Optimal. Leaf size=197 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac{c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6} \]
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Rubi [A] time = 0.151656, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac{c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^6}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^5}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^4}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^3}+\frac{c^2 (-5 B d+A e)}{e^5 (d+e x)^2}+\frac{B c^2}{e^5 (d+e x)}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )^2}{5 e^6 (d+e x)^5}-\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{4 e^6 (d+e x)^4}+\frac{2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{3 e^6 (d+e x)^3}-\frac{c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 (d+e x)^2}+\frac{c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.119099, size = 212, normalized size = 1.08 \[ \frac{-4 A e \left (3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-3 a^2 e^4 (d+5 e x)-6 a c e^2 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+c^2 d \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 362, normalized size = 1.8 \begin{align*} -{\frac{A{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+5\,{\frac{B{c}^{2}d}{{e}^{6} \left ( ex+d \right ) }}+2\,{\frac{A{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{aBc}{{e}^{4} \left ( ex+d \right ) ^{2}}}-5\,{\frac{B{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{2\,aAc}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-2\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{aBcd}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{10\,B{c}^{2}{d}^{3}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{B{c}^{2}\ln \left ( ex+d \right ) }{{e}^{6}}}+{\frac{Adac}{{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{B{a}^{2}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,aBc{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{5\,B{c}^{2}{d}^{4}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{A{a}^{2}}{5\,e \left ( ex+d \right ) ^{5}}}-{\frac{2\,A{d}^{2}ac}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{A{d}^{4}{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{B{a}^{2}d}{5\,{e}^{2} \left ( ex+d \right ) ^{5}}}+{\frac{2\,aBc{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}+{\frac{B{c}^{2}{d}^{5}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0695, size = 402, normalized size = 2.04 \begin{align*} \frac{137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \,{\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \,{\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \,{\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac{B c^{2} \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 [A] time = 1.83082, size = 783, normalized size = 3.97 \begin{align*} \frac{137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \,{\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \,{\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \,{\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x + 60 \,{\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28804, size = 323, normalized size = 1.64 \begin{align*} B c^{2} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (60 \,{\left (5 \, B c^{2} d e^{3} - A c^{2} e^{4}\right )} x^{4} + 60 \,{\left (15 \, B c^{2} d^{2} e^{2} - 2 \, A c^{2} d e^{3} - B a c e^{4}\right )} x^{3} + 20 \,{\left (55 \, B c^{2} d^{3} e - 6 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} - 2 \, A a c e^{4}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} - 12 \, A c^{2} d^{3} e - 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4}\right )} x +{\left (137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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