Optimal. Leaf size=121 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \sqrt{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.0767999, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {140, 139, 138} \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \sqrt{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x} (e+f x)^n}{\sqrt{a+b x}} \, dx &=\frac{\sqrt{c+d x} \int \frac{\sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} (e+f x)^n}{\sqrt{a+b x}} \, dx}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\\ &=\frac{\left (\sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n}\right ) \int \frac{\sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^n}{\sqrt{a+b x}} \, dx}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\\ &=\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \sqrt{\frac{b (c+d x)}{b c-a d}}}\\ \end{align*}
Mathematica [A] time = 0.0786572, size = 119, normalized size = 0.98 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b \sqrt{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx+e \right ) ^{n}\sqrt{dx+c}{\frac{1}{\sqrt{bx+a}}}}\, 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{\sqrt{d x + c}{\left (f x + e\right )}^{n}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x + c}{\left (f x + e\right )}^{n}}{\sqrt{b x + a}}, x\right ) \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{\sqrt{d x + c}{\left (f x + e\right )}^{n}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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