Integrand size = 30, antiderivative size = 291 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {b \left (a b c f^2-a^2 d f^2-2 b^2 e (d e-c f)\right ) x}{2 a (b c-a d) e (b e-a f)^2 (d e-c f) \sqrt {a+b x^2}}+\frac {f^2 x}{2 e (b e-a f) (d e-c f) \sqrt {a+b x^2} \left (e+f x^2\right )}-\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2} (d e-c f)^2}+\frac {f^2 (2 b e (3 d e-2 c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{5/2} (d e-c f)^2} \] Output:
-1/2*b*(a*b*c*f^2-a^2*d*f^2-2*b^2*e*(-c*f+d*e))*x/a/(-a*d+b*c)/e/(-a*f+b*e )^2/(-c*f+d*e)/(b*x^2+a)^(1/2)+1/2*f^2*x/e/(-a*f+b*e)/(-c*f+d*e)/(b*x^2+a) ^(1/2)/(f*x^2+e)-d^3*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c ^(1/2)/(-a*d+b*c)^(3/2)/(-c*f+d*e)^2+1/2*f^2*(2*b*e*(-2*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)/(-a *f+b*e)^(5/2)/(-c*f+d*e)^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 15.53 (sec) , antiderivative size = 1386, normalized size of antiderivative = 4.76 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2),x]
Output:
(x*((42*d^2*(-15*c*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 10*d*x^2*Sqrt [((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 15*c*ArcTanh[Sqrt[((b*c - a*d)*x^2)/ (c*(a + b*x^2))]] + 10*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2) )]] + 2*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5 /2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 2*d*x^2*(((b*c - a*d)*x^2)/( c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*c - a*d)*x^2)/(c* (a + b*x^2))]))/(c^2*(d*e - c*f)^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/ 2)) - (42*d*f*(-15*e*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - 10*f*x^2*Sq rt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] + 15*e*ArcTanh[Sqrt[((b*e - a*f)*x^2 )/(e*(a + b*x^2))]] + 10*f*x^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^ 2))]] + 2*e*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))] + 2*f*x^2*(((b*e - a*f)*x^2) /(e*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/( e*(a + b*x^2))]))/(e^2*(d*e - c*f)^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^( 3/2)) + (f*(-2625*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - (5250*f*x^2*Sq rt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e - (2310*f^2*x^4*Sqrt[((b*e - a*f) *x^2)/(e*(a + b*x^2))])/e^2 + 70*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2) + (560*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2))/e + (280*f^2*x^4* (((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2))/e^2 + 2625*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + (5250*f*x^2*ArcTanh[Sqrt[((b*e - a*f)*x^...
Time = 0.85 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.73, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {421, 25, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int \frac {a (b d e+b c f-2 a d f)-2 b (b c-a d) f x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {\int \frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}-\frac {\int -\frac {f \left (2 b d e x^2+a (3 d e-c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e f}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b d e x^2+a (3 d e-c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b d e x^2+a (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 b d e \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 b d e \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 \sqrt {b} d e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 \sqrt {b} d e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\left (2 b d e^2-a f (3 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+7 d e)+2 b^2 e (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (d e-c f)+2 b^2 c e\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \left (\frac {d^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2}-\frac {f \left (\frac {\frac {2 \sqrt {b} d e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 b d e^2-a f (3 d e-c f)\right )}{\sqrt {e} f \sqrt {b e-a f}}}{2 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right )}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}\) |
Input:
Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2),x]
Output:
(b*((b*(b*c - a*d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)) - (-1/2* (f*(2*b^2*c*e - 2*a^2*d*f - a*b*(d*e - c*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) + (a*(2*a^2*d*f^2 - a*b*f*(7*d*e + c*f) + 2*b^2*e*(d*e + 2*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)))/(a*(b*e - a*f))))/(b*c - a*d)^2 + (d^2*((d^2*((Sqrt[ b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b*c - a*d]*ArcTanh[(Sqr t[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[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*Ar cTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - ((2*b*d*e^2 - a*f*(3*d*e - c*f))*A rcTanh[(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))/(b*c - a*d)^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 = 1.61 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {-\left (a d -b c \right ) \left (\left (c \,f^{2}-3 d e f \right ) a +\left (-4 c e f +6 d \,e^{2}\right ) b \right ) a \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right ) f^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (-2 a \,d^{3} e \sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (c f -d e \right ) \left (a^{3} d \,f^{3}-b \,f^{3} \left (-x^{2} d +c \right ) a^{2}-a \,b^{2} c \,f^{3} x^{2}-2 b^{3} e \left (f \,x^{2}+e \right ) \left (c f -d e \right )\right ) x \right )}{2 \sqrt {\left (a f -b e \right ) e}\, \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, \left (c f -d e \right )^{2} \left (a d -b c \right ) e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{2} a}\) | \(335\) |
| default | \(\text {Expression too large to display}\) | \(2899\) |
Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/2/((a*f-b*e)*e)^(1/2)/((a*d-b*c)*c)^(1/2)/(b*x^2+a)^(1/2)*(-(a*d-b*c)*(( c*f^2-3*d*e*f)*a+(-4*c*e*f+6*d*e^2)*b)*a*((a*d-b*c)*c)^(1/2)*(b*x^2+a)^(1/ 2)*(f*x^2+e)*f^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e )*e)^(1/2)*(-2*a*d^3*e*(b*x^2+a)^(1/2)*(f*x^2+e)*(a*f-b*e)^2*arctan(c*(b*x ^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b*c)*c)^(1/2)*(c*f-d*e)*(a^3*d*f^ 3-b*f^3*(-d*x^2+c)*a^2-a*b^2*c*f^3*x^2-2*b^3*e*(f*x^2+e)*(c*f-d*e))*x))/(c *f-d*e)^2/(a*d-b*c)/e/(f*x^2+e)/(a*f-b*e)^2/a
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right ) \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)/(f*x**2+e)**2,x)
Output:
Integral(1/((a + b*x**2)**(3/2)*(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)*(f*x^2 + e)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (261) = 522\).
