Optimal. Leaf size=240 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac{c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac{B c^2 x}{e^5} \]
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Rubi [A] time = 0.281976, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac{c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac{B c^2 x}{e^5} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac{B c^2}{e^5}-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^5}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^4}+\frac{A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)^3}+\frac{-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^5 (d+e x)^2}+\frac{c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac{B c^2 x}{e^5}+\frac{d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac{A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{2 e^6 (d+e x)^2}+\frac{2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^6 (d+e x)}-\frac{c (5 B c d-2 b B e-A c e) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.152736, size = 275, normalized size = 1.15 \[ -\frac{A e \left (b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 b c e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+c^2 (-d) \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) (-A c e-2 b B e+5 B c d)+B \left (3 b^2 e^2 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )-2 b c d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+c^2 \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 465, normalized size = 1.9 \begin{align*} 2\,{\frac{c\ln \left ( ex+d \right ) bB}{{e}^{5}}}-5\,{\frac{{c}^{2}\ln \left ( ex+d \right ) Bd}{{e}^{6}}}+{\frac{B{c}^{2}x}{{e}^{5}}}+3\,{\frac{Abcd}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{4}Bbc}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-2\,{\frac{A{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{8\,B{d}^{3}bc}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-6\,{\frac{Bcb{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+8\,{\frac{Bcbd}{{e}^{5} \left ( ex+d \right ) }}+{\frac{A{d}^{3}bc}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}B}{{e}^{4} \left ( ex+d \right ) }}-{\frac{A{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) A}{{e}^{5}}}+{\frac{B{d}^{3}{b}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{B{c}^{2}{d}^{5}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{5\,B{c}^{2}{d}^{4}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-3\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{3\,{b}^{2}Bd}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{2}A{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{4}A{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-2\,{\frac{Abc}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{A{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-10\,{\frac{B{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}+5\,{\frac{B{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{2\,dA{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,A{d}^{3}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{2}B{b}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04323, size = 433, normalized size = 1.8 \begin{align*} -\frac{77 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} - 25 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 12 \,{\left (10 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 6 \,{\left (50 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (65 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac{B c^{2} x}{e^{5}} - \frac{{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\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 = 1.48591, size = 994, normalized size = 4.14 \begin{align*} \frac{12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} + 25 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 12 \,{\left (4 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 6 \,{\left (42 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 4 \,{\left (62 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 12 \,{\left (5 \, B c^{2} d^{5} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B c^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} + 6 \,{\left (5 \, B c^{2} d^{3} e^{2} -{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (5 \, B c^{2} d^{4} e -{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 99.1153, size = 381, normalized size = 1.59 \begin{align*} \frac{B c^{2} x}{e^{5}} + \frac{c \left (A c e + 2 B b e - 5 B c d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{A b^{2} d^{2} e^{3} + 6 A b c d^{3} e^{2} - 25 A c^{2} d^{4} e + 3 B b^{2} d^{3} e^{2} - 50 B b c d^{4} e + 77 B c^{2} d^{5} + x^{3} \left (24 A b c e^{5} - 48 A c^{2} d e^{4} + 12 B b^{2} e^{5} - 96 B b c d e^{4} + 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (6 A b^{2} e^{5} + 36 A b c d e^{4} - 108 A c^{2} d^{2} e^{3} + 18 B b^{2} d e^{4} - 216 B b c d^{2} e^{3} + 300 B c^{2} d^{3} e^{2}\right ) + x \left (4 A b^{2} d e^{4} + 24 A b c d^{2} e^{3} - 88 A c^{2} d^{3} e^{2} + 12 B b^{2} d^{2} e^{3} - 176 B b c d^{3} e^{2} + 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27512, size = 640, normalized size = 2.67 \begin{align*}{\left (x e + d\right )} B c^{2} e^{\left (-6\right )} +{\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{1}{12} \,{\left (\frac{120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac{60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac{20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac{96 \, B b c d e^{23}}{x e + d} - \frac{48 \, A c^{2} d e^{23}}{x e + d} + \frac{72 \, B b c d^{2} e^{23}}{{\left (x e + d\right )}^{2}} + \frac{36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac{32 \, B b c d^{3} e^{23}}{{\left (x e + d\right )}^{3}} - \frac{16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac{6 \, B b c d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B b^{2} e^{24}}{x e + d} + \frac{24 \, A b c e^{24}}{x e + d} - \frac{18 \, B b^{2} d e^{24}}{{\left (x e + d\right )}^{2}} - \frac{36 \, A b c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B b^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac{24 \, A b c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B b^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac{6 \, A b c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac{6 \, A b^{2} e^{25}}{{\left (x e + d\right )}^{2}} - \frac{8 \, A b^{2} d e^{25}}{{\left (x e + d\right )}^{3}} + \frac{3 \, A b^{2} d^{2} e^{25}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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