Optimal. Leaf size=78 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{b (c x)^{3/2}}{a}+1}} \]
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Rubi [A] time = 0.0827579, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {367, 343, 341, 365, 364} \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{b (c x)^{3/2}}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 367
Rule 343
Rule 341
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \sqrt{a+b (c x)^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d x}{c}\right )^m \sqrt{a+b x^{3/2}} \, dx,x,c x\right )}{c}\\ &=\frac{\left ((c x)^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int x^m \sqrt{a+b x^{3/2}} \, dx,x,c x\right )}{c}\\ &=\frac{\left (2 (c x)^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int x^{-1+2 (1+m)} \sqrt{a+b x^3} \, dx,x,\sqrt{c x}\right )}{c}\\ &=\frac{\left (2 (c x)^{-m} (d x)^m \sqrt{a+b (c x)^{3/2}}\right ) \operatorname{Subst}\left (\int x^{-1+2 (1+m)} \sqrt{1+\frac{b x^3}{a}} \, dx,x,\sqrt{c x}\right )}{c \sqrt{1+\frac{b (c x)^{3/2}}{a}}}\\ &=\frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (1+m)}{3};\frac{1}{3} (5+2 m);-\frac{b (c x)^{3/2}}{a}\right )}{(1+m) \sqrt{1+\frac{b (c x)^{3/2}}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0997827, size = 79, normalized size = 1.01 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{a+b (c x)^{3/2}}{a}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{a+b \left ( cx \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x\right )^{\frac{3}{2}} b + a} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \sqrt{a + b \left (c x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x\right )^{\frac{3}{2}} b + a} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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