Optimal. Leaf size=114 \[ \frac {2 (e x)^{9/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {2 B e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{5/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}} \]
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Rubi [A] time = 0.08, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {452, 288, 329, 275, 217, 206} \[ \frac {2 (e x)^{9/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {2 B e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 288
Rule 329
Rule 452
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {2 (A b-a B) (e x)^{9/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {B \int \frac {(e x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx}{b}\\ &=\frac {2 (A b-a B) (e x)^{9/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {\left (B e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{b^2}\\ &=\frac {2 (A b-a B) (e x)^{9/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {\left (2 B e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{b^2}\\ &=\frac {2 (A b-a B) (e x)^{9/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {\left (2 B e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{3 b^2}\\ &=\frac {2 (A b-a B) (e x)^{9/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {\left (2 B e^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {b x^2}{e^3}} \, dx,x,\frac {(e x)^{3/2}}{\sqrt {a+b x^3}}\right )}{3 b^2}\\ &=\frac {2 (A b-a B) (e x)^{9/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {2 B e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 119, normalized size = 1.04 \[ \frac {2 e^3 \sqrt {e x} \left (3 a^{3/2} B \left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )+\sqrt {b} x^{3/2} \left (-3 a^2 B-4 a b B x^3+A b^2 x^3\right )\right )}{9 a b^{5/2} \sqrt {x} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 345, normalized size = 3.03 \[ \left [\frac {3 \, {\left (B a b^{2} e^{3} x^{6} + 2 \, B a^{2} b e^{3} x^{3} + B a^{3} e^{3}\right )} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right ) - 4 \, {\left ({\left (4 \, B a b - A b^{2}\right )} e^{3} x^{4} + 3 \, B a^{2} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{18 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}, -\frac {3 \, {\left (B a b^{2} e^{3} x^{6} + 2 \, B a^{2} b e^{3} x^{3} + B a^{3} e^{3}\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right ) + 2 \, {\left ({\left (4 \, B a b - A b^{2}\right )} e^{3} x^{4} + 3 \, B a^{2} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{9 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 102, normalized size = 0.89 \[ -\frac {2 \, x^{\frac {3}{2}} {\left (\frac {3 \, B a e^{8}}{b^{2}} + \frac {{\left (4 \, B a^{5} b^{6} e^{24} - A a^{4} b^{7} e^{24}\right )} x^{3} e^{\left (-16\right )}}{a^{5} b^{7}}\right )} e^{\frac {3}{2}}}{9 \, {\left (b x^{3} e^{4} + a e^{4}\right )}^{\frac {3}{2}}} - \frac {2 \, B e^{\frac {7}{2}} \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} e^{2} + \sqrt {b x^{3} e^{4} + a e^{4}} \right |}\right )}{3 \, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.05, size = 7081, normalized size = 62.11 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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