Optimal. Leaf size=66 \[ \frac {x^{1+m} \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a (1+m) \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, 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 = {372, 371}
\begin {gather*} \frac {x^{m+1} \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a (m+1) \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {b x^3}{a}} \int \frac {x^m}{\left (1+\frac {b x^3}{a}\right )^{3/2}} \, dx}{a \sqrt {a+b x^3}}\\ &=\frac {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 )}{a (1+m) \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 68, normalized size = 1.03 \begin {gather*} \frac {x^{1+m} \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{3};1+\frac {1+m}{3};-\frac {b x^3}{a}\right )}{a (1+m) \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (b \,x^{3}+a \right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.37, size = 35, normalized size = 0.53 \begin {gather*} {\rm integral}\left (\frac {\sqrt {b x^{3} + a} x^{m}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.67, size = 53, normalized size = 0.80 \begin {gather*} \frac {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 a^{\frac {3}{2}} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {x^m}{{\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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