Optimal. Leaf size=53 \[ -\frac {a^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^4}{4 b^2}+\frac {x^6}{6 b} \]
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Rubi [A] time = 0.04, antiderivative size = 53, 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} \[ \frac {a^2 x^2}{2 b^3}-\frac {a^3 \log \left (a+b x^2\right )}{2 b^4}-\frac {a x^4}{4 b^2}+\frac {x^6}{6 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^2}{2 b^3}-\frac {a x^4}{4 b^2}+\frac {x^6}{6 b}-\frac {a^3 \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 1.00 \[ -\frac {a^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^4}{4 b^2}+\frac {x^6}{6 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 45, normalized size = 0.85 \[ \frac {2 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} - 6 \, a^{3} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 47, normalized size = 0.89 \[ -\frac {a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac {2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.87 \[ \frac {x^{6}}{6 b}-\frac {a \,x^{4}}{4 b^{2}}+\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 46, normalized size = 0.87 \[ -\frac {a^{3} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac {2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 45, normalized size = 0.85 \[ \frac {x^6}{6\,b}-\frac {a\,x^4}{4\,b^2}-\frac {a^3\,\ln \left (b\,x^2+a\right )}{2\,b^4}+\frac {a^2\,x^2}{2\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 44, normalized size = 0.83 \[ - \frac {a^{3} \log {\left (a + b x^{2} \right )}}{2 b^{4}} + \frac {a^{2} x^{2}}{2 b^{3}} - \frac {a x^{4}}{4 b^{2}} + \frac {x^{6}}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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