Optimal. Leaf size=96 \[ \frac{10 a^2 (a+b x)^{11}}{11 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^5 (a+b x)^8}{8 b^6}+\frac{(a+b x)^{13}}{13 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6} \]
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Rubi [A] time = 0.0402941, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{10 a^2 (a+b x)^{11}}{11 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^5 (a+b x)^8}{8 b^6}+\frac{(a+b x)^{13}}{13 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^5 (a+b x)^7 \, dx &=\int \left (-\frac{a^5 (a+b x)^7}{b^5}+\frac{5 a^4 (a+b x)^8}{b^5}-\frac{10 a^3 (a+b x)^9}{b^5}+\frac{10 a^2 (a+b x)^{10}}{b^5}-\frac{5 a (a+b x)^{11}}{b^5}+\frac{(a+b x)^{12}}{b^5}\right ) \, dx\\ &=-\frac{a^5 (a+b x)^8}{8 b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{10 a^2 (a+b x)^{11}}{11 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6}+\frac{(a+b x)^{13}}{13 b^6}\\ \end{align*}
Mathematica [A] time = 0.0024743, size = 92, normalized size = 0.96 \[ \frac{21}{11} a^2 b^5 x^{11}+\frac{7}{2} a^3 b^4 x^{10}+\frac{35}{9} a^4 b^3 x^9+\frac{21}{8} a^5 b^2 x^8+a^6 b x^7+\frac{a^7 x^6}{6}+\frac{7}{12} a b^6 x^{12}+\frac{b^7 x^{13}}{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 79, normalized size = 0.8 \begin{align*}{\frac{{b}^{7}{x}^{13}}{13}}+{\frac{7\,a{b}^{6}{x}^{12}}{12}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{11}}{11}}+{\frac{7\,{a}^{3}{b}^{4}{x}^{10}}{2}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{9}}{9}}+{\frac{21\,{a}^{5}{b}^{2}{x}^{8}}{8}}+{a}^{6}b{x}^{7}+{\frac{{a}^{7}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06887, size = 105, normalized size = 1.09 \begin{align*} \frac{1}{13} \, b^{7} x^{13} + \frac{7}{12} \, a b^{6} x^{12} + \frac{21}{11} \, a^{2} b^{5} x^{11} + \frac{7}{2} \, a^{3} b^{4} x^{10} + \frac{35}{9} \, a^{4} b^{3} x^{9} + \frac{21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac{1}{6} \, a^{7} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35283, size = 185, normalized size = 1.93 \begin{align*} \frac{1}{13} x^{13} b^{7} + \frac{7}{12} x^{12} b^{6} a + \frac{21}{11} x^{11} b^{5} a^{2} + \frac{7}{2} x^{10} b^{4} a^{3} + \frac{35}{9} x^{9} b^{3} a^{4} + \frac{21}{8} x^{8} b^{2} a^{5} + x^{7} b a^{6} + \frac{1}{6} x^{6} a^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.08211, size = 90, normalized size = 0.94 \begin{align*} \frac{a^{7} x^{6}}{6} + a^{6} b x^{7} + \frac{21 a^{5} b^{2} x^{8}}{8} + \frac{35 a^{4} b^{3} x^{9}}{9} + \frac{7 a^{3} b^{4} x^{10}}{2} + \frac{21 a^{2} b^{5} x^{11}}{11} + \frac{7 a b^{6} x^{12}}{12} + \frac{b^{7} x^{13}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22312, size = 105, normalized size = 1.09 \begin{align*} \frac{1}{13} \, b^{7} x^{13} + \frac{7}{12} \, a b^{6} x^{12} + \frac{21}{11} \, a^{2} b^{5} x^{11} + \frac{7}{2} \, a^{3} b^{4} x^{10} + \frac{35}{9} \, a^{4} b^{3} x^{9} + \frac{21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac{1}{6} \, a^{7} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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