Optimal. Leaf size=131 \[ \frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac{b (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{b (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rubi [A] time = 0.0620819, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 44} \[ \frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac{b (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{b (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^2} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{b}{(b d-a e)^2 (a+b x)}-\frac{e}{b (b d-a e) (d+e x)^2}-\frac{e}{(b d-a e)^2 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{a+b x}{(b d-a e) (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x) \log (a+b x)}{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) \log (d+e x)}{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0439544, size = 69, normalized size = 0.53 \[ \frac{(a+b x) (b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d)}{\sqrt{(a+b x)^2} (d+e x) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.162, size = 82, normalized size = 0.6 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( \ln \left ( ex+d \right ) xbe-\ln \left ( bx+a \right ) xbe+\ln \left ( ex+d \right ) bd-\ln \left ( bx+a \right ) bd+ae-bd \right ) }{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{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 = 1.59596, size = 198, normalized size = 1.51 \begin{align*} \frac{b d - a e +{\left (b e x + b d\right )} \log \left (b x + a\right ) -{\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} +{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.95312, size = 233, normalized size = 1.78 \begin{align*} - \frac{b \log{\left (x + \frac{- \frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} - \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e + \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} + \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e - \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} - \frac{1}{a d e - b d^{2} + x \left (a e^{2} - b d e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19021, size = 139, normalized size = 1.06 \begin{align*}{\left (\frac{b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} - \frac{b e \log \left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac{1}{{\left (b d - a e\right )}{\left (x e + d\right )}}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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