Optimal. Leaf size=96 \[ \frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]
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Rubi [A] time = 0.034538, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {646, 36, 31} \[ \frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx &=\frac{\left (\frac{b}{2}+c x\right ) \int \frac{1}{\left (\frac{b}{2}+c x\right ) (d+e x)} \, dx}{\sqrt{\frac{b^2}{4 c}+b x+c x^2}}\\ &=\frac{\left (2 c \left (\frac{b}{2}+c x\right )\right ) \int \frac{1}{\frac{b}{2}+c x} \, dx}{(2 c d-b e) \sqrt{\frac{b^2}{4 c}+b x+c x^2}}-\frac{\left (2 e \left (\frac{b}{2}+c x\right )\right ) \int \frac{1}{d+e x} \, dx}{(2 c d-b e) \sqrt{\frac{b^2}{4 c}+b x+c x^2}}\\ &=\frac{2 (b+2 c x) \log (b+2 c x)}{(2 c d-b e) \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}-\frac{2 (b+2 c x) \log (d+e x)}{(2 c d-b e) \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0317777, size = 51, normalized size = 0.53 \[ \frac{2 (b+2 c x) (\log (b+2 c x)-\log (d+e x))}{\sqrt{\frac{(b+2 c x)^2}{c}} (2 c d-b e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.187, size = 58, normalized size = 0.6 \begin{align*} 2\,{\frac{ \left ( 2\,cx+b \right ) \left ( \ln \left ( ex+d \right ) -\ln \left ( 2\,cx+b \right ) \right ) }{be-2\,cd}{\frac{1}{\sqrt{{\frac{4\,{c}^{2}{x}^{2}+4\,bcx+{b}^{2}}{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 = 2.18694, size = 626, normalized size = 6.52 \begin{align*} \left [-\frac{2 \, \sqrt{c} \log \left (\frac{16 \, c^{3} e^{2} x^{3} + 4 \, b c^{2} d^{2} + b^{3} e^{2} + 16 \,{\left (c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (4 \, c^{3} d^{2} + 4 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x +{\left (4 \, c^{2} d^{2} - b^{2} e^{2} + 4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{c} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{3} + b^{2} d + 4 \,{\left (c^{2} d + b c e\right )} x^{2} +{\left (4 \, b c d + b^{2} e\right )} x}\right )}{2 \, c d - b e}, -\frac{4 \, \sqrt{-c} \arctan \left (-\frac{{\left (4 \, c e x + 2 \, c d + b e\right )} \sqrt{-c} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{2 \, b c d - b^{2} e + 2 \,{\left (2 \, c^{2} d - b c e\right )} x}\right )}{2 \, c d - b e}\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*} 2 \int \frac{1}{d \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} + e x \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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