Optimal. Leaf size=134 \[ \frac{(a+b x) (A b-a B) (b d-a e) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e (a+b x) (A+B x)^2}{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x) (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.078713, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 77} \[ \frac{(a+b x) (A b-a B) (b d-a e) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e (a+b x) (A+B x)^2}{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x) (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
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) (d+e x)}{a b+b^2 x} \, 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)}{b^3}+\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{e (A+B x)}{b^2}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{B (b d-a e) x (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e (a+b x) (A+B x)^2}{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (b d-a e) (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0469599, size = 72, normalized size = 0.54 \[ \frac{(a+b x) (b x (b (2 A e+2 B d+B e x)-2 a B e)+2 (A b-a B) (b d-a e) \log (a+b x))}{2 b^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 104, normalized size = 0.8 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( -B{x}^{2}{b}^{2}e+2\,A\ln \left ( bx+a \right ) abe-2\,A\ln \left ( bx+a \right ){b}^{2}d-2\,Ax{b}^{2}e-2\,B\ln \left ( bx+a \right ){a}^{2}e+2\,B\ln \left ( bx+a \right ) abd+2\,Bxabe-2\,Bx{b}^{2}d \right ) }{2\,{b}^{3}}{\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 [A] time = 0.965489, size = 159, normalized size = 1.19 \begin{align*} \frac{B a^{2} b^{2} e \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{B a b e x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{B e x^{2}}{2 \, \sqrt{b^{2}}} + A \sqrt{\frac{1}{b^{2}}} d \log \left (x + \frac{a}{b}\right ) - \frac{{\left (B d + A e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}{\left (B d + A e\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36369, size = 157, normalized size = 1.17 \begin{align*} \frac{B b^{2} e x^{2} + 2 \,{\left (B b^{2} d -{\left (B a b - A b^{2}\right )} e\right )} x - 2 \,{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.474171, size = 53, normalized size = 0.4 \begin{align*} \frac{B e x^{2}}{2 b} - \frac{x \left (- A b e + B a e - B b d\right )}{b^{2}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09746, size = 165, normalized size = 1.23 \begin{align*} \frac{B b x^{2} e \mathrm{sgn}\left (b x + a\right ) + 2 \, B b d x \mathrm{sgn}\left (b x + a\right ) - 2 \, B a x e \mathrm{sgn}\left (b x + a\right ) + 2 \, A b x e \mathrm{sgn}\left (b x + a\right )}{2 \, b^{2}} - \frac{{\left (B a b d \mathrm{sgn}\left (b x + a\right ) - A b^{2} d \mathrm{sgn}\left (b x + a\right ) - B a^{2} e \mathrm{sgn}\left (b x + a\right ) + A a b e \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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