Optimal. Leaf size=99 \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.0358537, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {640, 612, 620, 206} \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int (d+e x) \sqrt{b x+c x^2} \, dx &=\frac{e \left (b x+c x^2\right )^{3/2}}{3 c}+\frac{(2 c d-b e) \int \sqrt{b x+c x^2} \, dx}{2 c}\\ &=\frac{(2 c d-b e) (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac{\left (b^2 (2 c d-b e)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 c^2}\\ &=\frac{(2 c d-b e) (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac{\left (b^2 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 c^2}\\ &=\frac{(2 c d-b e) (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.190187, size = 108, normalized size = 1.09 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-3 b^2 e+2 b c (3 d+e x)+4 c^2 x (3 d+2 e x)\right )+\frac{3 b^{3/2} (b e-2 c d) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{24 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 157, normalized size = 1.6 \begin{align*}{\frac{e}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bxe}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{e{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{dx}{2}\sqrt{c{x}^{2}+bx}}+{\frac{bd}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}d}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48892, size = 467, normalized size = 4.72 \begin{align*} \left [-\frac{3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (8 \, c^{3} e x^{2} + 6 \, b c^{2} d - 3 \, b^{2} c e + 2 \,{\left (6 \, c^{3} d + b c^{2} e\right )} x\right )} \sqrt{c x^{2} + b x}}{48 \, c^{3}}, \frac{3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (8 \, c^{3} e x^{2} + 6 \, b c^{2} d - 3 \, b^{2} c e + 2 \,{\left (6 \, c^{3} d + b c^{2} e\right )} x\right )} \sqrt{c x^{2} + b x}}{24 \, c^{3}}\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 )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31288, size = 146, normalized size = 1.47 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, x e + \frac{6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac{3 \,{\left (2 \, b c d - b^{2} e\right )}}{c^{2}}\right )} + \frac{{\left (2 \, b^{2} c d - b^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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