Optimal. Leaf size=114 \[ -\frac{10 a^3 x^2}{b^7}+\frac{10 a^2 x^3}{3 b^6}-\frac{a^8}{3 b^9 (a+b x)^3}+\frac{4 a^7}{b^9 (a+b x)^2}-\frac{28 a^6}{b^9 (a+b x)}+\frac{35 a^4 x}{b^8}-\frac{56 a^5 \log (a+b x)}{b^9}-\frac{a x^4}{b^5}+\frac{x^5}{5 b^4} \]
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Rubi [A] time = 0.08529, antiderivative size = 114, 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} \[ -\frac{10 a^3 x^2}{b^7}+\frac{10 a^2 x^3}{3 b^6}-\frac{a^8}{3 b^9 (a+b x)^3}+\frac{4 a^7}{b^9 (a+b x)^2}-\frac{28 a^6}{b^9 (a+b x)}+\frac{35 a^4 x}{b^8}-\frac{56 a^5 \log (a+b x)}{b^9}-\frac{a x^4}{b^5}+\frac{x^5}{5 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{(a+b x)^4} \, dx &=\int \left (\frac{35 a^4}{b^8}-\frac{20 a^3 x}{b^7}+\frac{10 a^2 x^2}{b^6}-\frac{4 a x^3}{b^5}+\frac{x^4}{b^4}+\frac{a^8}{b^8 (a+b x)^4}-\frac{8 a^7}{b^8 (a+b x)^3}+\frac{28 a^6}{b^8 (a+b x)^2}-\frac{56 a^5}{b^8 (a+b x)}\right ) \, dx\\ &=\frac{35 a^4 x}{b^8}-\frac{10 a^3 x^2}{b^7}+\frac{10 a^2 x^3}{3 b^6}-\frac{a x^4}{b^5}+\frac{x^5}{5 b^4}-\frac{a^8}{3 b^9 (a+b x)^3}+\frac{4 a^7}{b^9 (a+b x)^2}-\frac{28 a^6}{b^9 (a+b x)}-\frac{56 a^5 \log (a+b x)}{b^9}\\ \end{align*}
Mathematica [A] time = 0.0735772, size = 101, normalized size = 0.89 \[ \frac{-150 a^3 b^2 x^2+50 a^2 b^3 x^3-\frac{5 a^8}{(a+b x)^3}+\frac{60 a^7}{(a+b x)^2}-\frac{420 a^6}{a+b x}+525 a^4 b x-840 a^5 \log (a+b x)-15 a b^4 x^4+3 b^5 x^5}{15 b^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 109, normalized size = 1. \begin{align*} 35\,{\frac{{a}^{4}x}{{b}^{8}}}-10\,{\frac{{a}^{3}{x}^{2}}{{b}^{7}}}+{\frac{10\,{a}^{2}{x}^{3}}{3\,{b}^{6}}}-{\frac{a{x}^{4}}{{b}^{5}}}+{\frac{{x}^{5}}{5\,{b}^{4}}}-{\frac{{a}^{8}}{3\,{b}^{9} \left ( bx+a \right ) ^{3}}}+4\,{\frac{{a}^{7}}{{b}^{9} \left ( bx+a \right ) ^{2}}}-28\,{\frac{{a}^{6}}{{b}^{9} \left ( bx+a \right ) }}-56\,{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04087, size = 169, normalized size = 1.48 \begin{align*} -\frac{84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \,{\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} - \frac{56 \, a^{5} \log \left (b x + a\right )}{b^{9}} + \frac{3 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 50 \, a^{2} b^{2} x^{3} - 150 \, a^{3} b x^{2} + 525 \, a^{4} x}{15 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56293, size = 356, normalized size = 3.12 \begin{align*} \frac{3 \, b^{8} x^{8} - 6 \, a b^{7} x^{7} + 14 \, a^{2} b^{6} x^{6} - 42 \, a^{3} b^{5} x^{5} + 210 \, a^{4} b^{4} x^{4} + 1175 \, a^{5} b^{3} x^{3} + 1005 \, a^{6} b^{2} x^{2} - 255 \, a^{7} b x - 365 \, a^{8} - 840 \,{\left (a^{5} b^{3} x^{3} + 3 \, a^{6} b^{2} x^{2} + 3 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{15 \,{\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.885249, size = 129, normalized size = 1.13 \begin{align*} - \frac{56 a^{5} \log{\left (a + b x \right )}}{b^{9}} + \frac{35 a^{4} x}{b^{8}} - \frac{10 a^{3} x^{2}}{b^{7}} + \frac{10 a^{2} x^{3}}{3 b^{6}} - \frac{a x^{4}}{b^{5}} - \frac{73 a^{8} + 156 a^{7} b x + 84 a^{6} b^{2} x^{2}}{3 a^{3} b^{9} + 9 a^{2} b^{10} x + 9 a b^{11} x^{2} + 3 b^{12} x^{3}} + \frac{x^{5}}{5 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14936, size = 143, normalized size = 1.25 \begin{align*} -\frac{56 \, a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{9}} - \frac{84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \,{\left (b x + a\right )}^{3} b^{9}} + \frac{3 \, b^{16} x^{5} - 15 \, a b^{15} x^{4} + 50 \, a^{2} b^{14} x^{3} - 150 \, a^{3} b^{13} x^{2} + 525 \, a^{4} b^{12} x}{15 \, b^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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