Optimal. Leaf size=120 \[ \frac{b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{b^2 B}{4 e^4 (d+e x)^4} \]
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Rubi [A] time = 0.0779358, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{b^2 B}{4 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx &=\int \left (\frac{(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^8}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^7}+\frac{b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^6}+\frac{b^2 B}{e^3 (d+e x)^5}\right ) \, dx\\ &=\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{(b d-a e) (3 b B d-2 A b e-a B e)}{6 e^4 (d+e x)^6}+\frac{b (3 b B d-A b e-2 a B e)}{5 e^4 (d+e x)^5}-\frac{b^2 B}{4 e^4 (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.0608273, size = 129, normalized size = 1.08 \[ -\frac{10 a^2 e^2 (6 A e+B (d+7 e x))+4 a b e \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+b^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )}{420 e^4 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 166, normalized size = 1.4 \begin{align*} -{\frac{2\,Aba{e}^{2}-2\,Ad{b}^{2}e+B{a}^{2}{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{2}A{e}^{3}-2\,Adab{e}^{2}+A{d}^{2}{b}^{2}e-Bd{a}^{2}{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{7\,{e}^{4} \left ( ex+d \right ) ^{7}}}-{\frac{b \left ( Abe+2\,Bae-3\,Bbd \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{B{b}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15272, size = 304, normalized size = 2.53 \begin{align*} -\frac{105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \,{\left (3 \, B b^{2} d e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \,{\left (3 \, B b^{2} d^{2} e + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78207, size = 486, normalized size = 4.05 \begin{align*} -\frac{105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \,{\left (3 \, B b^{2} d e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \,{\left (3 \, B b^{2} d^{2} e + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 170.943, size = 260, normalized size = 2.17 \begin{align*} - \frac{60 A a^{2} e^{3} + 20 A a b d e^{2} + 4 A b^{2} d^{2} e + 10 B a^{2} d e^{2} + 8 B a b d^{2} e + 3 B b^{2} d^{3} + 105 B b^{2} e^{3} x^{3} + x^{2} \left (84 A b^{2} e^{3} + 168 B a b e^{3} + 63 B b^{2} d e^{2}\right ) + x \left (140 A a b e^{3} + 28 A b^{2} d e^{2} + 70 B a^{2} e^{3} + 56 B a b d e^{2} + 21 B b^{2} d^{2} e\right )}{420 d^{7} e^{4} + 2940 d^{6} e^{5} x + 8820 d^{5} e^{6} x^{2} + 14700 d^{4} e^{7} x^{3} + 14700 d^{3} e^{8} x^{4} + 8820 d^{2} e^{9} x^{5} + 2940 d e^{10} x^{6} + 420 e^{11} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.9777, size = 216, normalized size = 1.8 \begin{align*} -\frac{{\left (105 \, B b^{2} x^{3} e^{3} + 63 \, B b^{2} d x^{2} e^{2} + 21 \, B b^{2} d^{2} x e + 3 \, B b^{2} d^{3} + 168 \, B a b x^{2} e^{3} + 84 \, A b^{2} x^{2} e^{3} + 56 \, B a b d x e^{2} + 28 \, A b^{2} d x e^{2} + 8 \, B a b d^{2} e + 4 \, A b^{2} d^{2} e + 70 \, B a^{2} x e^{3} + 140 \, A a b x e^{3} + 10 \, B a^{2} d e^{2} + 20 \, A a b d e^{2} + 60 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{420 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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