Integrand size = 30, antiderivative size = 284 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx=-\frac {d (8 b d e-8 b c f-5 a d f) x \sqrt {a+b x^2}}{8 f^3}+\frac {b d^2 x^3 \sqrt {a+b x^2}}{4 f^2}-\frac {(b e-a f) (d e-c f)^2 x \sqrt {a+b x^2}}{2 e f^3 \left (e+f x^2\right )}+\frac {\left (3 a^2 d^2 f^2-24 a b d f (d e-c f)+8 b^2 \left (3 d^2 e^2-4 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} f^4}-\frac {\sqrt {b e-a f} (d e-c f) (2 b e (3 d e-c f)-a f (3 d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^4} \] Output:
-1/8*d*(-5*a*d*f-8*b*c*f+8*b*d*e)*x*(b*x^2+a)^(1/2)/f^3+1/4*b*d^2*x^3*(b*x ^2+a)^(1/2)/f^2-1/2*(-a*f+b*e)*(-c*f+d*e)^2*x*(b*x^2+a)^(1/2)/e/f^3/(f*x^2 +e)+1/8*(3*a^2*d^2*f^2-24*a*b*d*f*(-c*f+d*e)+8*b^2*(c^2*f^2-4*c*d*e*f+3*d^ 2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/f^4-1/2*(-a*f+b*e)^(1/2 )*(-c*f+d*e)*(2*b*e*(-c*f+3*d*e)-a*f*(c*f+3*d*e))*arctanh((-a*f+b*e)^(1/2) *x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/f^4
Time = 2.00 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx=\frac {\frac {f x \sqrt {a+b x^2} \left (a f \left (-8 c d e f+4 c^2 f^2+d^2 e \left (9 e+5 f x^2\right )\right )-2 b e \left (2 c^2 f^2-4 c d f \left (2 e+f x^2\right )+d^2 \left (6 e^2+3 e f x^2-f^2 x^4\right )\right )\right )}{e \left (e+f x^2\right )}-\frac {4 \sqrt {-b e+a f} (d e-c f) (2 b e (3 d e-c f)-a f (3 d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2}}-\frac {\left (3 a^2 d^2 f^2-24 a b d f (d e-c f)+8 b^2 \left (3 d^2 e^2-4 c d e f+c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{8 f^4} \] Input:
Integrate[((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^2,x]
Output:
((f*x*Sqrt[a + b*x^2]*(a*f*(-8*c*d*e*f + 4*c^2*f^2 + d^2*e*(9*e + 5*f*x^2) ) - 2*b*e*(2*c^2*f^2 - 4*c*d*f*(2*e + f*x^2) + d^2*(6*e^2 + 3*e*f*x^2 - f^ 2*x^4))))/(e*(e + f*x^2)) - (4*Sqrt[-(b*e) + a*f]*(d*e - c*f)*(2*b*e*(3*d* e - c*f) - a*f*(3*d*e + c*f))*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/e^(3/2) - ((3*a^2*d^2*f^2 - 24*a* b*d*f*(d*e - c*f) + 8*b^2*(3*d^2*e^2 - 4*c*d*e*f + c^2*f^2))*Log[-(Sqrt[b] *x) + Sqrt[a + b*x^2]])/Sqrt[b])/(8*f^4)
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {(4 b c-a d) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {\int -\frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}+\frac {d x \sqrt {a+b x^2}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\int \frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
Input:
Int[((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^2,x]
Output:
$Aborted
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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 + 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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b 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] && GtQ[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]
