Optimal. Leaf size=104 \[ -\frac {4 (e x)^{3/2} (4 A b-a B)}{9 a^3 e^4 \sqrt {a+b x^3}}-\frac {2 (e x)^{3/2} (4 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {453, 273, 264} \[ -\frac {4 (e x)^{3/2} (4 A b-a B)}{9 a^3 e^4 \sqrt {a+b x^3}}-\frac {2 (e x)^{3/2} (4 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^3}{(e x)^{5/2} \left (a+b x^3\right )^{5/2}} \, dx &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}}-\frac {(4 A b-a B) \int \frac {\sqrt {e x}}{\left (a+b x^3\right )^{5/2}} \, dx}{a e^3}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}}-\frac {2 (4 A b-a B) (e x)^{3/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {(2 (4 A b-a B)) \int \frac {\sqrt {e x}}{\left (a+b x^3\right )^{3/2}} \, dx}{3 a^2 e^3}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}}-\frac {2 (4 A b-a B) (e x)^{3/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {4 (4 A b-a B) (e x)^{3/2}}{9 a^3 e^4 \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.62 \[ \frac {x \left (-6 a^2 \left (A-B x^3\right )+4 a b x^3 \left (B x^3-6 A\right )-16 A b^2 x^6\right )}{9 a^3 (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 93, normalized size = 0.89 \[ \frac {2 \, {\left (2 \, {\left (B a b - 4 \, A b^{2}\right )} x^{6} + 3 \, {\left (B a^{2} - 4 \, A a b\right )} x^{3} - 3 \, A a^{2}\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{9 \, {\left (a^{3} b^{2} e^{3} x^{8} + 2 \, a^{4} b e^{3} x^{5} + a^{5} e^{3} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 0.60 \[ -\frac {2 \left (8 A \,b^{2} x^{6}-2 B a b \,x^{6}+12 A a b \,x^{3}-3 B \,a^{2} x^{3}+3 A \,a^{2}\right ) x}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (e x \right )^{\frac {5}{2}} a^{3}} \]
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 \[ \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.80, size = 115, normalized size = 1.11 \[ -\frac {\sqrt {b\,x^3+a}\,\left (\frac {2\,A}{3\,a\,b^2\,e^2}-\frac {x^3\,\left (6\,B\,a^2-24\,A\,a\,b\right )}{9\,a^3\,b^2\,e^2}+\frac {x^6\,\left (16\,A\,b^2-4\,B\,a\,b\right )}{9\,a^3\,b^2\,e^2}\right )}{x^7\,\sqrt {e\,x}+\frac {a^2\,x\,\sqrt {e\,x}}{b^2}+\frac {2\,a\,x^4\,\sqrt {e\,x}}{b}} \]
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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