Optimal. Leaf size=171 \[ -\frac{\log (d+e x) \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{d^3 (c d-b e)^3}+\frac{c^2 (b B-A c) \log (b+c x)}{b (c d-b e)^3}+\frac{B c d^2-A e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}+\frac{B d-A e}{2 d (d+e x)^2 (c d-b e)}+\frac{A \log (x)}{b d^3} \]
[Out]
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Rubi [A] time = 0.525408, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\log (d+e x) \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{d^3 (c d-b e)^3}+\frac{c^2 (b B-A c) \log (b+c x)}{b (c d-b e)^3}+\frac{B c d^2-A e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}+\frac{B d-A e}{2 d (d+e x)^2 (c d-b e)}+\frac{A \log (x)}{b d^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^3*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 77.8118, size = 158, normalized size = 0.92 \[ \frac{A \log{\left (x \right )}}{b d^{3}} + \frac{A e - B d}{2 d \left (d + e x\right )^{2} \left (b e - c d\right )} + \frac{A b e^{2} - 2 A c d e + B c d^{2}}{d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2}} - \frac{\left (A b^{2} e^{3} - 3 A b c d e^{2} + 3 A c^{2} d^{2} e - B c^{2} d^{3}\right ) \log{\left (d + e x \right )}}{d^{3} \left (b e - c d\right )^{3}} + \frac{c^{2} \left (A c - B b\right ) \log{\left (b + c x \right )}}{b \left (b e - c d\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**3/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.366302, size = 169, normalized size = 0.99 \[ -\frac{\log (d+e x) \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{d^3 (c d-b e)^3}+\frac{c^2 (A c-b B) \log (b+c x)}{b (b e-c d)^3}+\frac{A e (b e-2 c d)+B c d^2}{d^2 (d+e x) (c d-b e)^2}+\frac{B d-A e}{2 d (d+e x)^2 (c d-b e)}+\frac{A \log (x)}{b d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^3*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.02, size = 275, normalized size = 1.6 \[{\frac{A\ln \left ( x \right ) }{{d}^{3}b}}+{\frac{{c}^{3}\ln \left ( cx+b \right ) A}{b \left ( be-cd \right ) ^{3}}}-{\frac{{c}^{2}\ln \left ( cx+b \right ) B}{ \left ( be-cd \right ) ^{3}}}+{\frac{Ae}{2\,d \left ( be-cd \right ) \left ( ex+d \right ) ^{2}}}-{\frac{B}{ \left ( 2\,be-2\,cd \right ) \left ( ex+d \right ) ^{2}}}+{\frac{Ab{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{Ace}{d \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}+{\frac{Bc}{ \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ) A{b}^{2}{e}^{3}}{{d}^{3} \left ( be-cd \right ) ^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) Abc{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}e}{d \left ( be-cd \right ) ^{3}}}+{\frac{\ln \left ( ex+d \right ) B{c}^{2}}{ \left ( be-cd \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^3/(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.747984, size = 421, normalized size = 2.46 \[ \frac{{\left (B b c^{2} - A c^{3}\right )} \log \left (c x + b\right )}{b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}} - \frac{{\left (B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, A b c d e^{2} - A b^{2} e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} + \frac{3 \, B c d^{3} + 3 \, A b d e^{2} -{\left (B b + 5 \, A c\right )} d^{2} e + 2 \,{\left (B c d^{2} e - 2 \, A c d e^{2} + A b e^{3}\right )} x}{2 \,{\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} +{\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}} + \frac{A \log \left (x\right )}{b d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 40.2774, size = 869, normalized size = 5.08 \[ \frac{3 \, B b c^{2} d^{5} - 3 \, A b^{3} d^{2} e^{3} -{\left (4 \, B b^{2} c + 5 \, A b c^{2}\right )} d^{4} e +{\left (B b^{3} + 8 \, A b^{2} c\right )} d^{3} e^{2} + 2 \,{\left (B b c^{2} d^{4} e + 3 \, A b^{2} c d^{2} e^{3} - A b^{3} d e^{4} -{\left (B b^{2} c + 2 \, A b c^{2}\right )} d^{3} e^{2}\right )} x + 2 \,{\left ({\left (B b c^{2} - A c^{3}\right )} d^{3} e^{2} x^{2} + 2 \,{\left (B b c^{2} - A c^{3}\right )} d^{4} e x +{\left (B b c^{2} - A c^{3}\right )} d^{5}\right )} \log \left (c x + b\right ) - 2 \,{\left (B b c^{2} d^{5} - 3 \, A b c^{2} d^{4} e + 3 \, A b^{2} c d^{3} e^{2} - A b^{3} d^{2} e^{3} +{\left (B b c^{2} d^{3} e^{2} - 3 \, A b c^{2} d^{2} e^{3} + 3 \, A b^{2} c d e^{4} - A b^{3} e^{5}\right )} x^{2} + 2 \,{\left (B b c^{2} d^{4} e - 3 \, A b c^{2} d^{3} e^{2} + 3 \, A b^{2} c d^{2} e^{3} - A b^{3} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \,{\left (A c^{3} d^{5} - 3 \, A b c^{2} d^{4} e + 3 \, A b^{2} c d^{3} e^{2} - A b^{3} d^{2} e^{3} +{\left (A c^{3} d^{3} e^{2} - 3 \, A b c^{2} d^{2} e^{3} + 3 \, A b^{2} c d e^{4} - A b^{3} e^{5}\right )} x^{2} + 2 \,{\left (A c^{3} d^{4} e - 3 \, A b c^{2} d^{3} e^{2} + 3 \, A b^{2} c d^{2} e^{3} - A b^{3} d e^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (b c^{3} d^{8} - 3 \, b^{2} c^{2} d^{7} e + 3 \, b^{3} c d^{6} e^{2} - b^{4} d^{5} e^{3} +{\left (b c^{3} d^{6} e^{2} - 3 \, b^{2} c^{2} d^{5} e^{3} + 3 \, b^{3} c d^{4} e^{4} - b^{4} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (b c^{3} d^{7} e - 3 \, b^{2} c^{2} d^{6} e^{2} + 3 \, b^{3} c d^{5} e^{3} - b^{4} d^{4} e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*(e*x + d)^3),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((B*x+A)/(e*x+d)**3/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.284272, size = 416, normalized size = 2.43 \[ \frac{{\left (B b c^{3} - A c^{4}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} - b^{4} c e^{3}} - \frac{{\left (B c^{2} d^{3} e - 3 \, A c^{2} d^{2} e^{2} + 3 \, A b c d e^{3} - A b^{2} e^{4}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b d^{3}} + \frac{3 \, B c^{2} d^{5} - 4 \, B b c d^{4} e - 5 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 8 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} + 2 \,{\left (B c^{2} d^{4} e - B b c d^{3} e^{2} - 2 \, A c^{2} d^{3} e^{2} + 3 \, A b c d^{2} e^{3} - A b^{2} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (x e + d\right )}^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*(e*x + d)^3),x, algorithm="giac")
[Out]