Optimal. Leaf size=27 \[ \frac {2 \left (a x^3-2 b\right ) \sqrt {a x^3+b}}{9 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \begin {gather*} \frac {2 \left (a x^3+b\right )^{3/2}}{9 a^2}-\frac {2 b \sqrt {a x^3+b}}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {b+a x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt {b+a x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {b}{a \sqrt {b+a x}}+\frac {\sqrt {b+a x}}{a}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 b \sqrt {b+a x^3}}{3 a^2}+\frac {2 \left (b+a x^3\right )^{3/2}}{9 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (a x^3-2 b\right ) \sqrt {a x^3+b}}{9 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (a x^3-2 b\right ) \sqrt {a x^3+b}}{9 a^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 23, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {a x^{3} + b} {\left (a x^{3} - 2 \, b\right )}}{9 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 30, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{9 \, a^{2}} - \frac {2 \, \sqrt {a x^{3} + b} b}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} \frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 30, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{9 \, a^{2}} - \frac {2 \, \sqrt {a x^{3} + b} b}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2\,\sqrt {a\,x^3+b}\,\left (2\,b-a\,x^3\right )}{9\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 46, normalized size = 1.70 \begin {gather*} \begin {cases} \frac {2 x^{3} \sqrt {a x^{3} + b}}{9 a} - \frac {4 b \sqrt {a x^{3} + b}}{9 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6 \sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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