Optimal. Leaf size=167 \[ \frac{b^3 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{3 a b^2 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac{3 a^2 b x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.042637, antiderivative size = 167, normalized size of antiderivative = 1., 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 1355
Rule 270
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*}
Mathematica [A] time = 0.0204177, size = 61, normalized size = 0.37 \[ \frac{x^{10} \sqrt{\left (a+b x^3\right )^2} \left (4560 a^2 b x^3+1976 a^3+3705 a b^2 x^6+1040 b^3 x^9\right )}{19760 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.4 \begin{align*}{\frac{{x}^{10} \left ( 1040\,{b}^{3}{x}^{9}+3705\,a{b}^{2}{x}^{6}+4560\,{a}^{2}b{x}^{3}+1976\,{a}^{3} \right ) }{19760\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02062, size = 47, normalized size = 0.28 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65848, size = 90, normalized size = 0.54 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{9} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12727, size = 90, normalized size = 0.54 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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