Optimal. Leaf size=107 \[ \frac{2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}-\frac{2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt{a x^j+b x^n}} \]
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Rubi [A] time = 0.187196, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2031, 2030, 2029, 206} \[ \frac{2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}-\frac{2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Rule 2031
Rule 2030
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{(c x)^{-1+\frac{3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx &=\frac{\left (x^{-3 j/2} (c x)^{3 j/2}\right ) \int \frac{x^{-1+\frac{3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx}{c}\\ &=-\frac{2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt{a x^j+b x^n}}+\frac{\left (x^{-3 j/2} (c x)^{3 j/2}\right ) \int \frac{x^{-1+\frac{j}{2}}}{\sqrt{a x^j+b x^n}} \, dx}{a c}\\ &=-\frac{2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt{a x^j+b x^n}}+\frac{\left (2 x^{-3 j/2} (c x)^{3 j/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{a c (j-n)}\\ &=-\frac{2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt{a x^j+b x^n}}+\frac{2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}\\ \end{align*}
Mathematica [A] time = 0.17613, size = 117, normalized size = 1.09 \[ -\frac{2 x^{-3 j/2} (c x)^{3 j/2} \left (\sqrt{a} x^{j/2}-\sqrt{b} x^{n/2} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )\right )}{a^{3/2} c (j-n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.37, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1+{\frac{3\,j}{2}}} \left ( a{x}^{j}+b{x}^{n} \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 \frac{\left (c x\right )^{\frac{3}{2} \, j - 1}}{{\left (a x^{j} + b x^{n}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (c x\right )^{\frac{3}{2} \, j - 1}}{{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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