Optimal. Leaf size=167 \[ \frac {3 a b^2 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {3 a^2 b x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {b^3 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {a^3 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 167, 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, 270} \[ \frac {b^3 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {3 a b^2 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {3 a^2 b x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {a^3 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int x^9 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^9 \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^3 b^3 x^9+3 a^2 b^4 x^{12}+3 a b^5 x^{15}+b^6 x^{18}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {a^3 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {3 a^2 b x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {3 a b^2 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {b^3 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.37 \[ \frac {x^{10} \sqrt {\left (a+b x^3\right )^2} \left (1976 a^3+4560 a^2 b x^3+3705 a b^2 x^6+1040 b^3 x^9\right )}{19760 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 35, normalized size = 0.21 \[ \frac {1}{19} \, b^{3} x^{19} + \frac {3}{16} \, a b^{2} x^{16} + \frac {3}{13} \, a^{2} b x^{13} + \frac {1}{10} \, a^{3} x^{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 67, normalized size = 0.40 \[ \frac {1}{19} \, b^{3} x^{19} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{16} \, a b^{2} x^{16} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{13} \, a^{2} b x^{13} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{10} \, a^{3} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.35 \[ \frac {\left (1040 b^{3} x^{9}+3705 a \,b^{2} x^{6}+4560 a^{2} b \,x^{3}+1976 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} x^{10}}{19760 \left (b \,x^{3}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 35, normalized size = 0.21 \[ \frac {1}{19} \, b^{3} x^{19} + \frac {3}{16} \, a b^{2} x^{16} + \frac {3}{13} \, a^{2} b x^{13} + \frac {1}{10} \, a^{3} x^{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^9\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{9} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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