Optimal. Leaf size=29 \[ \frac {2 \sqrt {a x^3-b} \left (a x^3+2 b\right )}{9 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.45, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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 = 29, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b} \left (a x^3+2 b\right )}{9 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 29, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b} \left (a x^3+2 b\right )}{9 a^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 25, normalized size = 0.86 \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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giac [A] time = 0.41, size = 34, normalized size = 1.17 \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 = 0.00, size = 26, normalized size = 0.90 \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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maxima [A] time = 0.34, size = 34, normalized size = 1.17 \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 = 0.40, size = 25, normalized size = 0.86 \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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sympy [A] time = 0.63, size = 48, normalized size = 1.66 \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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