Optimal. Leaf size=110 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c} \]
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Rubi [A] time = 0.0773418, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {742, 640, 620, 206} \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\sqrt{b x+c x^2}} \, dx &=\frac{e (d+e x) \sqrt{b x+c x^2}}{2 c}+\frac{\int \frac{\frac{1}{2} d (4 c d-b e)+\frac{3}{2} e (2 c d-b e) x}{\sqrt{b x+c x^2}} \, dx}{2 c}\\ &=\frac{3 e (2 c d-b e) \sqrt{b x+c x^2}}{4 c^2}+\frac{e (d+e x) \sqrt{b x+c x^2}}{2 c}+\frac{\left (-\frac{3}{2} b e (2 c d-b e)+c d (4 c d-b e)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{4 c^2}\\ &=\frac{3 e (2 c d-b e) \sqrt{b x+c x^2}}{4 c^2}+\frac{e (d+e x) \sqrt{b x+c x^2}}{2 c}+\frac{\left (-\frac{3}{2} b e (2 c d-b e)+c d (4 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 )}{2 c^2}\\ &=\frac{3 e (2 c d-b e) \sqrt{b x+c x^2}}{4 c^2}+\frac{e (d+e x) \sqrt{b x+c x^2}}{2 c}+\frac{\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.105496, size = 111, normalized size = 1.01 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )+\sqrt{c} e x (b+c x) (-3 b e+8 c d+2 c e x)}{4 c^{5/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 158, normalized size = 1.4 \begin{align*}{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{e}^{2}b}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}{e}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+bx}}{c}}-{bde\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{{d}^{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \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 = 2.11732, size = 432, normalized size = 3.93 \begin{align*} \left [\frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (2 \, c^{2} e^{2} x + 8 \, c^{2} d e - 3 \, b c e^{2}\right )} \sqrt{c x^{2} + b x}}{8 \, c^{3}}, -\frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (2 \, c^{2} e^{2} x + 8 \, c^{2} d e - 3 \, b c e^{2}\right )} \sqrt{c x^{2} + b x}}{4 \, 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 \frac{\left (d + e x\right )^{2}}{\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 [A] time = 1.44893, size = 131, normalized size = 1.19 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (\frac{2 \, x e^{2}}{c} + \frac{8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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