Optimal. Leaf size=305 \[ \frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{(m+3 n+1) \sqrt{a+b x^n}} \]
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Rubi [A] time = 0.231849, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1844, 365, 364, 20} \[ \frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{(m+3 n+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 1844
Rule 365
Rule 364
Rule 20
Rubi steps
\begin{align*} \int \frac{(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt{a+b x^n}} \, dx &=\int \left (\frac{d (c x)^m}{\sqrt{a+b x^n}}+\frac{e x^n (c x)^m}{\sqrt{a+b x^n}}+\frac{f x^{2 n} (c x)^m}{\sqrt{a+b x^n}}+\frac{g x^{3 n} (c x)^m}{\sqrt{a+b x^n}}\right ) \, dx\\ &=d \int \frac{(c x)^m}{\sqrt{a+b x^n}} \, dx+e \int \frac{x^n (c x)^m}{\sqrt{a+b x^n}} \, dx+f \int \frac{x^{2 n} (c x)^m}{\sqrt{a+b x^n}} \, dx+g \int \frac{x^{3 n} (c x)^m}{\sqrt{a+b x^n}} \, dx\\ &=\left (e x^{-m} (c x)^m\right ) \int \frac{x^{m+n}}{\sqrt{a+b x^n}} \, dx+\left (f x^{-m} (c x)^m\right ) \int \frac{x^{m+2 n}}{\sqrt{a+b x^n}} \, dx+\left (g x^{-m} (c x)^m\right ) \int \frac{x^{m+3 n}}{\sqrt{a+b x^n}} \, dx+\frac{\left (d \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{(c x)^m}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{d (c x)^{1+m} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{c (1+m) \sqrt{a+b x^n}}+\frac{\left (e x^{-m} (c x)^m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{m+n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}+\frac{\left (f x^{-m} (c x)^m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{m+2 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}+\frac{\left (g x^{-m} (c x)^m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{m+3 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{d (c x)^{1+m} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{c (1+m) \sqrt{a+b x^n}}+\frac{e x^{1+n} (c x)^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+n}{n};\frac{1+m+2 n}{n};-\frac{b x^n}{a}\right )}{(1+m+n) \sqrt{a+b x^n}}+\frac{f x^{1+2 n} (c x)^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+2 n}{n};\frac{1+m+3 n}{n};-\frac{b x^n}{a}\right )}{(1+m+2 n) \sqrt{a+b x^n}}+\frac{g x^{1+3 n} (c x)^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+3 n}{n};\frac{1+m+4 n}{n};-\frac{b x^n}{a}\right )}{(1+m+3 n) \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.373558, size = 206, normalized size = 0.68 \[ \frac{x (c x)^m \sqrt{\frac{b x^n}{a}+1} \left (\frac{d \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}+x^n \left (\frac{e \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+x^n \left (\frac{f \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}+\frac{g x^n \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{m+3 n+1}\right )\right )\right )}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.464, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{m} \left ( d+e{x}^{n}+f{x}^{2\,n}+g{x}^{3\,n} \right ){\frac{1}{\sqrt{a+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{{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt{b x^{n} + a}}\,{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 (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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