Optimal. Leaf size=188 \[ \frac{b^2 (d+e x)^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}-\frac{b^3 (d+e x)^3 (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac{(b d-a e)^4 (B d-A e)}{e^6 (d+e x)}-\frac{2 b x (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^5}+\frac{(b d-a e)^3 \log (d+e x) (-a B e-4 A b e+5 b B d)}{e^6}+\frac{b^4 B (d+e x)^4}{4 e^6} \]
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Rubi [A] time = 0.316468, antiderivative size = 188, 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} \[ \frac{b^2 (d+e x)^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}-\frac{b^3 (d+e x)^3 (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac{(b d-a e)^4 (B d-A e)}{e^6 (d+e x)}-\frac{2 b x (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^5}+\frac{(b d-a e)^3 \log (d+e x) (-a B e-4 A b e+5 b B d)}{e^6}+\frac{b^4 B (d+e x)^4}{4 e^6} \]
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)^2} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^2} \, dx\\ &=\int \left (\frac{2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e)}{e^5}+\frac{(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^2}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 (d+e x)}-\frac{2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)}{e^5}+\frac{b^3 (-5 b B d+A b e+4 a B e) (d+e x)^2}{e^5}+\frac{b^4 B (d+e x)^3}{e^5}\right ) \, dx\\ &=-\frac{2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) x}{e^5}+\frac{(b d-a e)^4 (B d-A e)}{e^6 (d+e x)}+\frac{b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^2}{e^6}-\frac{b^3 (5 b B d-A b e-4 a B e) (d+e x)^3}{3 e^6}+\frac{b^4 B (d+e x)^4}{4 e^6}+\frac{(b d-a e)^3 (5 b B d-4 A b e-a B e) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.155323, size = 354, normalized size = 1.88 \[ \frac{36 a^2 b^2 e^2 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )+48 a^3 b e^3 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+12 a^4 e^4 (B d-A e)+8 a b^3 e \left (3 A e \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+2 B \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )\right )+12 (d+e x) (b d-a e)^3 \log (d+e x) (-a B e-4 A b e+5 b B d)+b^4 \left (4 A e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+B \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )}{12 e^6 (d+e x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 564, normalized size = 3. \begin{align*}{\frac{4\,B{x}^{3}a{b}^{3}}{3\,{e}^{2}}}-{\frac{2\,{b}^{4}B{x}^{3}d}{3\,{e}^{3}}}+4\,{\frac{A{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-4\,{\frac{{a}^{3}bB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{{b}^{2}B{a}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-4\,{\frac{Ba{b}^{3}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-12\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}{b}^{2}d}{{e}^{3}}}+12\,{\frac{\ln \left ( ex+d \right ) Aa{b}^{3}{d}^{2}}{{e}^{4}}}-4\,{\frac{{b}^{3}B{x}^{2}ad}{{e}^{3}}}-8\,{\frac{Ada{b}^{3}x}{{e}^{3}}}-12\,{\frac{{b}^{2}B{a}^{2}dx}{{e}^{3}}}+12\,{\frac{Ba{b}^{3}{d}^{2}x}{{e}^{4}}}-8\,{\frac{\ln \left ( ex+d \right ) B{a}^{3}bd}{{e}^{3}}}+18\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}{b}^{2}{d}^{2}}{{e}^{4}}}-16\,{\frac{\ln \left ( ex+d \right ) Ba{b}^{3}{d}^{3}}{{e}^{5}}}+4\,{\frac{Ad{a}^{3}b}{{e}^{2} \left ( ex+d \right ) }}-6\,{\frac{A{d}^{2}{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{\ln \left ( ex+d \right ) B{a}^{4}}{{e}^{2}}}-{\frac{A{a}^{4}}{e \left ( ex+d \right ) }}+{\frac{{b}^{4}B{x}^{4}}{4\,{e}^{2}}}+{\frac{A{x}^{3}{b}^{4}}{3\,{e}^{2}}}+{\frac{Bd{a}^{4}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{B{b}^{4}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+2\,{\frac{A{x}^{2}a{b}^{3}}{{e}^{2}}}+5\,{\frac{\ln \left ( ex+d \right ) B{b}^{4}{d}^{4}}{{e}^{6}}}-{\frac{A{d}^{4}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-4\,{\frac{\ln \left ( ex+d \right ) A{b}^{4}{d}^{3}}{{e}^{5}}}+4\,{\frac{\ln \left ( ex+d \right ) A{a}^{3}b}{{e}^{2}}}-{\frac{A{b}^{4}{x}^{2}d}{{e}^{3}}}+3\,{\frac{B{x}^{2}{a}^{2}{b}^{2}}{{e}^{2}}}+{\frac{3\,{b}^{4}B{x}^{2}{d}^{2}}{2\,{e}^{4}}}+6\,{\frac{A{a}^{2}{b}^{2}x}{{e}^{2}}}+3\,{\frac{A{d}^{2}{b}^{4}x}{{e}^{4}}}+4\,{\frac{B{a}^{3}bx}{{e}^{2}}}-4\,{\frac{B{b}^{4}{d}^{3}x}{{e}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00813, size = 554, normalized size = 2.95 \begin{align*} \frac{B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\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}}{e^{7} x + d e^{6}} + \frac{3 \, B b^{4} e^{3} x^{4} - 4 \,{\left (2 \, B b^{4} d e^{2} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B b^{4} d^{2} e - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{2} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 12 \,{\left (4 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{12 \, e^{5}} + \frac{{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\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 = 1.5375, size = 1249, normalized size = 6.64 \begin{align*} \frac{3 \, B b^{4} e^{5} x^{5} + 12 \, B b^{4} d^{5} - 12 \, A a^{4} e^{5} - 12 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 24 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 24 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 12 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} -{\left (5 \, B b^{4} d e^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \,{\left (5 \, B b^{4} d^{2} e^{3} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 6 \,{\left (5 \, B b^{4} d^{3} e^{2} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} - 12 \,{\left (4 \, B b^{4} d^{4} e - 3 \,{\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} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4}\right )} x + 12 \,{\left (5 \, B b^{4} d^{5} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\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} +{\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\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.96167, size = 384, normalized size = 2.04 \begin{align*} \frac{B b^{4} x^{4}}{4 e^{2}} + \frac{- A a^{4} e^{5} + 4 A a^{3} b d e^{4} - 6 A a^{2} b^{2} d^{2} e^{3} + 4 A a b^{3} d^{3} e^{2} - A b^{4} d^{4} e + B a^{4} d e^{4} - 4 B a^{3} b d^{2} e^{3} + 6 B a^{2} b^{2} d^{3} e^{2} - 4 B a b^{3} d^{4} e + B b^{4} d^{5}}{d e^{6} + e^{7} x} + \frac{x^{3} \left (A b^{4} e + 4 B a b^{3} e - 2 B b^{4} d\right )}{3 e^{3}} + \frac{x^{2} \left (4 A a b^{3} e^{2} - 2 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 8 B a b^{3} d e + 3 B b^{4} d^{2}\right )}{2 e^{4}} + \frac{x \left (6 A a^{2} b^{2} e^{3} - 8 A a b^{3} d e^{2} + 3 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 12 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 4 B b^{4} d^{3}\right )}{e^{5}} + \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13109, size = 710, normalized size = 3.78 \begin{align*} \frac{1}{12} \,{\left (3 \, B b^{4} - \frac{4 \,{\left (5 \, B b^{4} d e - 4 \, B a b^{3} e^{2} - A b^{4} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{12 \,{\left (5 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} - 2 \, A b^{4} d e^{3} + 3 \, B a^{2} b^{2} e^{4} + 2 \, A a b^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{24 \,{\left (5 \, B b^{4} d^{3} e^{3} - 12 \, B a b^{3} d^{2} e^{4} - 3 \, A b^{4} d^{2} e^{4} + 9 \, B a^{2} b^{2} d e^{5} + 6 \, A a b^{3} d e^{5} - 2 \, B a^{3} b e^{6} - 3 \, A a^{2} b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{4} e^{\left (-6\right )} -{\left (5 \, B b^{4} d^{4} - 16 \, B a b^{3} d^{3} e - 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 12 \, A a b^{3} d^{2} e^{2} - 8 \, B a^{3} b d e^{3} - 12 \, A a^{2} b^{2} d e^{3} + B a^{4} e^{4} + 4 \, A a^{3} b e^{4}\right )} e^{\left (-6\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B b^{4} d^{5} e^{4}}{x e + d} - \frac{4 \, B a b^{3} d^{4} e^{5}}{x e + d} - \frac{A b^{4} d^{4} e^{5}}{x e + d} + \frac{6 \, B a^{2} b^{2} d^{3} e^{6}}{x e + d} + \frac{4 \, A a b^{3} d^{3} e^{6}}{x e + d} - \frac{4 \, B a^{3} b d^{2} e^{7}}{x e + d} - \frac{6 \, A a^{2} b^{2} d^{2} e^{7}}{x e + d} + \frac{B a^{4} d e^{8}}{x e + d} + \frac{4 \, A a^{3} b d e^{8}}{x e + d} - \frac{A a^{4} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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