Optimal. Leaf size=119 \[ \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3}-\frac {2 a \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^4}{15 b^3}+\frac {a^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}{12 b^3} \]
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Rubi [A] time = 0.05, antiderivative size = 167, normalized size of antiderivative = 1.40, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \begin {gather*} \frac {b^3 x^{18} \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 \left (a+b x^3\right )}+\frac {a b^2 x^{15} \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {a^2 b x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {a^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int x^8 \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^8 \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} \operatorname {Subst}\left (\int x^2 \left (a b+b^2 x\right )^3 \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \left (a^3 b^3 x^2+3 a^2 b^4 x^3+3 a b^5 x^4+b^6 x^5\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {a^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {a^2 b x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {a b^2 x^{15} \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {b^3 x^{18} \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.51 \begin {gather*} \frac {x^9 \sqrt {\left (a+b x^3\right )^2} \left (20 a^3+45 a^2 b x^3+36 a b^2 x^6+10 b^3 x^9\right )}{180 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 10.09, size = 61, normalized size = 0.51 \begin {gather*} \frac {x^9 \sqrt {\left (a+b x^3\right )^2} \left (20 a^3+45 a^2 b x^3+36 a b^2 x^6+10 b^3 x^9\right )}{180 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 35, normalized size = 0.29 \begin {gather*} \frac {1}{18} \, b^{3} x^{18} + \frac {1}{5} \, a b^{2} x^{15} + \frac {1}{4} \, a^{2} b x^{12} + \frac {1}{9} \, a^{3} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 67, normalized size = 0.56 \begin {gather*} \frac {1}{18} \, b^{3} x^{18} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{5} \, a b^{2} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{4} \, a^{2} b x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{9} \, a^{3} x^{9} \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 = 0.49 \begin {gather*} \frac {\left (10 b^{3} x^{9}+36 a \,b^{2} x^{6}+45 a^{2} b \,x^{3}+20 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} x^{9}}{180 \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 = 0.46, size = 114, normalized size = 0.96 \begin {gather*} \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{2} x^{3}}{12 \, b^{2}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} x^{3}}{18 \, b^{2}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{3}}{12 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} a}{90 \, b^{3}} \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^8\,{\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^{8} \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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