Optimal. Leaf size=121 \[ -\frac {a e^{7/2} (4 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{12 b^{5/2}}+\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3} (4 A b-3 a B)}{12 b^2}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e} \]
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Rubi [A] time = 0.09, antiderivative size = 121, 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 = {459, 321, 329, 275, 217, 206} \[ \frac {e^2 (e x)^{3/2} \sqrt {a+b x^3} (4 A b-3 a B)}{12 b^2}-\frac {a e^{7/2} (4 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{12 b^{5/2}}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 321
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx &=\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e}-\frac {\left (-6 A b+\frac {9 a B}{2}\right ) \int \frac {(e x)^{7/2}}{\sqrt {a+b x^3}} \, dx}{6 b}\\ &=\frac {(4 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{12 b^2}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e}-\frac {\left (a (4 A b-3 a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{8 b^2}\\ &=\frac {(4 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{12 b^2}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e}-\frac {\left (a (4 A b-3 a 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 )}{4 b^2}\\ &=\frac {(4 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{12 b^2}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e}-\frac {\left (a (4 A b-3 a 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 )}{12 b^2}\\ &=\frac {(4 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{12 b^2}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e}-\frac {\left (a (4 A b-3 a 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 )}{12 b^2}\\ &=\frac {(4 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{12 b^2}+\frac {B (e x)^{9/2} \sqrt {a+b x^3}}{6 b e}-\frac {a (4 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{12 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 97, normalized size = 0.80 \[ \frac {e^3 \sqrt {e x} \left (\sqrt {b} x^{3/2} \sqrt {a+b x^3} \left (-3 a B+4 A b+2 b B x^3\right )+a (3 a B-4 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}\right )\right )}{12 b^{5/2} \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.10, size = 245, normalized size = 2.02 \[ \left [-\frac {{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{3} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right ) - 4 \, {\left (2 \, B b e^{3} x^{4} - {\left (3 \, B a - 4 \, A b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{48 \, b^{2}}, -\frac {{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{3} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right ) - 2 \, {\left (2 \, B b e^{3} x^{4} - {\left (3 \, B a - 4 \, A b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{24 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 114, normalized size = 0.94 \[ \frac {1}{12} \, \sqrt {b x^{3} e^{4} + a e^{4}} {\left (\frac {2 \, B x^{3} e^{\left (-2\right )}}{b} - \frac {{\left (3 \, B a b^{3} e^{5} - 4 \, A b^{4} e^{5}\right )} e^{\left (-7\right )}}{b^{5}}\right )} x^{\frac {3}{2}} e^{\frac {7}{2}} - \frac {{\left (3 \, B a^{2} b^{3} e^{9} - 4 \, A a b^{4} e^{9}\right )} e^{\left (-\frac {11}{2}\right )} \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} e^{2} + \sqrt {b x^{3} e^{4} + a e^{4}} \right |}\right )}{12 \, b^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.01, size = 6861, normalized size = 56.70 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {7}{2}}}{\sqrt {b x^{3} + a}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}}{\sqrt {b\,x^3+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 95.30, size = 194, normalized size = 1.60 \[ \frac {A \sqrt {a} e^{\frac {7}{2}} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 b} - \frac {A a e^{\frac {7}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{3 b^{\frac {3}{2}}} - \frac {B a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {3}{2}}}{4 b^{2} \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {B \sqrt {a} e^{\frac {7}{2}} x^{\frac {9}{2}}}{12 b \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B a^{2} e^{\frac {7}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {B e^{\frac {7}{2}} x^{\frac {15}{2}}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{3}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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