Optimal. Leaf size=79 \[ \frac {a x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {b x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 14} \[ \frac {b x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {a x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int x^5 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^5 \left (a b+b^2 x^3\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a b x^5+b^2 x^8\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {a x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {b x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ \frac {\sqrt {\left (a+b x^3\right )^2} \left (3 a x^6+2 b x^9\right )}{18 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 13, normalized size = 0.16 \[ \frac {1}{9} \, b x^{9} + \frac {1}{6} \, a x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 23, normalized size = 0.29 \[ \frac {1}{18} \, {\left (2 \, b x^{9} + 3 \, a x^{6}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.46 \[ \frac {\left (2 b \,x^{3}+3 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}\, x^{6}}{18 b \,x^{3}+18 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 83, normalized size = 1.05 \[ -\frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a x^{3}}{6 \, b} - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}{9 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 59, normalized size = 0.75 \[ \frac {\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}\,\left (8\,b^2\,\left (a^2+b^2\,x^6\right )-12\,a^2\,b^2+4\,a\,b^3\,x^3\right )}{72\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 12, normalized size = 0.15 \[ \frac {a x^{6}}{6} + \frac {b x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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