Optimal. Leaf size=84 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \sqrt{c x^3}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^3}}{a}+1}} \]
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Rubi [A] time = 0.0702241, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {369, 343, 341, 365, 364} \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \sqrt{c x^3}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^3}}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 343
Rule 341
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \sqrt{a+b \sqrt{c x^3}} \, dx &=\operatorname{Subst}\left (\int (d x)^m \sqrt{a+b \sqrt{c} x^{3/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\left (x^{-m} (d x)^m\right ) \int x^m \sqrt{a+b \sqrt{c} x^{3/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int x^{-1+2 (1+m)} \sqrt{a+b \sqrt{c} x^3} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{\left (2 x^{-m} (d x)^m \sqrt{a+b \sqrt{c} x^{3/2}}\right ) \operatorname{Subst}\left (\int x^{-1+2 (1+m)} \sqrt{1+\frac{b \sqrt{c} x^3}{a}} \, dx,x,\sqrt{x}\right )}{\sqrt{1+\frac{b \sqrt{c} x^{3/2}}{a}}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (1+m)}{3};\frac{1}{3} (5+2 m);-\frac{b \sqrt{c x^3}}{a}\right )}{(1+m) \sqrt{1+\frac{b \sqrt{c x^3}}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0664947, size = 84, normalized size = 1. \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \sqrt{c x^3}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^3}}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{c{x}^{3}}}\, 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{\sqrt{c x^{3}} 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 \sqrt{c x^{3}}}\, 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{\sqrt{c x^{3}} 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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