Optimal. Leaf size=193 \[ -\frac{b^3 (d+e x)^2 (-4 a B e-A b e+5 b B d)}{2 e^6}+\frac{2 b^2 x (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^5}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 (d+e x)}+\frac{(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac{2 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)}{e^6}+\frac{b^4 B (d+e x)^3}{3 e^6} \]
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Rubi [A] time = 0.245668, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{b^3 (d+e x)^2 (-4 a B e-A b e+5 b B d)}{2 e^6}+\frac{2 b^2 x (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^5}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 (d+e x)}+\frac{(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac{2 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)}{e^6}+\frac{b^4 B (d+e x)^3}{3 e^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^3} \, dx\\ &=\int \left (-\frac{2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e)}{e^5}+\frac{(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^3}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 (d+e x)^2}+\frac{2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e)}{e^5 (d+e x)}+\frac{b^3 (-5 b B d+A b e+4 a B e) (d+e x)}{e^5}+\frac{b^4 B (d+e x)^2}{e^5}\right ) \, dx\\ &=\frac{2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) x}{e^5}+\frac{(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac{(b d-a e)^3 (5 b B d-4 A b e-a B e)}{e^6 (d+e x)}-\frac{b^3 (5 b B d-A b e-4 a B e) (d+e x)^2}{2 e^6}+\frac{b^4 B (d+e x)^3}{3 e^6}-\frac{2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.119338, size = 187, normalized size = 0.97 \[ \frac{-6 b^2 e x \left (-6 a^2 B e^2-4 a b e (A e-3 B d)+3 b^2 d (A e-2 B d)\right )+3 b^3 e^2 x^2 (4 a B e+A b e-3 b B d)-\frac{6 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{d+e x}+\frac{3 (b d-a e)^4 (B d-A e)}{(d+e x)^2}-12 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)+2 b^4 B e^3 x^3}{6 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 601, normalized size = 3.1 \begin{align*} 12\,{\frac{Ad{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-12\,{\frac{A{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+8\,{\frac{{a}^{3}bBd}{{e}^{3} \left ( ex+d \right ) }}-18\,{\frac{{b}^{2}B{a}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+16\,{\frac{Ba{b}^{3}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-12\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Aad}{{e}^{4}}}-18\,{\frac{{b}^{2}\ln \left ( ex+d \right ) B{a}^{2}d}{{e}^{4}}}+24\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Ba{d}^{2}}{{e}^{5}}}-2\,{\frac{B{d}^{2}{a}^{3}b}{{e}^{3} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}B{a}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-2\,{\frac{Ba{b}^{3}{d}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-3\,{\frac{A{d}^{2}{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{A{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{Ad{a}^{3}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}-12\,{\frac{Ba{b}^{3}dx}{{e}^{4}}}-{\frac{B{a}^{4}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{A{b}^{4}{x}^{2}}{2\,{e}^{3}}}-{\frac{A{a}^{4}}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{4}B{x}^{3}}{3\,{e}^{3}}}-3\,{\frac{Ad{b}^{4}x}{{e}^{4}}}+6\,{\frac{{b}^{2}B{a}^{2}x}{{e}^{3}}}+6\,{\frac{B{b}^{4}{d}^{2}x}{{e}^{5}}}+2\,{\frac{{b}^{3}B{x}^{2}a}{{e}^{3}}}-{\frac{3\,{b}^{4}B{x}^{2}d}{2\,{e}^{4}}}-10\,{\frac{{b}^{4}\ln \left ( ex+d \right ) B{d}^{3}}{{e}^{6}}}-4\,{\frac{A{a}^{3}b}{{e}^{2} \left ( ex+d \right ) }}+4\,{\frac{A{d}^{3}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-5\,{\frac{B{b}^{4}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+{\frac{Bd{a}^{4}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{B{b}^{4}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{A{d}^{4}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+4\,{\frac{Aa{b}^{3}x}{{e}^{3}}}+4\,{\frac{b\ln \left ( ex+d \right ) B{a}^{3}}{{e}^{3}}}+6\,{\frac{{b}^{2}\ln \left ( ex+d \right ) A{a}^{2}}{{e}^{3}}}+6\,{\frac{{b}^{4}\ln \left ( ex+d \right ) A{d}^{2}}{{e}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12044, size = 566, normalized size = 2.93 \begin{align*} -\frac{9 \, B b^{4} d^{5} + A a^{4} e^{5} - 7 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 6 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 2 \,{\left (5 \, B b^{4} d^{4} e - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B b^{4} e^{2} x^{3} - 3 \,{\left (3 \, B b^{4} d e -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B b^{4} d^{2} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{2 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60197, size = 1343, normalized size = 6.96 \begin{align*} \frac{2 \, B b^{4} e^{5} x^{5} - 27 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 21 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 30 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 18 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} -{\left (5 \, B b^{4} d e^{4} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B b^{4} d^{2} e^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \,{\left (21 \, B b^{4} d^{3} e^{2} - 11 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 8 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 6 \,{\left (B b^{4} d^{4} e +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 12 \,{\left (5 \, B b^{4} d^{5} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (5 \, B b^{4} d^{3} e^{2} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 2 \,{\left (5 \, B b^{4} d^{4} e - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.0693, size = 435, normalized size = 2.25 \begin{align*} \frac{B b^{4} x^{3}}{3 e^{3}} + \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{A a^{4} e^{5} + 4 A a^{3} b d e^{4} - 18 A a^{2} b^{2} d^{2} e^{3} + 20 A a b^{3} d^{3} e^{2} - 7 A b^{4} d^{4} e + B a^{4} d e^{4} - 12 B a^{3} b d^{2} e^{3} + 30 B a^{2} b^{2} d^{3} e^{2} - 28 B a b^{3} d^{4} e + 9 B b^{4} d^{5} + x \left (8 A a^{3} b e^{5} - 24 A a^{2} b^{2} d e^{4} + 24 A a b^{3} d^{2} e^{3} - 8 A b^{4} d^{3} e^{2} + 2 B a^{4} e^{5} - 16 B a^{3} b d e^{4} + 36 B a^{2} b^{2} d^{2} e^{3} - 32 B a b^{3} d^{3} e^{2} + 10 B b^{4} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (A b^{4} e + 4 B a b^{3} e - 3 B b^{4} d\right )}{2 e^{4}} + \frac{x \left (4 A a b^{3} e^{2} - 3 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 12 B a b^{3} d e + 6 B b^{4} d^{2}\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14419, size = 564, normalized size = 2.92 \begin{align*} -2 \,{\left (5 \, B b^{4} d^{3} - 12 \, B a b^{3} d^{2} e - 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} + 6 \, A a b^{3} d e^{2} - 2 \, B a^{3} b e^{3} - 3 \, A a^{2} b^{2} e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B b^{4} x^{3} e^{6} - 9 \, B b^{4} d x^{2} e^{5} + 36 \, B b^{4} d^{2} x e^{4} + 12 \, B a b^{3} x^{2} e^{6} + 3 \, A b^{4} x^{2} e^{6} - 72 \, B a b^{3} d x e^{5} - 18 \, A b^{4} d x e^{5} + 36 \, B a^{2} b^{2} x e^{6} + 24 \, A a b^{3} x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B b^{4} d^{5} - 28 \, B a b^{3} d^{4} e - 7 \, A b^{4} d^{4} e + 30 \, B a^{2} b^{2} d^{3} e^{2} + 20 \, A a b^{3} d^{3} e^{2} - 12 \, B a^{3} b d^{2} e^{3} - 18 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} + A a^{4} e^{5} + 2 \,{\left (5 \, B b^{4} d^{4} e - 16 \, B a b^{3} d^{3} e^{2} - 4 \, A b^{4} d^{3} e^{2} + 18 \, B a^{2} b^{2} d^{2} e^{3} + 12 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} + B a^{4} e^{5} + 4 \, A a^{3} b e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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