Optimal. Leaf size=111 \[ \frac{2 x^{1-n} (c x)^m \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{m-\frac{3 n}{2}+1}{j-n};\frac{m-\frac{3 n}{2}+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{b (2 m-3 n+2) \sqrt{a x^j+b x^n}} \]
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Rubi [A] time = 0.107765, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2032, 365, 364} \[ \frac{2 x^{1-n} (c x)^m \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{m-\frac{3 n}{2}+1}{j-n};\frac{m-\frac{3 n}{2}+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{b (2 m-3 n+2) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Rule 2032
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^m}{\left (a x^j+b x^n\right )^{3/2}} \, dx &=\frac{\left (x^{-m+\frac{n}{2}} (c x)^m \sqrt{b+a x^{j-n}}\right ) \int \frac{x^{m-\frac{3 n}{2}}}{\left (b+a x^{j-n}\right )^{3/2}} \, dx}{\sqrt{a x^j+b x^n}}\\ &=\frac{\left (x^{-m+\frac{n}{2}} (c x)^m \sqrt{1+\frac{a x^{j-n}}{b}}\right ) \int \frac{x^{m-\frac{3 n}{2}}}{\left (1+\frac{a x^{j-n}}{b}\right )^{3/2}} \, dx}{b \sqrt{a x^j+b x^n}}\\ &=\frac{2 x^{1-n} (c x)^m \sqrt{1+\frac{a x^{j-n}}{b}} \, _2F_1\left (\frac{3}{2},\frac{1+m-\frac{3 n}{2}}{j-n};1+\frac{1+m-\frac{3 n}{2}}{j-n};-\frac{a x^{j-n}}{b}\right )}{b (2+2 m-3 n) \sqrt{a x^j+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.128102, size = 116, normalized size = 1.05 \[ \frac{2 x^{1-j} (c x)^m \left (\sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{-2 j+2 m-n+2}{2 j-2 n};\frac{2 m-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )-1\right )}{a (j-n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.372, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{m} \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 )^{m}}{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m}}{\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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m}}{{\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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