Optimal. Leaf size=101 \[ \frac{2 e \sqrt{b x+c x^2} (2 c d-b e)}{b^2 c}-\frac{2 (d+e x) (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}} \]
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Rubi [A] time = 0.061544, antiderivative size = 101, 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 = {738, 640, 620, 206} \[ \frac{2 e \sqrt{b x+c x^2} (2 c d-b e)}{b^2 c}-\frac{2 (d+e x) (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 738
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x) (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}-\frac{2 \int \frac{-b d e-e (2 c d-b e) x}{\sqrt{b x+c x^2}} \, dx}{b^2}\\ &=-\frac{2 (d+e x) (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e (2 c d-b e) \sqrt{b x+c x^2}}{b^2 c}+\frac{e^2 \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{c}\\ &=-\frac{2 (d+e x) (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e (2 c d-b e) \sqrt{b x+c x^2}}{b^2 c}+\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{c}\\ &=-\frac{2 (d+e x) (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e (2 c d-b e) \sqrt{b x+c x^2}}{b^2 c}+\frac{2 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0960489, size = 100, normalized size = 0.99 \[ \frac{2 b^{5/2} e^2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )-2 \sqrt{c} \left (b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )}{b^2 c^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 97, normalized size = 1. \begin{align*} -2\,{\frac{{e}^{2}x}{c\sqrt{c{x}^{2}+bx}}}+{{e}^{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+4\,{\frac{dex}{b\sqrt{c{x}^{2}+bx}}}-2\,{\frac{{d}^{2} \left ( 2\,cx+b \right ) }{{b}^{2}\sqrt{c{x}^{2}+bx}}} \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 = 2.05542, size = 505, normalized size = 5. \begin{align*} \left [\frac{{\left (b^{2} c e^{2} x^{2} + b^{3} e^{2} x\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (b c^{2} d^{2} +{\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e + b^{2} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{b^{2} c^{3} x^{2} + b^{3} c^{2} x}, -\frac{2 \,{\left ({\left (b^{2} c e^{2} x^{2} + b^{3} e^{2} x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (b c^{2} d^{2} +{\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e + b^{2} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}\right )}}{b^{2} c^{3} x^{2} + b^{3} c^{2} 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 )^{2}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26464, size = 120, normalized size = 1.19 \begin{align*} -\frac{2 \,{\left (\frac{d^{2}}{b} + \frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x}{b^{2} c}\right )}}{\sqrt{c x^{2} + b x}} - \frac{e^{2} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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