Optimal. Leaf size=132 \[ \frac{21 a^2 (a+b x)^{16}}{16 b^8}-\frac{7 a^3 (a+b x)^{15}}{3 b^8}+\frac{5 a^4 (a+b x)^{14}}{2 b^8}-\frac{21 a^5 (a+b x)^{13}}{13 b^8}+\frac{7 a^6 (a+b x)^{12}}{12 b^8}-\frac{a^7 (a+b x)^{11}}{11 b^8}+\frac{(a+b x)^{18}}{18 b^8}-\frac{7 a (a+b x)^{17}}{17 b^8} \]
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Rubi [A] time = 0.0566174, antiderivative size = 132, 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} \[ \frac{21 a^2 (a+b x)^{16}}{16 b^8}-\frac{7 a^3 (a+b x)^{15}}{3 b^8}+\frac{5 a^4 (a+b x)^{14}}{2 b^8}-\frac{21 a^5 (a+b x)^{13}}{13 b^8}+\frac{7 a^6 (a+b x)^{12}}{12 b^8}-\frac{a^7 (a+b x)^{11}}{11 b^8}+\frac{(a+b x)^{18}}{18 b^8}-\frac{7 a (a+b x)^{17}}{17 b^8} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^7 (a+b x)^{10} \, dx &=\int \left (-\frac{a^7 (a+b x)^{10}}{b^7}+\frac{7 a^6 (a+b x)^{11}}{b^7}-\frac{21 a^5 (a+b x)^{12}}{b^7}+\frac{35 a^4 (a+b x)^{13}}{b^7}-\frac{35 a^3 (a+b x)^{14}}{b^7}+\frac{21 a^2 (a+b x)^{15}}{b^7}-\frac{7 a (a+b x)^{16}}{b^7}+\frac{(a+b x)^{17}}{b^7}\right ) \, dx\\ &=-\frac{a^7 (a+b x)^{11}}{11 b^8}+\frac{7 a^6 (a+b x)^{12}}{12 b^8}-\frac{21 a^5 (a+b x)^{13}}{13 b^8}+\frac{5 a^4 (a+b x)^{14}}{2 b^8}-\frac{7 a^3 (a+b x)^{15}}{3 b^8}+\frac{21 a^2 (a+b x)^{16}}{16 b^8}-\frac{7 a (a+b x)^{17}}{17 b^8}+\frac{(a+b x)^{18}}{18 b^8}\\ \end{align*}
Mathematica [A] time = 0.0134137, size = 130, normalized size = 0.98 \[ \frac{45}{16} a^2 b^8 x^{16}+8 a^3 b^7 x^{15}+15 a^4 b^6 x^{14}+\frac{252}{13} a^5 b^5 x^{13}+\frac{35}{2} a^6 b^4 x^{12}+\frac{120}{11} a^7 b^3 x^{11}+\frac{9}{2} a^8 b^2 x^{10}+\frac{10}{9} a^9 b x^9+\frac{a^{10} x^8}{8}+\frac{10}{17} a b^9 x^{17}+\frac{b^{10} x^{18}}{18} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 113, normalized size = 0.9 \begin{align*}{\frac{{b}^{10}{x}^{18}}{18}}+{\frac{10\,a{b}^{9}{x}^{17}}{17}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{16}}{16}}+8\,{a}^{3}{b}^{7}{x}^{15}+15\,{a}^{4}{b}^{6}{x}^{14}+{\frac{252\,{a}^{5}{b}^{5}{x}^{13}}{13}}+{\frac{35\,{a}^{6}{b}^{4}{x}^{12}}{2}}+{\frac{120\,{a}^{7}{b}^{3}{x}^{11}}{11}}+{\frac{9\,{a}^{8}{b}^{2}{x}^{10}}{2}}+{\frac{10\,{a}^{9}b{x}^{9}}{9}}+{\frac{{a}^{10}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08172, size = 151, normalized size = 1.14 \begin{align*} \frac{1}{18} \, b^{10} x^{18} + \frac{10}{17} \, a b^{9} x^{17} + \frac{45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac{252}{13} \, a^{5} b^{5} x^{13} + \frac{35}{2} \, a^{6} b^{4} x^{12} + \frac{120}{11} \, a^{7} b^{3} x^{11} + \frac{9}{2} \, a^{8} b^{2} x^{10} + \frac{10}{9} \, a^{9} b x^{9} + \frac{1}{8} \, a^{10} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56997, size = 278, normalized size = 2.11 \begin{align*} \frac{1}{18} x^{18} b^{10} + \frac{10}{17} x^{17} b^{9} a + \frac{45}{16} x^{16} b^{8} a^{2} + 8 x^{15} b^{7} a^{3} + 15 x^{14} b^{6} a^{4} + \frac{252}{13} x^{13} b^{5} a^{5} + \frac{35}{2} x^{12} b^{4} a^{6} + \frac{120}{11} x^{11} b^{3} a^{7} + \frac{9}{2} x^{10} b^{2} a^{8} + \frac{10}{9} x^{9} b a^{9} + \frac{1}{8} x^{8} a^{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.125765, size = 131, normalized size = 0.99 \begin{align*} \frac{a^{10} x^{8}}{8} + \frac{10 a^{9} b x^{9}}{9} + \frac{9 a^{8} b^{2} x^{10}}{2} + \frac{120 a^{7} b^{3} x^{11}}{11} + \frac{35 a^{6} b^{4} x^{12}}{2} + \frac{252 a^{5} b^{5} x^{13}}{13} + 15 a^{4} b^{6} x^{14} + 8 a^{3} b^{7} x^{15} + \frac{45 a^{2} b^{8} x^{16}}{16} + \frac{10 a b^{9} x^{17}}{17} + \frac{b^{10} x^{18}}{18} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19444, size = 151, normalized size = 1.14 \begin{align*} \frac{1}{18} \, b^{10} x^{18} + \frac{10}{17} \, a b^{9} x^{17} + \frac{45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac{252}{13} \, a^{5} b^{5} x^{13} + \frac{35}{2} \, a^{6} b^{4} x^{12} + \frac{120}{11} \, a^{7} b^{3} x^{11} + \frac{9}{2} \, a^{8} b^{2} x^{10} + \frac{10}{9} \, a^{9} b x^{9} + \frac{1}{8} \, a^{10} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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