Optimal. Leaf size=117 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}-\frac{e \sqrt{b x+c x^2}}{d (d+e x) (c d-b e)} \]
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Rubi [A] time = 0.0746605, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {730, 724, 206} \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}-\frac{e \sqrt{b x+c x^2}}{d (d+e x) (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 730
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx &=-\frac{e \sqrt{b x+c x^2}}{d (c d-b e) (d+e x)}+\frac{(2 c d-b e) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 d (c d-b e)}\\ &=-\frac{e \sqrt{b x+c x^2}}{d (c d-b e) (d+e x)}-\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{d (c d-b e)}\\ &=-\frac{e \sqrt{b x+c x^2}}{d (c d-b e) (d+e x)}+\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.143699, size = 122, normalized size = 1.04 \[ \frac{\sqrt{x} \left (\frac{\sqrt{d} e \sqrt{x} (b+c x)}{(d+e x) (b e-c d)}-\frac{\sqrt{b+c x} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b e-c d)^{3/2}}\right )}{d^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.257, size = 355, normalized size = 3. \begin{align*}{\frac{1}{d \left ( be-cd \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{b}{2\,d \left ( be-cd \right ) }\ln \left ({ \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+{\frac{c}{e \left ( be-cd \right ) }\ln \left ({ \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{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 = 2.04603, size = 716, normalized size = 6.12 \begin{align*} \left [\frac{{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt{c d^{2} - b d e} \log \left (\frac{b d +{\left (2 \, c d - b e\right )} x + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}{e x + d}\right ) - 2 \,{\left (c d^{2} e - b d e^{2}\right )} \sqrt{c x^{2} + b x}}{2 \,{\left (c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x\right )}}, \frac{{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt{-c d^{2} + b d e} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) -{\left (c d^{2} e - b d e^{2}\right )} \sqrt{c x^{2} + b x}}{c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} 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{1}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.00405, size = 540, normalized size = 4.62 \begin{align*} \frac{{\left (2 \, c d \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt{c d^{2} - b d e} \sqrt{c} \right |}\right ) - b e \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt{c d^{2} - b d e} \sqrt{c} \right |}\right ) + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c}\right )} \mathrm{sgn}\left (\frac{1}{x e + d}\right )}{2 \,{\left (\sqrt{c d^{2} - b d e} c d^{2} - \sqrt{c d^{2} - b d e} b d e\right )}} - \frac{\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}}}{c d^{2} \mathrm{sgn}\left (\frac{1}{x e + d}\right ) - b d e \mathrm{sgn}\left (\frac{1}{x e + d}\right )} - \frac{{\left (2 \, c d e - b e^{2}\right )} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt{c d^{2} - b d e}{\left (\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}} + \frac{\sqrt{c d^{2} e^{2} - b d e^{3}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{2 \,{\left (c d^{2} e - b d e^{2}\right )} \sqrt{c d^{2} - b d e} \mathrm{sgn}\left (\frac{1}{x e + d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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