Optimal. Leaf size=120 \[ \frac {e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{5/2}}-\frac {e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt {a+b x^3}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}} \]
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Rubi [A] time = 0.09, antiderivative size = 120, 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, 288, 329, 275, 217, 206} \begin {gather*} -\frac {e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt {a+b x^3}}+\frac {e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{5/2}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 288
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}}-\frac {\left (-3 A b+\frac {9 a B}{2}\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}}+\frac {\left ((2 A b-3 a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{2 b^2}\\ &=-\frac {(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}}+\frac {\left ((2 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 )}{b^2}\\ &=-\frac {(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}}+\frac {\left ((2 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 )}{3 b^2}\\ &=-\frac {(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}}+\frac {\left ((2 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 )}{3 b^2}\\ &=-\frac {(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt {a+b x^3}}+\frac {B (e x)^{9/2}}{3 b e \sqrt {a+b x^3}}+\frac {(2 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 )}{3 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 109, normalized size = 0.91 \begin {gather*} \frac {e^3 \sqrt {e x} \left (\sqrt {b} x^{3/2} \left (3 a B-2 A b+b B x^3\right )-\sqrt {a} \sqrt {\frac {b x^3}{a}+1} (3 a B-2 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )\right )}{3 b^{5/2} \sqrt {x} \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.76, size = 136, normalized size = 1.13 \begin {gather*} \frac {\sqrt {a+b x^3} \left (3 a B e^5 (e x)^{3/2}-2 A b e^5 (e x)^{3/2}+b B e^2 (e x)^{9/2}\right )}{3 b^2 \left (a e^3+b e^3 x^3\right )}-\frac {e^5 \sqrt {\frac {b}{e^3}} (2 A b-3 a B) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.92, size = 307, normalized size = 2.56 \begin {gather*} \left [-\frac {{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} + {\left (3 \, B a^{2} - 2 \, A a b\right )} 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 (B b e^{3} x^{4} + {\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{12 \, {\left (b^{3} x^{3} + a b^{2}\right )}}, \frac {{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} + {\left (3 \, B a^{2} - 2 \, A a b\right )} 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 (B b e^{3} x^{4} + {\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{6 \, {\left (b^{3} x^{3} + a b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 107, normalized size = 0.89 \begin {gather*} \frac {{\left (\frac {B x^{3} e^{4}}{b} + \frac {3 \, B a b^{3} e^{4} - 2 \, A b^{4} e^{4}}{b^{5}}\right )} x^{\frac {3}{2}} e^{\frac {3}{2}}}{3 \, \sqrt {b x^{3} e^{4} + a e^{4}}} + \frac {{\left (3 \, B a b^{3} e^{4} - 2 \, A b^{4} e^{4}\right )} e^{\left (-\frac {1}{2}\right )} \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 {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.13, size = 7016, normalized size = 58.47 \begin {gather*} \text {output too large to display} \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 )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \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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