Optimal. Leaf size=161 \[ \frac {a b^2 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^3 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 161, 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} \begin {gather*} \frac {b^3 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {a b^2 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}
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^4} \, 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^4} \, 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^2} \, 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 b^5+\frac {a^3 b^3}{x^2}+\frac {3 a^2 b^4}{x}+b^6 x\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {a b^2 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^3 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {3 a^2 b \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 = 62, normalized size = 0.39 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-2 a^3+18 a^2 b x^3 \log (x)+6 a b^2 x^6+b^3 x^9\right )}{6 x^3 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.74, size = 317, normalized size = 1.97 \begin {gather*} -\frac {1}{2} a^2 \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}-a-\sqrt {b^2} x^3\right )-\frac {1}{2} a^2 \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}+a-\sqrt {b^2} x^3\right )+a^2 b \tanh ^{-1}\left (\frac {\sqrt {b^2} x^3}{a}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{a}\right )+\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (-8 a^3 b-21 a^2 b^2 x^3+24 a b^3 x^6+4 b^4 x^9\right )+\sqrt {b^2} \left (8 a^4+29 a^3 b x^3-3 a^2 b^2 x^6-28 a b^3 x^9-4 b^4 x^{12}\right )}{24 x^3 \left (a b+b^2 x^3\right )-24 \sqrt {b^2} x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 38, normalized size = 0.24 \begin {gather*} \frac {b^{3} x^{9} + 6 \, a b^{2} x^{6} + 18 \, a^{2} b x^{3} \log \relax (x) - 2 \, a^{3}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 85, normalized size = 0.53 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + a b^{2} x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {3 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.37 \begin {gather*} \frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} x^{9}+6 a \,b^{2} x^{6}+18 a^{2} b \,x^{3} \ln \relax (x )-2 a^{3}\right )}{6 \left (b \,x^{3}+a \right )^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 156, normalized size = 0.97 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{2} x^{3} + \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a^{2} b \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a^{2} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a b - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}{3 \, x^{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 \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^4} \,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 \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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