Optimal. Leaf size=163 \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{4 e^5 (d+e x)^4}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (d+e x)^5}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac{(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}-\frac{b^3 B}{3 e^5 (d+e x)^3} \]
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Rubi [A] time = 0.131143, antiderivative size = 163, 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} \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{4 e^5 (d+e x)^4}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (d+e x)^5}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac{(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}-\frac{b^3 B}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^8}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^7}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^6}+\frac{b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^5}+\frac{b^3 B}{e^4 (d+e x)^4}\right ) \, dx\\ &=-\frac{(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}+\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e)}{6 e^5 (d+e x)^6}-\frac{3 b (b d-a e) (2 b B d-A b e-a B e)}{5 e^5 (d+e x)^5}+\frac{b^2 (4 b B d-A b e-3 a B e)}{4 e^5 (d+e x)^4}-\frac{b^3 B}{3 e^5 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.100017, size = 215, normalized size = 1.32 \[ -\frac{6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+10 a^3 e^3 (6 A e+B (d+7 e x))+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )}{420 e^5 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 281, normalized size = 1.7 \begin{align*} -{\frac{3\,Ab{a}^{2}{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{d}^{2}{b}^{3}e+B{a}^{3}{e}^{3}-6\,Bd{a}^{2}b{e}^{2}+9\,B{d}^{2}a{b}^{2}e-4\,{b}^{3}B{d}^{3}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{B{b}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{3}A{e}^{4}-3\,Ad{a}^{2}b{e}^{3}+3\,A{d}^{2}a{b}^{2}{e}^{2}-A{d}^{3}{b}^{3}e-Bd{a}^{3}{e}^{3}+3\,B{d}^{2}{a}^{2}b{e}^{2}-3\,B{d}^{3}a{b}^{2}e+{b}^{3}B{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{3\,b \left ( Aba{e}^{2}-A{b}^{2}de+B{a}^{2}{e}^{2}-3\,Bdabe+2\,{b}^{2}B{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{2} \left ( Abe+3\,Bae-4\,Bbd \right ) }{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.39134, size = 448, normalized size = 2.75 \begin{align*} -\frac{140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \,{\left (4 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \,{\left (4 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \,{\left (4 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7244, size = 706, normalized size = 4.33 \begin{align*} -\frac{140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \,{\left (4 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \,{\left (4 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \,{\left (4 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\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 = 2.37476, size = 382, normalized size = 2.34 \begin{align*} -\frac{{\left (140 \, B b^{3} x^{4} e^{4} + 140 \, B b^{3} d x^{3} e^{3} + 84 \, B b^{3} d^{2} x^{2} e^{2} + 28 \, B b^{3} d^{3} x e + 4 \, B b^{3} d^{4} + 315 \, B a b^{2} x^{3} e^{4} + 105 \, A b^{3} x^{3} e^{4} + 189 \, B a b^{2} d x^{2} e^{3} + 63 \, A b^{3} d x^{2} e^{3} + 63 \, B a b^{2} d^{2} x e^{2} + 21 \, A b^{3} d^{2} x e^{2} + 9 \, B a b^{2} d^{3} e + 3 \, A b^{3} d^{3} e + 252 \, B a^{2} b x^{2} e^{4} + 252 \, A a b^{2} x^{2} e^{4} + 84 \, B a^{2} b d x e^{3} + 84 \, A a b^{2} d x e^{3} + 12 \, B a^{2} b d^{2} e^{2} + 12 \, A a b^{2} d^{2} e^{2} + 70 \, B a^{3} x e^{4} + 210 \, A a^{2} b x e^{4} + 10 \, B a^{3} d e^{3} + 30 \, A a^{2} b d e^{3} + 60 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{420 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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