Time = 1.06 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \sqrt {b}\, \left (-3 b d \,e^{2}+f \left (b c +\frac {3 a d}{2}\right ) e +\frac {a c \,f^{2}}{2}\right ) \left (c f -d e \right ) \left (-a f +b e \right ) \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (-\frac {3 \left (8 b^{2} d^{2} e^{2}-8 \left (a d +\frac {4 b c}{3}\right ) d b f e +f^{2} \left (a^{2} d^{2}+8 a b c d +\frac {8}{3} b^{2} c^{2}\right )\right ) \left (f \,x^{2}+e \right ) e \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{4}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, \left (3 e^{3} b \,d^{2}-\frac {9 d \left (\left (-\frac {2 x^{2} d}{3}+\frac {16 c}{9}\right ) b +a d \right ) f \,e^{2}}{4}+2 \left (\left (-\frac {1}{4} d^{2} x^{4}-c d \,x^{2}+\frac {1}{2} c^{2}\right ) b +d a \left (-\frac {5 x^{2} d}{8}+c \right )\right ) f^{2} e -a \,c^{2} f^{3}\right ) x f \right )}{2 \sqrt {\left (a f -b e \right ) e}\, \sqrt {b}\, f^{4} e \left (f \,x^{2}+e \right )}\) | \(306\) |
| risch | \(\text {Expression too large to display}\) | \(1364\) |
| default | \(\text {Expression too large to display}\) | \(3499\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
-1/2/((a*f-b*e)*e)^(1/2)/b^(1/2)*(-2*b^(1/2)*(-3*b*d*e^2+f*(b*c+3/2*a*d)*e +1/2*a*c*f^2)*(c*f-d*e)*(-a*f+b*e)*(f*x^2+e)*arctan(e*(b*x^2+a)^(1/2)/x/(( a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(-3/4*(8*b^2*d^2*e^2-8*(a*d+4/3*b*c )*d*b*f*e+f^2*(a^2*d^2+8*a*b*c*d+8/3*b^2*c^2))*(f*x^2+e)*e*arctanh((b*x^2+ a)^(1/2)/x/b^(1/2))+b^(1/2)*(b*x^2+a)^(1/2)*(3*e^3*b*d^2-9/4*d*((-2/3*x^2* d+16/9*c)*b+a*d)*f*e^2+2*((-1/4*d^2*x^4-c*d*x^2+1/2*c^2)*b+d*a*(-5/8*x^2*d +c))*f^2*e-a*c^2*f^3)*x*f))/f^4/e/(f*x^2+e)
Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (252) = 504\).
Time = 11.62 (sec) , antiderivative size = 2263, normalized size of antiderivative = 7.97 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^2,x, algorithm="fricas")
Output:
[1/16*((24*b^2*d^2*e^4 - 8*(4*b^2*c*d + 3*a*b*d^2)*e^3*f + (8*b^2*c^2 + 24 *a*b*c*d + 3*a^2*d^2)*e^2*f^2 + (24*b^2*d^2*e^3*f - 8*(4*b^2*c*d + 3*a*b*d ^2)*e^2*f^2 + (8*b^2*c^2 + 24*a*b*c*d + 3*a^2*d^2)*e*f^3)*x^2)*sqrt(b)*log (-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(6*b^2*d^2*e^4 + a*b*c^2* e*f^3 - (8*b^2*c*d + 3*a*b*d^2)*e^3*f + 2*(b^2*c^2 + a*b*c*d)*e^2*f^2 + (6 *b^2*d^2*e^3*f + a*b*c^2*f^4 - (8*b^2*c*d + 3*a*b*d^2)*e^2*f^2 + 2*(b^2*c^ 2 + a*b*c*d)*e*f^3)*x^2)*sqrt((b*e - a*f)/e)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 - 4*(a*e^2*x + (2* b*e^2 - a*e*f)*x^3)*sqrt(b*x^2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f* x^2 + e^2)) + 2*(2*b^2*d^2*e*f^3*x^5 - (6*b^2*d^2*e^2*f^2 - (8*b^2*c*d + 5 *a*b*d^2)*e*f^3)*x^3 - (12*b^2*d^2*e^3*f - 4*a*b*c^2*f^4 - (16*b^2*c*d + 9 *a*b*d^2)*e^2*f^2 + 4*(b^2*c^2 + 2*a*b*c*d)*e*f^3)*x)*sqrt(b*x^2 + a))/(b* e*f^5*x^2 + b*e^2*f^4), -1/8*((24*b^2*d^2*e^4 - 8*(4*b^2*c*d + 3*a*b*d^2)* e^3*f + (8*b^2*c^2 + 24*a*b*c*d + 3*a^2*d^2)*e^2*f^2 + (24*b^2*d^2*e^3*f - 8*(4*b^2*c*d + 3*a*b*d^2)*e^2*f^2 + (8*b^2*c^2 + 24*a*b*c*d + 3*a^2*d^2)* e*f^3)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (6*b^2*d^2*e^4 + a*b*c^2*e*f^3 - (8*b^2*c*d + 3*a*b*d^2)*e^3*f + 2*(b^2*c^2 + a*b*c*d)*e^2 *f^2 + (6*b^2*d^2*e^3*f + a*b*c^2*f^4 - (8*b^2*c*d + 3*a*b*d^2)*e^2*f^2 + 2*(b^2*c^2 + a*b*c*d)*e*f^3)*x^2)*sqrt((b*e - a*f)/e)*log(((8*b^2*e^2 - 8* a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 - 4*(a...
