Optimal. Leaf size=54 \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{\sqrt{a} c (n+1) \sqrt{c x}} \]
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Rubi [A] time = 0.110832, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2031, 2029, 206} \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{\sqrt{a} c (n+1) \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 2031
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/2} \sqrt{\frac{a}{x}+b x^n}} \, dx &=\frac{\sqrt{x} \int \frac{1}{x^{3/2} \sqrt{\frac{a}{x}+b x^n}} \, dx}{c \sqrt{c x}}\\ &=-\frac{\left (2 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{c (1+n) \sqrt{c x}}\\ &=-\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{\sqrt{a} c (1+n) \sqrt{c x}}\\ \end{align*}
Mathematica [A] time = 0.0480262, size = 68, normalized size = 1.26 \[ -\frac{2 x \sqrt{a+b x^{n+1}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^{n+1}}}{\sqrt{a}}\right )}{\sqrt{a} (n+1) (c x)^{3/2} \sqrt{\frac{a}{x}+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.339, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{a}{x}}+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 \frac{1}{\sqrt{b x^{n} + \frac{a}{x}} \left (c x\right )^{\frac{3}{2}}}\,{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 \frac{1}{\left (c x\right )^{\frac{3}{2}} \sqrt{\frac{a}{x} + 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 \frac{1}{\sqrt{b x^{n} + \frac{a}{x}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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