Optimal. Leaf size=204 \[ -\frac{2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac{c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{2 e^6 (d+e x)^4}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6 (d+e x)^6}+\frac{c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac{B c^2}{e^6 (d+e x)} \]
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Rubi [A] time = 0.141267, antiderivative size = 204, 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} \[ -\frac{2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac{c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{2 e^6 (d+e x)^4}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6 (d+e x)^6}+\frac{c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac{B c^2}{e^6 (d+e x)} \]
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)^7} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^7}+\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)^6}+\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)^5}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^4}+\frac{c^2 (-5 B d+A e)}{e^5 (d+e x)^3}+\frac{B c^2}{e^5 (d+e x)^2}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )^2}{6 e^6 (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 )}{5 e^6 (d+e x)^5}+\frac{c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^6 (d+e x)^4}-\frac{2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{3 e^6 (d+e x)^3}+\frac{c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac{B c^2}{e^6 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.102317, size = 198, normalized size = 0.97 \[ -\frac{A e \left (5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (a^2 e^4 (d+6 e x)+a c e^2 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+5 c^2 \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )}{30 e^6 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 249, normalized size = 1.2 \begin{align*} -{\frac{B{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{c}^{2} \left ( Ae-5\,Bd \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{2\,c \left ( 2\,Acde-aB{e}^{2}-5\,Bc{d}^{2} \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( aA{e}^{3}+3\,Ac{d}^{2}e-3\,aBd{e}^{2}-5\,Bc{d}^{3} \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{-4\,Adac{e}^{3}-4\,A{c}^{2}{d}^{3}e+B{e}^{4}{a}^{2}+6\,aBc{d}^{2}{e}^{2}+5\,B{c}^{2}{d}^{4}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{2}{e}^{5}+2\,A{d}^{2}ac{e}^{3}+A{d}^{4}{c}^{2}e-B{a}^{2}d{e}^{4}-2\,aBc{d}^{3}{e}^{2}-B{c}^{2}{d}^{5}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04775, size = 394, normalized size = 1.93 \begin{align*} -\frac{30 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5} + 15 \,{\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{4} + 20 \,{\left (5 \, B c^{2} d^{2} e^{3} + A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 15 \,{\left (5 \, B c^{2} d^{3} e^{2} + A c^{2} d^{2} e^{3} + B a c d e^{4} + A a c e^{5}\right )} x^{2} + 6 \,{\left (5 \, B c^{2} d^{4} e + A c^{2} d^{3} e^{2} + B a c d^{2} e^{3} + A a c d e^{4} + B a^{2} e^{5}\right )} x}{30 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69315, size = 614, normalized size = 3.01 \begin{align*} -\frac{30 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5} + 15 \,{\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{4} + 20 \,{\left (5 \, B c^{2} d^{2} e^{3} + A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 15 \,{\left (5 \, B c^{2} d^{3} e^{2} + A c^{2} d^{2} e^{3} + B a c d e^{4} + A a c e^{5}\right )} x^{2} + 6 \,{\left (5 \, B c^{2} d^{4} e + A c^{2} d^{3} e^{2} + B a c d^{2} e^{3} + A a c d e^{4} + B a^{2} e^{5}\right )} x}{30 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} 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 = 1.39592, size = 321, normalized size = 1.57 \begin{align*} -\frac{{\left (30 \, B c^{2} x^{5} e^{5} + 75 \, B c^{2} d x^{4} e^{4} + 100 \, B c^{2} d^{2} x^{3} e^{3} + 75 \, B c^{2} d^{3} x^{2} e^{2} + 30 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 15 \, A c^{2} x^{4} e^{5} + 20 \, A c^{2} d x^{3} e^{4} + 15 \, A c^{2} d^{2} x^{2} e^{3} + 6 \, A c^{2} d^{3} x e^{2} + A c^{2} d^{4} e + 20 \, B a c x^{3} e^{5} + 15 \, B a c d x^{2} e^{4} + 6 \, B a c d^{2} x e^{3} + B a c d^{3} e^{2} + 15 \, A a c x^{2} e^{5} + 6 \, A a c d x e^{4} + A a c d^{2} e^{3} + 6 \, B a^{2} x e^{5} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{30 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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