Optimal. Leaf size=167 \[ \frac {3 a b^2 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {3 a^2 b x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {b^3 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {a^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]
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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} \begin {gather*} \frac {b^3 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {3 a b^2 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {3 a^2 b x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {a^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int x^7 \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^7 \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^7+3 a^2 b^4 x^{10}+3 a b^5 x^{13}+b^6 x^{16}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {a^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {3 a^2 b x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {3 a b^2 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {b^3 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.37 \begin {gather*} \frac {x^8 \sqrt {\left (a+b x^3\right )^2} \left (1309 a^3+2856 a^2 b x^3+2244 a b^2 x^6+616 b^3 x^9\right )}{10472 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.62, size = 61, normalized size = 0.37 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (1309 a^3 x^8+2856 a^2 b x^{11}+2244 a b^2 x^{14}+616 b^3 x^{17}\right )}{10472 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 35, normalized size = 0.21 \begin {gather*} \frac {1}{17} \, b^{3} x^{17} + \frac {3}{14} \, a b^{2} x^{14} + \frac {3}{11} \, a^{2} b x^{11} + \frac {1}{8} \, a^{3} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 67, normalized size = 0.40 \begin {gather*} \frac {1}{17} \, b^{3} x^{17} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{14} \, a b^{2} x^{14} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{11} \, a^{2} b x^{11} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{8} \, a^{3} x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}
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 \begin {gather*} \frac {\left (616 b^{3} x^{9}+2244 a \,b^{2} x^{6}+2856 a^{2} b \,x^{3}+1309 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} x^{8}}{10472 \left (b \,x^{3}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 35, normalized size = 0.21 \begin {gather*} \frac {1}{17} \, b^{3} x^{17} + \frac {3}{14} \, a b^{2} x^{14} + \frac {3}{11} \, a^{2} b x^{11} + \frac {1}{8} \, a^{3} x^{8} \end {gather*}
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 \begin {gather*} \int x^7\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2} \,d x \end {gather*}
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 \begin {gather*} \int x^{7} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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