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{2}}{\left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**2/(f*x**2+e)**2,x)
Output:
Integral((a + b*x**2)**(3/2)*(c + d*x**2)**2/(e + f*x**2)**2, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{2}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^2,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^2/(f*x^2 + e)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (252) = 504\).
Time = 0.17 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^2,x, algorithm="giac")
Output:
1/8*sqrt(b*x^2 + a)*(2*b*d^2*x^2/f^2 - (8*b^3*d^2*e*f^6 - 8*b^3*c*d*f^7 - 5*a*b^2*d^2*f^7)/(b^2*f^9))*x - 1/16*(24*b^2*d^2*e^2 - 32*b^2*c*d*e*f - 24 *a*b*d^2*e*f + 8*b^2*c^2*f^2 + 24*a*b*c*d*f^2 + 3*a^2*d^2*f^2)*log((sqrt(b )*x - sqrt(b*x^2 + a))^2)/(sqrt(b)*f^4) + 1/2*(6*b^(5/2)*d^2*e^4 - 8*b^(5/ 2)*c*d*e^3*f - 9*a*b^(3/2)*d^2*e^3*f + 2*b^(5/2)*c^2*e^2*f^2 + 10*a*b^(3/2 )*c*d*e^2*f^2 + 3*a^2*sqrt(b)*d^2*e^2*f^2 - a*b^(3/2)*c^2*e*f^3 - 2*a^2*sq rt(b)*c*d*e*f^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))/(sqrt(-b^2*e^2 + a*b*e *f)*e*f^4) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*d^2*e^4 - 4*(sqrt( b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c*d*e^3*f - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*d^2*e^3*f + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^2* e^2*f^2 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c*d*e^2*f^2 + (sqrt( b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*d^2*e^2*f^2 - 3*(sqrt(b)*x - sqrt(b* x^2 + a))^2*a*b^(3/2)*c^2*e*f^3 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sq rt(b)*c*d*e*f^3 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*c^2*f^4 + a^ 2*b^(3/2)*d^2*e^3*f - 2*a^2*b^(3/2)*c*d*e^2*f^2 - a^3*sqrt(b)*d^2*e^2*f^2 + a^2*b^(3/2)*c^2*e*f^3 + 2*a^3*sqrt(b)*c*d*e*f^3 - a^3*sqrt(b)*c^2*f^4)/( ((sqrt(b)*x - sqrt(b*x^2 + a))^4*f + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*e - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*f + a^2*f)*e*f^4)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^2}{{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^2,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^2, x)
Time = 0.30 (sec) , antiderivative size = 2161, normalized size of antiderivative = 7.61 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^2,x)
Output:
( - 4*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x **2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b*c**2*e*f**3 - 4*sqrt(e)*s qrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)* sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b*c**2*f**4*x**2 - 8*sqrt(e)*sqrt(a*f - b* e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/( sqrt(e)*sqrt(b)))*a*b*c*d*e**2*f**2 - 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt (a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt( b)))*a*b*c*d*e*f**3*x**2 + 12*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e ) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b*d **2*e**3*f + 12*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sq rt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b*d**2*e**2*f**2* x**2 - 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c**2*e**2*f**2 - 8*sq rt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - s qrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c**2*e*f**3*x**2 + 32*sqrt(e)*sq rt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*s qrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c*d*e**3*f + 32*sqrt(e)*sqrt(a*f - b*e)* atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqr t(e)*sqrt(b)))*b**2*c*d*e**2*f**2*x**2 - 24*sqrt(e)*sqrt(a*f - b*e)*atan(( sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e...