Optimal. Leaf size=55 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}+\frac{e \sqrt{b x+c x^2}}{c} \]
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Rubi [A] time = 0.0217423, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {640, 620, 206} \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}+\frac{e \sqrt{b x+c x^2}}{c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x}{\sqrt{b x+c x^2}} \, dx &=\frac{e \sqrt{b x+c x^2}}{c}+\frac{(2 c d-b e) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2 c}\\ &=\frac{e \sqrt{b x+c x^2}}{c}+\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{c}\\ &=\frac{e \sqrt{b x+c x^2}}{c}+\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0617841, size = 80, normalized size = 1.45 \[ \frac{\sqrt{c} e x (b+c x)-\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} (b e-2 c d) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 78, normalized size = 1.4 \begin{align*}{\frac{e}{c}\sqrt{c{x}^{2}+bx}}-{\frac{be}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{d\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 = 1.8986, size = 274, normalized size = 4.98 \begin{align*} \left [\frac{2 \, \sqrt{c x^{2} + b x} c e -{\left (2 \, c d - b e\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right )}{2 \, c^{2}}, \frac{\sqrt{c x^{2} + b x} c e -{\left (2 \, c d - b e\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{c^{2}}\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{d + e x}{\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 = 1.36431, size = 85, normalized size = 1.55 \begin{align*} \frac{\sqrt{c x^{2} + b x} e}{c} - \frac{{\left (2 \, c d - b e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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