Optimal. Leaf size=85 \[ \frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (n+3)}-\frac{2 \sqrt{a} c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}} \]
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Rubi [A] time = 0.198622, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2028, 2031, 2029, 206} \[ \frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (n+3)}-\frac{2 \sqrt{a} c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 2028
Rule 2031
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \sqrt{c x} \sqrt{\frac{a}{x^3}+b x^n} \, dx &=\frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (3+n)}+\left (a c^3\right ) \int \frac{1}{(c x)^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \, dx\\ &=\frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (3+n)}+\frac{\left (a c \sqrt{x}\right ) \int \frac{1}{x^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \, dx}{\sqrt{c x}}\\ &=\frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (3+n)}-\frac{\left (2 a c \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(3+n) \sqrt{c x}}\\ &=\frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (3+n)}-\frac{2 \sqrt{a} c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(3+n) \sqrt{c x}}\\ \end{align*}
Mathematica [A] time = 0.0676533, size = 85, normalized size = 1. \[ \frac{x \sqrt{c x} \sqrt{\frac{a}{x^3}+b x^n} \left (2 \sqrt{a+b x^{n+3}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^{n+3}}}{\sqrt{a}}\right )\right )}{(n+3) \sqrt{a+b x^{n+3}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.346, size = 0, normalized size = 0. \begin{align*} \int \sqrt{cx}\sqrt{{\frac{a}{{x}^{3}}}+b{x}^{n}}\, 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{b x^{n} + \frac{a}{x^{3}}} \sqrt{c x}\,{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 \sqrt{c x} \sqrt{\frac{a}{x^{3}} + b x^{n}}\, 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{b x^{n} + \frac{a}{x^{3}}} \sqrt{c x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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