Optimal. Leaf size=56 \[ \frac {a^4 x^7}{7}+\frac {4}{9} a^3 b x^9+\frac {6}{11} a^2 b^2 x^{11}+\frac {4}{13} a b^3 x^{13}+\frac {b^4 x^{15}}{15} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270} \begin {gather*} \frac {6}{11} a^2 b^2 x^{11}+\frac {4}{9} a^3 b x^9+\frac {a^4 x^7}{7}+\frac {4}{13} a b^3 x^{13}+\frac {b^4 x^{15}}{15} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac {\int x^6 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac {\int \left (a^4 b^4 x^6+4 a^3 b^5 x^8+6 a^2 b^6 x^{10}+4 a b^7 x^{12}+b^8 x^{14}\right ) \, dx}{b^4}\\ &=\frac {a^4 x^7}{7}+\frac {4}{9} a^3 b x^9+\frac {6}{11} a^2 b^2 x^{11}+\frac {4}{13} a b^3 x^{13}+\frac {b^4 x^{15}}{15}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 56, normalized size = 1.00 \begin {gather*} \frac {a^4 x^7}{7}+\frac {4}{9} a^3 b x^9+\frac {6}{11} a^2 b^2 x^{11}+\frac {4}{13} a b^3 x^{13}+\frac {b^4 x^{15}}{15} \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^6 \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 = 0.68, size = 46, normalized size = 0.82 \begin {gather*} \frac {1}{15} x^{15} b^{4} + \frac {4}{13} x^{13} b^{3} a + \frac {6}{11} x^{11} b^{2} a^{2} + \frac {4}{9} x^{9} b a^{3} + \frac {1}{7} x^{7} 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 = 0.82 \begin {gather*} \frac {1}{15} \, b^{4} x^{15} + \frac {4}{13} \, a b^{3} x^{13} + \frac {6}{11} \, a^{2} b^{2} x^{11} + \frac {4}{9} \, a^{3} b x^{9} + \frac {1}{7} \, a^{4} x^{7} \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 = 0.84 \begin {gather*} \frac {1}{15} b^{4} x^{15}+\frac {4}{13} a \,b^{3} x^{13}+\frac {6}{11} a^{2} b^{2} x^{11}+\frac {4}{9} a^{3} b \,x^{9}+\frac {1}{7} a^{4} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 46, normalized size = 0.82 \begin {gather*} \frac {1}{15} \, b^{4} x^{15} + \frac {4}{13} \, a b^{3} x^{13} + \frac {6}{11} \, a^{2} b^{2} x^{11} + \frac {4}{9} \, a^{3} b x^{9} + \frac {1}{7} \, a^{4} x^{7} \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 = 0.82 \begin {gather*} \frac {a^4\,x^7}{7}+\frac {4\,a^3\,b\,x^9}{9}+\frac {6\,a^2\,b^2\,x^{11}}{11}+\frac {4\,a\,b^3\,x^{13}}{13}+\frac {b^4\,x^{15}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 53, normalized size = 0.95 \begin {gather*} \frac {a^{4} x^{7}}{7} + \frac {4 a^{3} b x^{9}}{9} + \frac {6 a^{2} b^{2} x^{11}}{11} + \frac {4 a b^{3} x^{13}}{13} + \frac {b^{4} x^{15}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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