Optimal. Leaf size=107 \[ \frac{(a+b x) (A b-a B) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{(a+b x) (B d-A e) \log (d+e x)}{e \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.0770278, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 72} \[ \frac{(a+b x) (A b-a B) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{(a+b x) (B d-A e) \log (d+e x)}{e \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 72
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x) \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{A+B x}{\left (a b+b^2 x\right ) (d+e x)} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{A b-a B}{b (b d-a e) (a+b x)}+\frac{B d-A e}{b (b d-a e) (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) (a+b x) \log (a+b x)}{b (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(B d-A e) (a+b x) \log (d+e x)}{e (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.043808, size = 66, normalized size = 0.62 \[ \frac{(a+b x) (e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x))}{b e \sqrt{(a+b x)^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.7 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( A\ln \left ( ex+d \right ) be-A\ln \left ( bx+a \right ) be-B\ln \left ( ex+d \right ) bd+B\ln \left ( bx+a \right ) ae \right ) }{e \left ( ae-bd \right ) b}{\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 = 1.30536, size = 111, normalized size = 1.04 \begin{align*} -\frac{{\left (B a - A b\right )} e \log \left (b x + a\right ) -{\left (B b d - A b e\right )} \log \left (e x + d\right )}{b^{2} d e - a b e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.47977, size = 226, normalized size = 2.11 \begin{align*} - \frac{\left (- A e + B d\right ) \log{\left (x + \frac{- A a e - A b d + 2 B a d - \frac{a^{2} e \left (- A e + B d\right )}{a e - b d} + \frac{2 a b d \left (- A e + B d\right )}{a e - b d} - \frac{b^{2} d^{2} \left (- A e + B d\right )}{e \left (a e - b d\right )}}{- 2 A b e + B a e + B b d} \right )}}{e \left (a e - b d\right )} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a e - A b d + 2 B a d + \frac{a^{2} e^{2} \left (- A b + B a\right )}{b \left (a e - b d\right )} - \frac{2 a d e \left (- A b + B a\right )}{a e - b d} + \frac{b d^{2} \left (- A b + B a\right )}{a e - b d}}{- 2 A b e + B a e + B b d} \right )}}{b \left (a e - b d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14964, size = 197, normalized size = 1.84 \begin{align*} \frac{B e^{\left (-1\right )} \log \left ({\left | b x^{2} e + b d x + a x e + a d \right |}\right ) \mathrm{sgn}\left (b x + a\right )}{2 \, b} - \frac{{\left (B b d \mathrm{sgn}\left (b x + a\right ) + B a e \mathrm{sgn}\left (b x + a\right ) - 2 \, A b e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \log \left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{2 \, b{\left | b d - a e \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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