Optimal. Leaf size=48 \[ \frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{5/2} \, _2F_1\left (\frac {5}{2},-m;\frac {7}{2};1+\frac {b x}{a}\right )}{5 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {69, 67}
\begin {gather*} \frac {2 x^m (a+b x)^{5/2} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {5}{2},-m;\frac {7}{2};\frac {b x}{a}+1\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 69
Rubi steps
\begin {align*} \int x^m (a+b x)^{3/2} \, dx &=\left (x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \left (-\frac {b x}{a}\right )^m (a+b x)^{3/2} \, dx\\ &=\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{5/2} \, _2F_1\left (\frac {5}{2},-m;\frac {7}{2};1+\frac {b x}{a}\right )}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{5/2} \, _2F_1\left (\frac {5}{2},-m;\frac {7}{2};1+\frac {b x}{a}\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{m} \left (b x +a \right )^{\frac {3}{2}}\, dx\]
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 = 0.00, 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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Sympy [C] Result contains complex when optimal does not.
time = 1.67, size = 37, normalized size = 0.77 \begin {gather*} \frac {a^{\frac {3}{2}} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\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 x^m\,{\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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