Optimal. Leaf size=38 \[ \frac {2 \sqrt {a x^3+b} \left (3 a^2 x^6+a b x^3-2 b^2\right )}{45 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 )^{5/2}}{15 a^2}-\frac {2 b \left (a x^3+b\right )^{3/2}}{9 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \sqrt {b+a x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x \sqrt {b+a x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {b \sqrt {b+a x}}{a}+\frac {(b+a x)^{3/2}}{a}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 b \left (b+a x^3\right )^{3/2}}{9 a^2}+\frac {2 \left (b+a x^3\right )^{5/2}}{15 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.74 \begin {gather*} \frac {2 \left (a x^3+b\right )^{3/2} \left (3 a x^3-2 b\right )}{45 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 38, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3+b} \left (3 a^2 x^6+a b x^3-2 b^2\right )}{45 a^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 34, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} x^{6} + a b x^{3} - 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{45 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 29, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (3 \, {\left (a x^{3} + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} b\right )}}{45 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \begin {gather*} \frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}} \left (3 a \,x^{3}-2 b \right )}{45 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 30, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {5}{2}}}{15 \, a^{2}} - \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} b}{9 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 29, normalized size = 0.76 \begin {gather*} -\frac {10\,b\,{\left (a\,x^3+b\right )}^{3/2}-6\,{\left (a\,x^3+b\right )}^{5/2}}{45\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 66, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {2 x^{6} \sqrt {a x^{3} + b}}{15} + \frac {2 b x^{3} \sqrt {a x^{3} + b}}{45 a} - \frac {4 b^{2} \sqrt {a x^{3} + b}}{45 a^{2}} & \text {for}\: a \neq 0 \\\frac {\sqrt {b} x^{6}}{6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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