Optimal. Leaf size=59 \[ \frac {a^3}{5 b^4 \left (a+b x^5\right )}+\frac {3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac {2 a x^5}{5 b^3}+\frac {x^{10}}{10 b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 59, 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} \[ \frac {a^3}{5 b^4 \left (a+b x^5\right )}+\frac {3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac {2 a x^5}{5 b^3}+\frac {x^{10}}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{19}}{\left (a+b x^5\right )^2} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {2 a x^5}{5 b^3}+\frac {x^{10}}{10 b^2}+\frac {a^3}{5 b^4 \left (a+b x^5\right )}+\frac {3 a^2 \log \left (a+b x^5\right )}{5 b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.83 \[ \frac {\frac {2 a^3}{a+b x^5}+6 a^2 \log \left (a+b x^5\right )-4 a b x^5+b^2 x^{10}}{10 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 70, normalized size = 1.19 \[ \frac {b^{3} x^{15} - 3 \, a b^{2} x^{10} - 4 \, a^{2} b x^{5} + 2 \, a^{3} + 6 \, {\left (a^{2} b x^{5} + a^{3}\right )} \log \left (b x^{5} + a\right )}{10 \, {\left (b^{5} x^{5} + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 67, normalized size = 1.14 \[ \frac {3 \, a^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac {b^{2} x^{10} - 4 \, a b x^{5}}{10 \, b^{4}} - \frac {3 \, a^{2} b x^{5} + 2 \, a^{3}}{5 \, {\left (b x^{5} + a\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.88 \[ \frac {x^{10}}{10 b^{2}}-\frac {2 a \,x^{5}}{5 b^{3}}+\frac {a^{3}}{5 \left (b \,x^{5}+a \right ) b^{4}}+\frac {3 a^{2} \ln \left (b \,x^{5}+a \right )}{5 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 54, normalized size = 0.92 \[ \frac {a^{3}}{5 \, {\left (b^{5} x^{5} + a b^{4}\right )}} + \frac {3 \, a^{2} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac {b x^{10} - 4 \, a x^{5}}{10 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 57, normalized size = 0.97 \[ \frac {x^{10}}{10\,b^2}+\frac {a^3}{5\,b\,\left (b^4\,x^5+a\,b^3\right )}-\frac {2\,a\,x^5}{5\,b^3}+\frac {3\,a^2\,\ln \left (b\,x^5+a\right )}{5\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.97, size = 56, normalized size = 0.95 \[ \frac {a^{3}}{5 a b^{4} + 5 b^{5} x^{5}} + \frac {3 a^{2} \log {\left (a + b x^{5} \right )}}{5 b^{4}} - \frac {2 a x^{5}}{5 b^{3}} + \frac {x^{10}}{10 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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