Optimal. Leaf size=94 \[ -\frac {a^6}{2 b^7 \left (a+b x^2\right )}-\frac {3 a^5 \log \left (a+b x^2\right )}{b^7}+\frac {5 a^4 x^2}{2 b^6}-\frac {a^3 x^4}{b^5}+\frac {a^2 x^6}{2 b^4}-\frac {a x^8}{4 b^3}+\frac {x^{10}}{10 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 94, 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} \[ \frac {a^2 x^6}{2 b^4}-\frac {a^3 x^4}{b^5}+\frac {5 a^4 x^2}{2 b^6}-\frac {a^6}{2 b^7 \left (a+b x^2\right )}-\frac {3 a^5 \log \left (a+b x^2\right )}{b^7}-\frac {a x^8}{4 b^3}+\frac {x^{10}}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {5 a^4}{b^6}-\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{b^4}-\frac {2 a x^3}{b^3}+\frac {x^4}{b^2}+\frac {a^6}{b^6 (a+b x)^2}-\frac {6 a^5}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {5 a^4 x^2}{2 b^6}-\frac {a^3 x^4}{b^5}+\frac {a^2 x^6}{2 b^4}-\frac {a x^8}{4 b^3}+\frac {x^{10}}{10 b^2}-\frac {a^6}{2 b^7 \left (a+b x^2\right )}-\frac {3 a^5 \log \left (a+b x^2\right )}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 83, normalized size = 0.88 \[ \frac {-\frac {10 a^6}{a+b x^2}-60 a^5 \log \left (a+b x^2\right )+50 a^4 b x^2-20 a^3 b^2 x^4+10 a^2 b^3 x^6-5 a b^4 x^8+2 b^5 x^{10}}{20 b^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 104, normalized size = 1.11 \[ \frac {2 \, b^{6} x^{12} - 3 \, a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} - 10 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 50 \, a^{5} b x^{2} - 10 \, a^{6} - 60 \, {\left (a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{20 \, {\left (b^{8} x^{2} + a b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 103, normalized size = 1.10 \[ -\frac {3 \, a^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{7}} + \frac {6 \, a^{5} b x^{2} + 5 \, a^{6}}{2 \, {\left (b x^{2} + a\right )} b^{7}} + \frac {2 \, b^{8} x^{10} - 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} - 20 \, a^{3} b^{5} x^{4} + 50 \, a^{4} b^{4} x^{2}}{20 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 0.90 \[ \frac {x^{10}}{10 b^{2}}-\frac {a \,x^{8}}{4 b^{3}}+\frac {a^{2} x^{6}}{2 b^{4}}-\frac {a^{3} x^{4}}{b^{5}}+\frac {5 a^{4} x^{2}}{2 b^{6}}-\frac {a^{6}}{2 \left (b \,x^{2}+a \right ) b^{7}}-\frac {3 a^{5} \ln \left (b \,x^{2}+a \right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 88, normalized size = 0.94 \[ -\frac {a^{6}}{2 \, {\left (b^{8} x^{2} + a b^{7}\right )}} - \frac {3 \, a^{5} \log \left (b x^{2} + a\right )}{b^{7}} + \frac {2 \, b^{4} x^{10} - 5 \, a b^{3} x^{8} + 10 \, a^{2} b^{2} x^{6} - 20 \, a^{3} b x^{4} + 50 \, a^{4} x^{2}}{20 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 90, normalized size = 0.96 \[ \frac {x^{10}}{10\,b^2}-\frac {a^6}{2\,b\,\left (b^7\,x^2+a\,b^6\right )}-\frac {a\,x^8}{4\,b^3}-\frac {3\,a^5\,\ln \left (b\,x^2+a\right )}{b^7}+\frac {a^2\,x^6}{2\,b^4}-\frac {a^3\,x^4}{b^5}+\frac {5\,a^4\,x^2}{2\,b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 88, normalized size = 0.94 \[ - \frac {a^{6}}{2 a b^{7} + 2 b^{8} x^{2}} - \frac {3 a^{5} \log {\left (a + b x^{2} \right )}}{b^{7}} + \frac {5 a^{4} x^{2}}{2 b^{6}} - \frac {a^{3} x^{4}}{b^{5}} + \frac {a^{2} x^{6}}{2 b^{4}} - \frac {a x^{8}}{4 b^{3}} + \frac {x^{10}}{10 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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