} a^{2} d^{2}+\frac {8}{33} b^{2} c^{2}+\frac {82}{33} a b c d \right ) x^{4}+\left (-\frac {128}{33} a^{2} c d -\frac {70}{33} b \,c^{2} a \right ) x^{2}+a^{2} c^{2}\right ) f^{4} e^{2}+\frac {40 a c \left (\left (-\frac {6 a d}{5}-\frac {13 b c}{20}\right ) x^{2}+a c \right ) x^{2} f^{5} e}{33}+\frac {5 a^{2} c^{2} f^{6} x^{4}}{11}\right ) \left (c f -d e \right ) \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, x f}{16}\right )}{5}\right )}{16 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (a f -b e \right )^{2} \left (c f -d e \right )^{4} e^{3} \left (f \,x^{2}+e \right )^{3}}\) | \(622\) |
| default | \(\text {Expression too large to display}\) | \(7836\) |
Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
Output:
-5/16/((a*d-b*c)*c)^(1/2)*((16/5*b^3*d^3*e^6-72/5*a*b^2*d^3*e^5*f+18*a*d^2 *(a*d+8/15*b*c)*b*f^2*e^4-7*a*d*(a^2*d^2+16/7*a*b*c*d+32/35*b^2*c^2)*f^3*e ^3+7*a*c*(a^2*d^2+10/7*a*b*c*d+8/35*b^2*c^2)*f^4*e^2-21/5*(a*d+4/7*b*c)*a^ 2*c^2*f^5*e+a^3*c^3*f^6)*((a*d-b*c)*c)^(1/2)*(f*x^2+e)^3*arctan(e*(b*x^2+a )^(1/2)/x/((a*f-b*e)*e)^(1/2))+16/5*((a*f-b*e)*e)^(1/2)*(d^3*e^3*(f*x^2+e) ^3*(a*f-b*e)^2*(a*d-b*c)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))-1 1/16*(24/11*b^2*d^2*e^6-54/11*d*b*f*(-2/3*b*d*x^2+a*d+4/9*b*c)*e^5+29/11*( 44/87*b^2*d^2*x^4+(-250/87*a*b*d^2-28/29*b^2*c*d)*x^2+a^2*d^2+2*a*b*c*d+8/ 29*b^2*c^2)*f^2*e^4-32/11*((13/12*a*b*d^2+7/24*b^2*c*d)*x^4+(-17/12*a^2*d^ 2-7/3*a*b*c*d-1/4*b^2*c^2)*x^2+a*c*(a*d+5/8*b*c))*f^3*e^3+((19/11*a^2*d^2+ 8/33*b^2*c^2+82/33*a*b*c*d)*x^4+(-128/33*a^2*c*d-70/33*b*c^2*a)*x^2+a^2*c^ 2)*f^4*e^2+40/33*a*c*((-6/5*a*d-13/20*b*c)*x^2+a*c)*x^2*f^5*e+5/11*a^2*c^2 *f^6*x^4)*(c*f-d*e)*((a*d-b*c)*c)^(1/2)*(b*x^2+a)^(1/2)*x*f))/((a*f-b*e)*e )^(1/2)/(a*f-b*e)^2/(c*f-d*e)^4/e^3/(f*x^2+e)^3
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^4,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(1/2)/(d*x**2+c)/(f*x**2+e)**4,x)
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^4,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)*(f*x^2 + e)^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 3384 vs. \(2 (499) = 998\).
Time = 6.36 (sec) , antiderivative size = 3384, normalized size of antiderivative = 6.36 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^4,x, algorithm="giac")
Output:
(b^(3/2)*c*d^3 - a*sqrt(b)*d^4)*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))/((d^4*e^4 - 4*c*d^3*e^3*f + 6 *c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/ 16*(16*b^(7/2)*d^3*e^6 - 72*a*b^(5/2)*d^3*e^5*f + 48*a*b^(5/2)*c*d^2*e^4*f ^2 + 90*a^2*b^(3/2)*d^3*e^4*f^2 - 32*a*b^(5/2)*c^2*d*e^3*f^3 - 80*a^2*b^(3 /2)*c*d^2*e^3*f^3 - 35*a^3*sqrt(b)*d^3*e^3*f^3 + 8*a*b^(5/2)*c^3*e^2*f^4 + 50*a^2*b^(3/2)*c^2*d*e^2*f^4 + 35*a^3*sqrt(b)*c*d^2*e^2*f^4 - 12*a^2*b^(3 /2)*c^3*e*f^5 - 21*a^3*sqrt(b)*c^2*d*e*f^5 + 5*a^3*sqrt(b)*c^3*f^6)*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))/((b^2*d^4*e^9 - 4*b^2*c*d^3*e^8*f - 2*a*b*d^4*e^8*f + 6*b^2*c^2*d^2 *e^7*f^2 + 8*a*b*c*d^3*e^7*f^2 + a^2*d^4*e^7*f^2 - 4*b^2*c^3*d*e^6*f^3 - 1 2*a*b*c^2*d^2*e^6*f^3 - 4*a^2*c*d^3*e^6*f^3 + b^2*c^4*e^5*f^4 + 8*a*b*c^3* d*e^5*f^4 + 6*a^2*c^2*d^2*e^5*f^4 - 2*a*b*c^4*e^4*f^5 - 4*a^2*c^3*d*e^4*f^ 5 + a^2*c^4*e^3*f^6)*sqrt(-b^2*e^2 + a*b*e*f)) - 1/24*(48*(sqrt(b)*x - sqr t(b*x^2 + a))^10*b^(7/2)*d^2*e^5*f^2 - 168*(sqrt(b)*x - sqrt(b*x^2 + a))^1 0*a*b^(5/2)*d^2*e^4*f^3 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*c* d*e^3*f^4 + 174*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*d^2*e^3*f^4 - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*c^2*e^2*f^5 - 114*(sqrt(b)* x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*c*d*e^2*f^5 - 57*(sqrt(b)*x - sqrt(b*x ^2 + a))^10*a^3*sqrt(b)*d^2*e^2*f^5 + 36*(sqrt(b)*x - sqrt(b*x^2 + a))^...
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\left (d\,x^2+c\right )\,{\left (f\,x^2+e\right )}^4} \,d x \] Input:
int((a + b*x^2)^(1/2)/((c + d*x^2)*(e + f*x^2)^4),x)
Output:
int((a + b*x^2)^(1/2)/((c + d*x^2)*(e + f*x^2)^4), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^4} \, dx=\int \frac {\sqrt {b \,x^{2}+a}}{\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )^{4}}d x \] Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^4,x)
Output:
int((b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^4,x)