Time = 2.19 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.17 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {b} d^{3} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{{\left (b c d^{2} e^{2} - a d^{3} e^{2} - 2 \, b c^{2} d e f + 2 \, a c d^{2} e f + b c^{3} f^{2} - a c^{2} d f^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {b^{3} x}{{\left (a b^{3} c e^{2} - a^{2} b^{2} d e^{2} - 2 \, a^{2} b^{2} c e f + 2 \, a^{3} b d e f + a^{3} b c f^{2} - a^{4} d f^{2}\right )} \sqrt {b x^{2} + a}} - \frac {{\left (6 \, b^{\frac {3}{2}} d e^{2} f^{2} - 4 \, b^{\frac {3}{2}} c e f^{3} - 3 \, a \sqrt {b} d e f^{3} + a \sqrt {b} c f^{4}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, {\left (b^{2} d^{2} e^{5} - 2 \, b^{2} c d e^{4} f - 2 \, a b d^{2} e^{4} f + b^{2} c^{2} e^{3} f^{2} + 4 \, a b c d e^{3} f^{2} + a^{2} d^{2} e^{3} f^{2} - 2 \, a b c^{2} e^{2} f^{3} - 2 \, a^{2} c d e^{2} f^{3} + a^{2} c^{2} e f^{4}\right )} \sqrt {-b^{2} e^{2} + a b e f}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} e f^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} f^{3} + a^{2} \sqrt {b} f^{3}}{{\left (b^{2} d e^{4} - b^{2} c e^{3} f - 2 \, a b d e^{3} f + 2 \, a b c e^{2} f^{2} + a^{2} d e^{2} f^{2} - a^{2} c e f^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )}} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorithm="giac")
Output:
sqrt(b)*d^3*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/s qrt(-b^2*c^2 + a*b*c*d))/((b*c*d^2*e^2 - a*d^3*e^2 - 2*b*c^2*d*e*f + 2*a*c *d^2*e*f + b*c^3*f^2 - a*c^2*d*f^2)*sqrt(-b^2*c^2 + a*b*c*d)) + b^3*x/((a* b^3*c*e^2 - a^2*b^2*d*e^2 - 2*a^2*b^2*c*e*f + 2*a^3*b*d*e*f + a^3*b*c*f^2 - a^4*d*f^2)*sqrt(b*x^2 + a)) - 1/2*(6*b^(3/2)*d*e^2*f^2 - 4*b^(3/2)*c*e*f ^3 - 3*a*sqrt(b)*d*e*f^3 + a*sqrt(b)*c*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))/((b^2*d^2*e^5 - 2 *b^2*c*d*e^4*f - 2*a*b*d^2*e^4*f + b^2*c^2*e^3*f^2 + 4*a*b*c*d*e^3*f^2 + a ^2*d^2*e^3*f^2 - 2*a*b*c^2*e^2*f^3 - 2*a^2*c*d*e^2*f^3 + a^2*c^2*e*f^4)*sq rt(-b^2*e^2 + a*b*e*f)) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*e*f^2 - (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*f^3 + a^2*sqrt(b)*f^3)/((b^2* d*e^4 - b^2*c*e^3*f - 2*a*b*d*e^3*f + 2*a*b*c*e^2*f^2 + a^2*d*e^2*f^2 - a^ 2*c*e*f^3)*((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))
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^2), x)
Time = 5.95 (sec) , antiderivative size = 12518, normalized size of antiderivative = 43.02 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^2,x)
Output:
( - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x **2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**6*d**3*e**3*f**4 - 2*sqrt( c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt (d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**6*d**3*e**2*f**5*x**2 + 14*sqrt(c)*sq rt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*s qrt(b)*x)/(sqrt(c)*sqrt(b)))*a**5*b*d**3*e**4*f**3 + 12*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x )/(sqrt(c)*sqrt(b)))*a**5*b*d**3*e**3*f**4*x**2 - 2*sqrt(c)*sqrt(a*d - b*c )*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(s qrt(c)*sqrt(b)))*a**5*b*d**3*e**2*f**5*x**4 - 30*sqrt(c)*sqrt(a*d - b*c)*a tan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt (c)*sqrt(b)))*a**4*b**2*d**3*e**5*f**2 - 16*sqrt(c)*sqrt(a*d - b*c)*atan(( sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*s qrt(b)))*a**4*b**2*d**3*e**4*f**3*x**2 + 14*sqrt(c)*sqrt(a*d - b*c)*atan(( sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*s qrt(b)))*a**4*b**2*d**3*e**3*f**4*x**4 + 26*sqrt(c)*sqrt(a*d - b*c)*atan(( sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*s qrt(b)))*a**3*b**3*d**3*e**6*f - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))* a**3*b**3*d**3*e**5*f**2*x**2 - 30*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a...