Optimal. Leaf size=100 \[ \frac{2 x (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{m+\frac{n}{2}+1}{j-n};\frac{2 m+n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(2 m+n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]
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Rubi [A] time = 0.094578, antiderivative size = 100, 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} \[ \frac{2 x (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{m+\frac{n}{2}+1}{j-n};\frac{2 m+n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(2 m+n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]
Antiderivative was successfully verified.
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Rule 2032
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (c x)^m \sqrt{a x^j+b x^n} \, dx &=\frac{\left (x^{-m-\frac{n}{2}} (c x)^m \sqrt{a x^j+b x^n}\right ) \int x^{m+\frac{n}{2}} \sqrt{b+a x^{j-n}} \, dx}{\sqrt{b+a x^{j-n}}}\\ &=\frac{\left (x^{-m-\frac{n}{2}} (c x)^m \sqrt{a x^j+b x^n}\right ) \int x^{m+\frac{n}{2}} \sqrt{1+\frac{a x^{j-n}}{b}} \, dx}{\sqrt{1+\frac{a x^{j-n}}{b}}}\\ &=\frac{2 x (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1+m+\frac{n}{2}}{j-n};1+\frac{2+2 m+n}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )}{(2+2 m+n) \sqrt{1+\frac{a x^{j-n}}{b}}}\\ \end{align*}
Mathematica [A] time = 0.191334, size = 156, normalized size = 1.56 \[ \frac{2 x (c x)^m \left ((2 j+2 m-n+2) \left (a x^j+b x^n\right )-a (j-n) x^j \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2 j+2 m-n+2}{2 j-2 n};\frac{4 j+2 m-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )\right )}{(2 m+n+2) (2 j+2 m-n+2) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.512, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m}\sqrt{a{x}^{j}+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{a x^{j} + b x^{n}} \left (c 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 (c x\right )^{m} \sqrt{a x^{j} + 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{a x^{j} + b x^{n}} \left (c x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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