Optimal. Leaf size=79 \[ \frac {2 (e x)^{3/2} (a B+2 A b)}{9 a^2 b e \sqrt {a+b x^3}}+\frac {2 (e x)^{3/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {457, 264} \begin {gather*} \frac {2 (e x)^{3/2} (a B+2 A b)}{9 a^2 b e \sqrt {a+b x^3}}+\frac {2 (e x)^{3/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 457
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {2 (A b-a B) (e x)^{3/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {\left (2 \left (3 A b+\frac {3 a B}{2}\right )\right ) \int \frac {\sqrt {e x}}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b}\\ &=\frac {2 (A b-a B) (e x)^{3/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {2 (2 A b+a B) (e x)^{3/2}}{9 a^2 b e \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.56 \begin {gather*} \frac {2 x \sqrt {e x} \left (3 a A+a B x^3+2 A b x^3\right )}{9 a^2 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.26, size = 76, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (3 a A e^5 (e x)^{3/2}+a B e^2 (e x)^{9/2}+2 A b e^2 (e x)^{9/2}\right )}{9 a^2 \left (a e^3+b e^3 x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 59, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left ({\left (B a + 2 \, A b\right )} x^{4} + 3 \, A a x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{9 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 64, normalized size = 0.81 \begin {gather*} \frac {2 \, x^{\frac {3}{2}} {\left (\frac {3 \, A e^{5}}{a} + \frac {{\left (B a^{5} b^{5} e^{21} + 2 \, A a^{4} b^{6} e^{21}\right )} x^{3} e^{\left (-16\right )}}{a^{6} b^{5}}\right )} e^{\frac {3}{2}}}{9 \, {\left (b x^{3} e^{4} + a e^{4}\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.49 \begin {gather*} \frac {2 \left (2 A \,x^{3} b +B a \,x^{3}+3 A a \right ) \sqrt {e x}\, x}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x^{3} + A\right )} \sqrt {e x}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.63, size = 73, normalized size = 0.92 \begin {gather*} \frac {\left (\frac {2\,A\,x\,\sqrt {e\,x}}{3\,a\,b^2}+\frac {x^4\,\sqrt {e\,x}\,\left (4\,A\,b+2\,B\,a\right )}{9\,a^2\,b^2}\right )\,\sqrt {b\,x^3+a}}{x^6+\frac {a^2}{b^2}+\frac {2\,a\,x^3}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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