Optimal. Leaf size=128 \[ \frac{2 \sqrt{a+b x} (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{\sqrt{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.0769206, antiderivative size = 128, 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} \[ \frac{2 \sqrt{a+b x} (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{\sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{(e+f x)^n}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx &=\frac{\left (b \sqrt{\frac{b (c+d x)}{b c-a d}}\right ) \int \frac{(e+f x)^n}{\sqrt{a+b x} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{3/2}} \, dx}{(b c-a d) \sqrt{c+d x}}\\ &=\frac{\left (b \sqrt{\frac{b (c+d x)}{b c-a d}} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n}\right ) \int \frac{\left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^n}{\sqrt{a+b x} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{3/2}} \, dx}{(b c-a d) \sqrt{c+d x}}\\ &=\frac{2 \sqrt{a+b x} \sqrt{\frac{b (c+d x)}{b c-a d}} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{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 c-a d) \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.249498, size = 241, normalized size = 1.88 \[ -\frac{2 \sqrt{a+b x} (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left ((3 a d-3 b c) F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (a+b x) \left (F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+F_1\left (\frac{3}{2};\frac{3}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}{3 \sqrt{c+d x} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx+e \right ) ^{n}{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \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 (f x + e\right )}^{n}}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{2}}}\,{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{b x + a} \sqrt{d x + c}{\left (f x + e\right )}^{n}}{b d^{2} x^{3} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2} +{\left (b c^{2} + 2 \, a c d\right )} x}, x\right ) \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 (e + f x\right )^{n}}{\sqrt{a + b x} \left (c + d x\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 (f x + e\right )}^{n}}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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