Optimal. Leaf size=105 \[ \frac{\sqrt{b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}} \]
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Rubi [A] time = 0.0493167, 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} \[ \frac{\sqrt{b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}} \]
Antiderivative was successfully verified.
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Rule 759
Rule 133
Rubi steps
\begin{align*} \int (d+e x)^m \sqrt{b x+c x^2} \, dx &=\frac{\sqrt{b x+c x^2} \operatorname{Subst}\left (\int x^m \sqrt{1-\frac{x}{d}} \sqrt{1-\frac{c x}{c d-b e}} \, dx,x,d+e x\right )}{e \sqrt{1-\frac{d+e x}{d}} \sqrt{1-\frac{d+e x}{d-\frac{b e}{c}}}}\\ &=\frac{(d+e x)^{1+m} \sqrt{b x+c x^2} 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{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}}\\ \end{align*}
Mathematica [A] time = 0.0352619, size = 76, normalized size = 0.72 \[ \frac{2 x \sqrt{x (b+c x)} (d+e x)^m \left (\frac{d+e x}{d}\right )^{-m} F_1\left (\frac{3}{2};-\frac{1}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{3 \sqrt{\frac{b+c x}{b}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.614, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m}\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 \sqrt{c x^{2} + b x}{\left (e x + d\right )}^{m}\,{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 (\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{m}, 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 \sqrt{x \left (b + c x\right )} \left (d + e x\right )^{m}\, 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{c x^{2} + b x}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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