Optimal. Leaf size=105 \[ \frac{\sqrt{-\frac{e x}{d}} (d+e x)^{m+1} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0510385, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {759, 133} \[ \frac{\sqrt{-\frac{e x}{d}} (d+e x)^{m+1} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 759
Rule 133
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\sqrt{b x+c x^2}} \, dx &=\frac{\left (\sqrt{1-\frac{d+e x}{d}} \sqrt{1-\frac{d+e x}{d-\frac{b e}{c}}}\right ) \operatorname{Subst}\left (\int \frac{x^m}{\sqrt{1-\frac{x}{d}} \sqrt{1-\frac{c x}{c d-b e}}} \, dx,x,d+e x\right )}{e \sqrt{b x+c x^2}}\\ &=\frac{\sqrt{-\frac{e x}{d}} (d+e x)^{1+m} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left (1+m;\frac{1}{2},\frac{1}{2};2+m;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (1+m) \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0402688, size = 74, normalized size = 0.7 \[ \frac{2 x \sqrt{\frac{b+c x}{b}} (d+e x)^m \left (\frac{d+e x}{d}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{\sqrt{x (b+c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.613, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}\, 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 (e x + d\right )}^{m}}{\sqrt{c x^{2} + b x}}\,{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{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + b 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 (d + e x\right )^{m}}{\sqrt{x \left (b + c x\right )}}\, 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 (e x + d\right )}^{m}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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