Optimal. Leaf size=72 \[ -\frac {a^4}{5 b^5 \left (a+b x^5\right )}-\frac {4 a^3 \log \left (a+b x^5\right )}{5 b^5}+\frac {3 a^2 x^5}{5 b^4}-\frac {a x^{10}}{5 b^3}+\frac {x^{15}}{15 b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {3 a^2 x^5}{5 b^4}-\frac {a^4}{5 b^5 \left (a+b x^5\right )}-\frac {4 a^3 \log \left (a+b x^5\right )}{5 b^5}-\frac {a x^{10}}{5 b^3}+\frac {x^{15}}{15 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{24}}{\left (a+b x^5\right )^2} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {x^4}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \left (\frac {3 a^2}{b^4}-\frac {2 a x}{b^3}+\frac {x^2}{b^2}+\frac {a^4}{b^4 (a+b x)^2}-\frac {4 a^3}{b^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=\frac {3 a^2 x^5}{5 b^4}-\frac {a x^{10}}{5 b^3}+\frac {x^{15}}{15 b^2}-\frac {a^4}{5 b^5 \left (a+b x^5\right )}-\frac {4 a^3 \log \left (a+b x^5\right )}{5 b^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.83 \[ \frac {-\frac {3 a^4}{a+b x^5}-12 a^3 \log \left (a+b x^5\right )+9 a^2 b x^5-3 a b^2 x^{10}+b^3 x^{15}}{15 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 81, normalized size = 1.12 \[ \frac {b^{4} x^{20} - 2 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} + 9 \, a^{3} b x^{5} - 3 \, a^{4} - 12 \, {\left (a^{3} b x^{5} + a^{4}\right )} \log \left (b x^{5} + a\right )}{15 \, {\left (b^{6} x^{5} + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 80, normalized size = 1.11 \[ -\frac {4 \, a^{3} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac {b^{4} x^{15} - 3 \, a b^{3} x^{10} + 9 \, a^{2} b^{2} x^{5}}{15 \, b^{6}} + \frac {4 \, a^{3} b x^{5} + 3 \, a^{4}}{5 \, {\left (b x^{5} + a\right )} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.88 \[ \frac {x^{15}}{15 b^{2}}-\frac {a \,x^{10}}{5 b^{3}}+\frac {3 a^{2} x^{5}}{5 b^{4}}-\frac {a^{4}}{5 \left (b \,x^{5}+a \right ) b^{5}}-\frac {4 a^{3} \ln \left (b \,x^{5}+a \right )}{5 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 65, normalized size = 0.90 \[ -\frac {a^{4}}{5 \, {\left (b^{6} x^{5} + a b^{5}\right )}} - \frac {4 \, a^{3} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac {b^{2} x^{15} - 3 \, a b x^{10} + 9 \, a^{2} x^{5}}{15 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 68, normalized size = 0.94 \[ \frac {x^{15}}{15\,b^2}-\frac {a^4}{5\,b\,\left (b^5\,x^5+a\,b^4\right )}-\frac {a\,x^{10}}{5\,b^3}-\frac {4\,a^3\,\ln \left (b\,x^5+a\right )}{5\,b^5}+\frac {3\,a^2\,x^5}{5\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 68, normalized size = 0.94 \[ - \frac {a^{4}}{5 a b^{5} + 5 b^{6} x^{5}} - \frac {4 a^{3} \log {\left (a + b x^{5} \right )}}{5 b^{5}} + \frac {3 a^{2} x^{5}}{5 b^{4}} - \frac {a x^{10}}{5 b^{3}} + \frac {x^{15}}{15 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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