Optimal. Leaf size=41 \[ \frac {2 \sqrt {a x^3-b} \left (3 a^2 x^6-a b x^3-2 b^2\right )}{45 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.02, 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 )^{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 = 0.01, size = 30, normalized size = 0.73 \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 = 0.03, size = 41, 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 = 0.39, size = 37, normalized size = 0.90 \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 = 0.34, size = 33, normalized size = 0.80 \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 = 0.00, size = 27, 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 = 0.40, size = 34, normalized size = 0.83 \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 = 0.55, size = 33, normalized size = 0.80 \begin {gather*} \frac {6\,{\left (a\,x^3-b\right )}^{5/2}+10\,b\,{\left (a\,x^3-b\right )}^{3/2}}{45\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 68, normalized size = 1.66 \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 {x^{6} \sqrt {- b}}{6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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