Optimal. Leaf size=56 \[ \frac{6}{11} a^2 b^2 x^{11}+\frac{4}{9} a^3 b x^9+\frac{a^4 x^7}{7}+\frac{4}{13} a b^3 x^{13}+\frac{b^4 x^{15}}{15} \]
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Rubi [A] time = 0.0283449, antiderivative size = 56, normalized size of antiderivative = 1., 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} \[ \frac{6}{11} a^2 b^2 x^{11}+\frac{4}{9} a^3 b x^9+\frac{a^4 x^7}{7}+\frac{4}{13} a b^3 x^{13}+\frac{b^4 x^{15}}{15} \]
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*}
Mathematica [A] time = 0.0025841, size = 56, normalized size = 1. \[ \frac{6}{11} a^2 b^2 x^{11}+\frac{4}{9} a^3 b x^9+\frac{a^4 x^7}{7}+\frac{4}{13} a b^3 x^{13}+\frac{b^4 x^{15}}{15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 47, normalized size = 0.8 \begin{align*}{\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989778, size = 62, normalized size = 1.11 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3078, size = 112, normalized size = 2. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.073511, size = 53, normalized size = 0.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12032, size = 62, normalized size = 1.11 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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