Optimal. Leaf size=291 \[ \frac{\left (2 a^3 c d^2-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )+2 a b^2 e (b d+2 c e)+b^4 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{e^3}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{e^3 \log (d+e x) \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x} \]
[Out]
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Rubi [A] time = 1.20222, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (2 a^3 c d^2-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )+2 a b^2 e (b d+2 c e)+b^4 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{e^3}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{e^3 \log (d+e x) \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x} \]
Antiderivative was successfully verified.
[In] Int[1/((a + c/x^2 + b/x)*x^4*(d + e*x)^2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+c/x**2+b/x)/x**4/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.658234, size = 287, normalized size = 0.99 \[ \frac{\left (-2 a^3 c d^2+a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )-2 a b^2 e (b d+2 c e)+b^4 e^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c^2 \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac{(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log (x (a x+b)+c)}{2 c^2 \left (a d^2+e (c e-b d)\right )^2}-\frac{e^3}{d^2 (d+e x) \left (a d^2+e (c e-b d)\right )}+\frac{e^3 \log (d+e x) \left (4 a d^2+e (2 c e-3 b d)\right )}{d^3 \left (a d^2+e (c e-b d)\right )^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + c/x^2 + b/x)*x^4*(d + e*x)^2),x]
[Out]
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Maple [B] time = 0.023, size = 791, normalized size = 2.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+c/x^2+b/x)/x^4/(e*x+d)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((e*x + d)^2*(a + b/x + c/x^2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((e*x + d)^2*(a + b/x + c/x^2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+c/x**2+b/x)/x**4/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.284107, size = 657, normalized size = 2.26 \[ -\frac{{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} c d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 6 \, a^{2} b c d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} c^{2} d^{4} - 2 \, a b c^{2} d^{3} e + b^{2} c^{2} d^{2} e^{2} + 2 \, a c^{3} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + c^{4} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (a^{2} b d^{2} - 2 \, a b^{2} d e + 2 \, a^{2} c d e + b^{3} e^{2} - 2 \, a b c e^{2}\right )}{\rm ln}\left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (a^{2} c^{2} d^{4} - 2 \, a b c^{2} d^{3} e + b^{2} c^{2} d^{2} e^{2} + 2 \, a c^{3} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + c^{4} e^{4}\right )}} - \frac{e^{7}}{{\left (a d^{4} e^{4} - b d^{3} e^{5} + c d^{2} e^{6}\right )}{\left (x e + d\right )}} - \frac{{\left (b d e + 2 \, c e^{2}\right )} e^{\left (-1\right )}{\rm ln}\left ({\left | -\frac{d}{x e + d} + 1 \right |}\right )}{c^{2} d^{3}} + \frac{e}{c d^{3}{\left (\frac{d}{x e + d} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((e*x + d)^2*(a + b/x + c/x^2)*x^4),x, algorithm="giac")
[Out]