Optimal. Leaf size=429 \[ -\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]
[Out]
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Rubi [A] time = 2.35853, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(a + b*x + c*x^2)^3),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/(e*x+d)/(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 2.595, size = 429, normalized size = 1. \[ \frac{1}{2} \left (\frac{2 \left (-20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+30 a^2 b c^2 e^5-10 a b^3 c e^5+10 c^4 d^3 e (3 b d-4 a e)+b^5 e^5-12 c^5 d^5\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{4 c^2 \left (4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right )+b^2 c e \left (c d (2 e x-9 d)-15 a e^2\right )+2 b c^2 \left (7 a e^2 (d-e x)+3 c d^2 (d-3 e x)\right )+2 b^4 e^3+b^3 c e^2 (d+2 e x)}{\left (b^2-4 a c\right )^2 (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}+\frac{2 c (a e+c d x)+b^2 (-e)+b c (d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2 \left (e (b d-a e)-c d^2\right )}+\frac{2 e^5 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}-\frac{e^5 \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(a + b*x + c*x^2)^3),x]
[Out]
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Maple [B] time = 0.034, size = 5715, normalized size = 13.3 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*x^2+b*x+a)^3,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/((c*x^2 + b*x + a)^3*(e*x + d)),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/((c*x^2 + b*x + a)^3*(e*x + d)),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/(e*x+d)/(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.218215, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^3*(e*x + d)),x, algorithm="giac")
[Out]