Optimal. Leaf size=47 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}} \]
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Rubi [A] time = 0.0658729, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1981, 618, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}} \]
Antiderivative was successfully verified.
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Rule 1981
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{b x+c (d+e x)^2} \, dx &=\int \frac{1}{c d^2+(b+2 c d e) x+c e^2 x^2} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{b (b+4 c d e)-x^2} \, dx,x,b+2 c d e+2 c e^2 x\right )\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}}\\ \end{align*}
Mathematica [A] time = 0.0253848, size = 47, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 43, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{\sqrt{4\,bcde+{b}^{2}}}{\it Artanh} \left ({\frac{2\,c{e}^{2}x+2\,cde+b}{\sqrt{4\,bcde+{b}^{2}}}} \right ) } \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 = 1.02551, size = 431, normalized size = 9.17 \begin{align*} \left [\frac{\log \left (\frac{2 \, c^{2} e^{4} x^{2} + 2 \, c^{2} d^{2} e^{2} + 4 \, b c d e + b^{2} + 2 \,{\left (2 \, c^{2} d e^{3} + b c e^{2}\right )} x - \sqrt{4 \, b c d e + b^{2}}{\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{c e^{2} x^{2} + c d^{2} +{\left (2 \, c d e + b\right )} x}\right )}{\sqrt{4 \, b c d e + b^{2}}}, \frac{2 \, \sqrt{-4 \, b c d e - b^{2}} \arctan \left (\frac{\sqrt{-4 \, b c d e - b^{2}}{\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{4 \, b c d e + b^{2}}\right )}{4 \, b c d e + b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.268286, size = 151, normalized size = 3.21 \begin{align*} \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} \log{\left (x + \frac{- b^{2} \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} - 4 b c d e \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} + b + 2 c d e}{2 c e^{2}} \right )} - \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} \log{\left (x + \frac{b^{2} \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} + 4 b c d e \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} + b + 2 c d e}{2 c e^{2}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15207, size = 65, normalized size = 1.38 \begin{align*} \frac{2 \, \arctan \left (\frac{2 \, c x e^{2} + 2 \, c d e + b}{\sqrt{-4 \, b c d e - b^{2}}}\right )}{\sqrt{-4 \, b c d e - b^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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