Optimal. Leaf size=160 \[ \frac {a b^2 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {a^2 b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {a^3 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, 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} \[ \frac {b^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {a b^2 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {a^2 b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {a^3 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^3}{x} \, 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 \frac {\left (a b+b^2 x\right )^3}{x} \, 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 (3 a^2 b^4+\frac {a^3 b^3}{x}+3 a b^5 x+b^6 x^2\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {a^2 b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {a b^2 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {b^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.38 \[ \frac {\sqrt {\left (a+b x^3\right )^2} \left (18 a^3 \log (x)+b x^3 \left (18 a^2+9 a b x^3+2 b^2 x^6\right )\right )}{18 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 32, normalized size = 0.20 \[ \frac {1}{9} \, b^{3} x^{9} + \frac {1}{2} \, a b^{2} x^{6} + a^{2} b x^{3} + a^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 65, normalized size = 0.41 \[ \frac {1}{9} \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{2} \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.36 \[ \frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} \left (2 b^{3} x^{9}+9 a \,b^{2} x^{6}+18 a^{2} b \,x^{3}+18 a^{3} \ln \relax (x )\right )}{18 \left (b \,x^{3}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 152, normalized size = 0.95 \[ \frac {1}{6} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a b x^{3} + \frac {1}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a^{3} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {1}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{2} + \frac {1}{9} \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} \]
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 \[ \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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