Optimal. Leaf size=53 \[ \frac {a^2 \left (a+b x^2\right )^5}{10 b^3}+\frac {\left (a+b x^2\right )^7}{14 b^3}-\frac {a \left (a+b x^2\right )^6}{6 b^3} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \begin {gather*} \frac {a^2 \left (a+b x^2\right )^5}{10 b^3}+\frac {\left (a+b x^2\right )^7}{14 b^3}-\frac {a \left (a+b x^2\right )^6}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac {\int x^5 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \left (a b+b^2 x\right )^4 \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 \left (a b+b^2 x\right )^4}{b^2}-\frac {2 a \left (a b+b^2 x\right )^5}{b^3}+\frac {\left (a b+b^2 x\right )^6}{b^4}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {a^2 \left (a+b x^2\right )^5}{10 b^3}-\frac {a \left (a+b x^2\right )^6}{6 b^3}+\frac {\left (a+b x^2\right )^7}{14 b^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 56, normalized size = 1.06 \begin {gather*} \frac {a^4 x^6}{6}+\frac {1}{2} a^3 b x^8+\frac {3}{5} a^2 b^2 x^{10}+\frac {1}{3} a b^3 x^{12}+\frac {b^4 x^{14}}{14} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 46, normalized size = 0.87 \begin {gather*} \frac {1}{14} x^{14} b^{4} + \frac {1}{3} x^{12} b^{3} a + \frac {3}{5} x^{10} b^{2} a^{2} + \frac {1}{2} x^{8} b a^{3} + \frac {1}{6} x^{6} a^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 46, normalized size = 0.87 \begin {gather*} \frac {1}{14} \, b^{4} x^{14} + \frac {1}{3} \, a b^{3} x^{12} + \frac {3}{5} \, a^{2} b^{2} x^{10} + \frac {1}{2} \, a^{3} b x^{8} + \frac {1}{6} \, a^{4} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 47, normalized size = 0.89 \begin {gather*} \frac {1}{14} b^{4} x^{14}+\frac {1}{3} a \,b^{3} x^{12}+\frac {3}{5} a^{2} b^{2} x^{10}+\frac {1}{2} a^{3} b \,x^{8}+\frac {1}{6} a^{4} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 46, normalized size = 0.87 \begin {gather*} \frac {1}{14} \, b^{4} x^{14} + \frac {1}{3} \, a b^{3} x^{12} + \frac {3}{5} \, a^{2} b^{2} x^{10} + \frac {1}{2} \, a^{3} b x^{8} + \frac {1}{6} \, a^{4} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 46, normalized size = 0.87 \begin {gather*} \frac {a^4\,x^6}{6}+\frac {a^3\,b\,x^8}{2}+\frac {3\,a^2\,b^2\,x^{10}}{5}+\frac {a\,b^3\,x^{12}}{3}+\frac {b^4\,x^{14}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 49, normalized size = 0.92 \begin {gather*} \frac {a^{4} x^{6}}{6} + \frac {a^{3} b x^{8}}{2} + \frac {3 a^{2} b^{2} x^{10}}{5} + \frac {a b^{3} x^{12}}{3} + \frac {b^{4} x^{14}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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