Integrand size = 30, antiderivative size = 490 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=-\frac {b \left (2 a b^2 c e f^2 (7 d e-5 c f)+a^3 d f^3 (7 d e-3 c f)-8 b^3 e^2 (d e-c f)^2-a^2 b f^2 \left (14 d^2 e^2-3 c d e f-3 c^2 f^2\right )\right ) x}{8 a (b c-a d) e^2 (b e-a f)^3 (d e-c f)^2 \sqrt {a+b x^2}}+\frac {f^2 x}{4 e (b e-a f) (d e-c f) \sqrt {a+b x^2} \left (e+f x^2\right )^2}-\frac {f^2 (a f (7 d e-3 c f)-4 b e (3 d e-2 c f)) x}{8 e^2 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )}-\frac {d^4 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2} (d e-c f)^3}-\frac {f^2 \left (4 a b e f \left (12 d^2 e^2-11 c d e f+3 c^2 f^2\right )-a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )-8 b^2 e^2 \left (6 d^2 e^2-8 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{7/2} (d e-c f)^3} \] Output:
-1/8*b*(2*a*b^2*c*e*f^2*(-5*c*f+7*d*e)+a^3*d*f^3*(-3*c*f+7*d*e)-8*b^3*e^2* (-c*f+d*e)^2-a^2*b*f^2*(-3*c^2*f^2-3*c*d*e*f+14*d^2*e^2))*x/a/(-a*d+b*c)/e ^2/(-a*f+b*e)^3/(-c*f+d*e)^2/(b*x^2+a)^(1/2)+1/4*f^2*x/e/(-a*f+b*e)/(-c*f+ d*e)/(b*x^2+a)^(1/2)/(f*x^2+e)^2-1/8*f^2*(a*f*(-3*c*f+7*d*e)-4*b*e*(-2*c*f +3*d*e))*x/e^2/(-a*f+b*e)^2/(-c*f+d*e)^2/(b*x^2+a)^(1/2)/(f*x^2+e)-d^4*arc tanh((-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)^3-1/8*f^2*(4*a*b*e*f*(3*c^2*f^2-11*c*d*e*f+12*d^2*e^2)-a^2*f^2* (3*c^2*f^2-10*c*d*e*f+15*d^2*e^2)-8*b^2*e^2*(3*c^2*f^2-8*c*d*e*f+6*d^2*e^2 ))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/(-a*f+b*e)^ (7/2)/(-c*f+d*e)^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 20.50 (sec) , antiderivative size = 2885, normalized size of antiderivative = 5.89 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^3),x]
Output:
(d^3*x*(-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))]))/(5*a*c^2*(d*e - c*f)^3*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2 )*Sqrt[a + b*x^2]*(1 + (b*x^2)/a)) + (d^2*f*x*(-15*e*Sqrt[((b*e - a*f)*x^2 )/(e*(a + b*x^2))] - 10*f*x^2*Sqrt[((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[Sqr t[((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))]))/(5*a*e^2*(-(d*e) + c*f)^ 3*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*Sqrt[a + b*x^2]*(1 + (b*x^2)/a )) - (d*f*x*(-2625*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - (5250*f*x^2*S qrt[((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...
