Optimal. Leaf size=101 \[ -\frac{(a+b x)^3 (B d-A e)}{3 e (d+e x)^3 (b d-a e)}+\frac{2 b B (b d-a e)}{e^4 (d+e x)}-\frac{B (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{b^2 B \log (d+e x)}{e^4} \]
[Out]
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Rubi [A] time = 0.186009, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{(a+b x)^3 (B d-A e)}{3 e (d+e x)^3 (b d-a e)}+\frac{2 b B (b d-a e)}{e^4 (d+e x)}-\frac{B (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{b^2 B \log (d+e x)}{e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 49.6736, size = 87, normalized size = 0.86 \[ \frac{B b^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 B b \left (a e - b d\right )}{e^{4} \left (d + e x\right )} - \frac{B \left (a e - b d\right )^{2}}{2 e^{4} \left (d + e x\right )^{2}} - \frac{\left (a + b x\right )^{3} \left (A e - B d\right )}{3 e \left (d + e x\right )^{3} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.140777, size = 138, normalized size = 1.37 \[ \frac{-a^2 e^2 (2 A e+B (d+3 e x))-2 a b e \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+b^2 \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 b^2 B (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.01, size = 251, normalized size = 2.5 \[ -{\frac{A{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{2\,Adab}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{A{d}^{2}{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{Bd{a}^{2}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{2\,B{d}^{2}ab}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{B{b}^{2}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{abA}{{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Ad{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}B}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{Bdab}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{2}B{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}B\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{{b}^{2}A}{{e}^{3} \left ( ex+d \right ) }}-2\,{\frac{abB}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{{b}^{2}Bd}{{e}^{4} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.711175, size = 248, normalized size = 2.46 \[ \frac{11 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e -{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 3 \,{\left (9 \, B b^{2} d^{2} e - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} -{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{B b^{2} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267889, size = 298, normalized size = 2.95 \[ \frac{11 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e -{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 3 \,{\left (9 \, B b^{2} d^{2} e - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} -{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x + 6 \,{\left (B b^{2} e^{3} x^{3} + 3 \, B b^{2} d e^{2} x^{2} + 3 \, B b^{2} d^{2} e x + B b^{2} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.7393, size = 211, normalized size = 2.09 \[ \frac{B b^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 A a^{2} e^{3} + 2 A a b d e^{2} + 2 A b^{2} d^{2} e + B a^{2} d e^{2} + 4 B a b d^{2} e - 11 B b^{2} d^{3} + x^{2} \left (6 A b^{2} e^{3} + 12 B a b e^{3} - 18 B b^{2} d e^{2}\right ) + x \left (6 A a b e^{3} + 6 A b^{2} d e^{2} + 3 B a^{2} e^{3} + 12 B a b d e^{2} - 27 B b^{2} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.279452, size = 220, normalized size = 2.18 \[ B b^{2} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (6 \,{\left (3 \, B b^{2} d e - 2 \, B a b e^{2} - A b^{2} e^{2}\right )} x^{2} + 3 \,{\left (9 \, B b^{2} d^{2} - 4 \, B a b d e - 2 \, A b^{2} d e - B a^{2} e^{2} - 2 \, A a b e^{2}\right )} x +{\left (11 \, B b^{2} d^{3} - 4 \, B a b d^{2} e - 2 \, A b^{2} d^{2} e - B a^{2} d e^{2} - 2 \, A a b d e^{2} - 2 \, A a^{2} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^4,x, algorithm="giac")
[Out]