Optimal. Leaf size=133 \[ \frac {\sqrt {\frac {b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A b (7-2 m)) \, _2F_1\left (\frac {3}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{9 a^2 b e (m+1) \sqrt {a+b x^3}}+\frac {2 (e x)^{m+1} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {457, 365, 364} \[ \frac {\sqrt {\frac {b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A b (7-2 m)) \, _2F_1\left (\frac {3}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{9 a^2 b e (m+1) \sqrt {a+b x^3}}+\frac {2 (e x)^{m+1} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {2 (A b-a B) (e x)^{1+m}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {\left (2 \left (-A b \left (-\frac {7}{2}+m\right )+a B (1+m)\right )\right ) \int \frac {(e x)^m}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b}\\ &=\frac {2 (A b-a B) (e x)^{1+m}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {\left (2 \left (-A b \left (-\frac {7}{2}+m\right )+a B (1+m)\right ) \sqrt {1+\frac {b x^3}{a}}\right ) \int \frac {(e x)^m}{\left (1+\frac {b x^3}{a}\right )^{3/2}} \, dx}{9 a^2 b \sqrt {a+b x^3}}\\ &=\frac {2 (A b-a B) (e x)^{1+m}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {(A b (7-2 m)+2 a B (1+m)) (e x)^{1+m} \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{9 a^2 b e (1+m) \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 113, normalized size = 0.85 \[ \frac {x \sqrt {\frac {b x^3}{a}+1} (e x)^m \left (A (m+4) \, _2F_1\left (\frac {5}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )+B (m+1) x^3 \, _2F_1\left (\frac {5}{2},\frac {m+4}{3};\frac {m+7}{3};-\frac {b x^3}{a}\right )\right )}{a^2 (m+1) (m+4) \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} \left (e x\right )^{m}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{m}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{3}+A \right ) \left (e x \right )^{m}}{\left (b \,x^{3}+a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{m}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^3+A\right )\,{\left (e\,x\right )}^m}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \]
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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