Time = 1.30 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.60, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {421, 25, 402, 402, 402, 25, 27, 291, 221, 421, 401, 25, 27, 402, 25, 27, 291, 221, 422, 301, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int \frac {a (b d e+b c f-2 a d f)-4 b (b c-a d) f x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\int \frac {a \left (4 e (d e+2 c f) b^2-3 a f (5 d e+c f) b+6 a^2 d f^2\right )-2 b f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\frac {\int -\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {\int \frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\int -\frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b (3 d e-c f) x^2+a (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\int -\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {\int \frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 422 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}\) |
Input:
Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^3),x]
Output:
(d^2*((d^2*((d*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b *c - a*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c] *d)))/(d*e - c*f) - (f*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - (Sqrt[b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/ (Sqrt[e]*f)))/(d*e - c*f)))/(d*e - c*f)^2 - (f*(((d*e - c*f)*x*Sqrt[a + b* x^2])/(4*e*(e + f*x^2)^2) + (-1/2*((a*f*(7*d*e - 3*c*f) - 2*b*e*(3*d*e - c *f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) - (a*(a*f*(7*d*e - 3*c *f) - 4*b*e*(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)))/(d*e - c*f)^2))/(b*c - a*d )^2 + (b*((b*(b*c - a*d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)^2) - (-1/4*(f*(4*b^2*c*e - 2*a^2*d*f - a*b*(3*d*e - c*f))*x*Sqrt[a + b*x^2])/ (e*(b*e - a*f)*(e + f*x^2)^2) + (-1/2*(f*(8*b^3*c*e^2 + 6*a^3*d*f^2 - 2*a* b^2*e*(d*e - 5*c*f) - a^2*b*f*(19*d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) - (a*(6*a^3*d*f^3 - 8*b^3*e^2*(d*e + 3*c*f) + 4*a*b^2 *e*f*(10*d*e + 3*c*f) - a^2*b*f^2*(23*d*e + 3*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)))/(a*(b*e - a*f))))/(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[((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]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[-d/(b*c - a*d) Int[(c + d*x^2)^q*(e + f*x^2)^r, x], x] + Simp[b/(b*c - a*d) Int[(c + d*x^2)^(q + 1)*((e + f*x^2)^r/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LeQ[q, -1]
Time = 2.15 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \left (a d -b c \right ) a \sqrt {b \,x^{2}+a}\, \sqrt {\left (a d -b c \right ) c}\, \left (f^{2} \left (c^{2} f^{2}-\frac {10}{3} c d e f +5 d^{2} e^{2}\right ) a^{2}-4 \left (c^{2} f^{2}-\frac {11}{3} c d e f +4 d^{2} e^{2}\right ) b f e a +8 b^{2} \left (c^{2} f^{2}-\frac {8}{3} c d e f +2 d^{2} e^{2}\right ) e^{2}\right ) \left (f \,x^{2}+e \right )^{2} f^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \sqrt {\left (a f -b e \right ) e}\, \left (\frac {8 \sqrt {b \,x^{2}+a}\, a \,d^{4} e^{2} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right )^{3} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{5}+\sqrt {\left (a d -b c \right ) c}\, \left (c f -d e \right ) \left (\left (f \left (\frac {3 f \,x^{2}}{5}+e \right ) c -\frac {9 \left (\frac {7 f \,x^{2}}{9}+e \right ) d e}{5}\right ) d \,f^{4} a^{4}-\left (f^{2} \left (\frac {3 f \,x^{2}}{5}+e \right ) c^{2}+\frac {3 d f \left (-f^{2} x^{4}-\frac {2}{3} e f \,x^{2}+e^{2}\right ) c}{5}-\frac {16 \left (-\frac {7}{16} f^{2} x^{4}+\frac {5}{16} e f \,x^{2}+e^{2}\right ) d^{2} e}{5}\right ) b \,f^{3} a^{3}+\frac {12 \left (\left (\frac {3 f \,x^{2}}{4}+e \right ) f \left (-\frac {f \,x^{2}}{3}+e \right ) c^{2}-\frac {4 d e \left (\frac {3}{16} f^{2} x^{4}+\frac {17}{16} e f \,x^{2}+e^{2}\right ) c}{3}+\frac {4 \left (\frac {7 f \,x^{2}}{8}+e \right ) d^{2} x^{2} e^{2}}{3}\right ) b^{2} f^{3} a^{2}}{5}+\frac {12 c \left (f \left (\frac {5 f \,x^{2}}{6}+e \right ) c -\frac {4 \left (\frac {7 f \,x^{2}}{8}+e \right ) d e}{3}\right ) b^{3} x^{2} f^{3} e a}{5}+\frac {8 b^{4} e^{2} \left (f \,x^{2}+e \right )^{2} \left (c f -d e \right )^{2}}{5}\right ) x \right )}{8}}{\left (f \,x^{2}+e \right )^{2} \left (c f -d e \right )^{3} \left (a f -b e \right )^{3} e^{2} \sqrt {\left (a f -b e \right ) e}\, \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, a}\) | \(589\) |
| default | \(\text {Expression too large to display}\) | \(5160\) |
Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
5/8*(-3/5*(a*d-b*c)*a*(b*x^2+a)^(1/2)*((a*d-b*c)*c)^(1/2)*(f^2*(c^2*f^2-10 /3*c*d*e*f+5*d^2*e^2)*a^2-4*(c^2*f^2-11/3*c*d*e*f+4*d^2*e^2)*b*f*e*a+8*b^2 *(c^2*f^2-8/3*c*d*e*f+2*d^2*e^2)*e^2)*(f*x^2+e)^2*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)*(8/5*(b*x^2+a)^(1/2)*a*d^4 *e^2*(f*x^2+e)^2*(a*f-b*e)^3*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)*((f*(3/5*f*x^2+e)*c-9/5*(7/9*f*x^2+e)*d*e )*d*f^4*a^4-(f^2*(3/5*f*x^2+e)*c^2+3/5*d*f*(-f^2*x^4-2/3*e*f*x^2+e^2)*c-16 /5*(-7/16*f^2*x^4+5/16*e*f*x^2+e^2)*d^2*e)*b*f^3*a^3+12/5*((3/4*f*x^2+e)*f *(-1/3*f*x^2+e)*c^2-4/3*d*e*(3/16*f^2*x^4+17/16*e*f*x^2+e^2)*c+4/3*(7/8*f* x^2+e)*d^2*x^2*e^2)*b^2*f^3*a^2+12/5*c*(f*(5/6*f*x^2+e)*c-4/3*(7/8*f*x^2+e )*d*e)*b^3*x^2*f^3*e*a+8/5*b^4*e^2*(f*x^2+e)^2*(c*f-d*e)^2)*x))/((a*f-b*e) *e)^(1/2)/((a*d-b*c)*c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^2/(c*f-d*e)^3/(a*f -b*e)^3/e^2/(a*d-b*c)/a
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)*(f*x^2 + e)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 1630 vs. \(2 (456) = 912\).
Time = 9.21 (sec) , antiderivative size = 1630, normalized size of antiderivative = 3.33 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="giac")
Output:
sqrt(b)*d^4*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^3*e^3 - a*d^4*e^3 - 3*b*c^2*d^2*e^2*f + 3 *a*c*d^3*e^2*f + 3*b*c^3*d*e*f^2 - 3*a*c^2*d^2*e*f^2 - b*c^4*f^3 + a*c^3*d *f^3)*sqrt(-b^2*c^2 + a*b*c*d)) + b^4*x/((a*b^4*c*e^3 - a^2*b^3*d*e^3 - 3* a^2*b^3*c*e^2*f + 3*a^3*b^2*d*e^2*f + 3*a^3*b^2*c*e*f^2 - 3*a^4*b*d*e*f^2 - a^4*b*c*f^3 + a^5*d*f^3)*sqrt(b*x^2 + a)) - 1/8*(48*b^(5/2)*d^2*e^4*f^2 - 64*b^(5/2)*c*d*e^3*f^3 - 48*a*b^(3/2)*d^2*e^3*f^3 + 24*b^(5/2)*c^2*e^2*f ^4 + 44*a*b^(3/2)*c*d*e^2*f^4 + 15*a^2*sqrt(b)*d^2*e^2*f^4 - 12*a*b^(3/2)* c^2*e*f^5 - 10*a^2*sqrt(b)*c*d*e*f^5 + 3*a^2*sqrt(b)*c^2*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^3*d^3*e^8 - 3*b^3*c*d^2*e^7*f - 3*a*b^2*d^3*e^7*f + 3*b^3*c^2*d*e^6*f ^2 + 9*a*b^2*c*d^2*e^6*f^2 + 3*a^2*b*d^3*e^6*f^2 - b^3*c^3*e^5*f^3 - 9*a*b ^2*c^2*d*e^5*f^3 - 9*a^2*b*c*d^2*e^5*f^3 - a^3*d^3*e^5*f^3 + 3*a*b^2*c^3*e ^4*f^4 + 9*a^2*b*c^2*d*e^4*f^4 + 3*a^3*c*d^2*e^4*f^4 - 3*a^2*b*c^3*e^3*f^5 - 3*a^3*c^2*d*e^3*f^5 + a^3*c^3*e^2*f^6)*sqrt(-b^2*e^2 + a*b*e*f)) - 1/4* (24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d*e^3*f^3 - 16*(sqrt(b)*x - sq rt(b*x^2 + a))^6*b^(5/2)*c*e^2*f^4 - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a* b^(3/2)*d*e^2*f^4 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*e*f^5 + 7*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d*e*f^5 - 3*(sqrt(b)*x - sq rt(b*x^2 + a))^6*a^2*sqrt(b)*c*f^6 + 112*(sqrt(b)*x - sqrt(b*x^2 + a))^...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^3),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)^3), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^3,x)
Output:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e)^3,x)