Optimal. Leaf size=64 \[ \frac {a x^{m+1} \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{(m+1) \sqrt {\frac {b x^3}{a}+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac {a x^{m+1} \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{(m+1) \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int x^m \left (a+b x^3\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a+b x^3}\right ) \int x^m \left (1+\frac {b x^3}{a}\right )^{3/2} \, dx}{\sqrt {1+\frac {b x^3}{a}}}\\ &=\frac {a x^{1+m} \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{(1+m) \sqrt {1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 1.03 \[ \frac {a x^{m+1} \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{3};\frac {m+1}{3}+1;-\frac {b x^3}{a}\right )}{(m+1) \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{m}, x\right ) \]
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 \[ \int {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{3}+a \right )^{\frac {3}{2}} x^{m}\, dx \]
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 \[ \int {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (b\,x^3+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.35, size = 54, normalized size = 0.84 \[ \frac {a^{\frac {3}{2}} x x^{m} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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