Optimal. Leaf size=132 \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{e^2 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) \log (d+e x)}{e^3 (a+b x)}+\frac{B (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b e} \]
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Rubi [A] time = 0.0858667, antiderivative size = 132, 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, 77} \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{e^2 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) \log (d+e x)}{e^3 (a+b x)}+\frac{B (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b e} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{d+e x} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right ) (A+B x)}{d+e x} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{b^2 (-B d+A e)}{e^2}+\frac{B \left (a b+b^2 x\right )}{e}-\frac{b (b d-a e) (-B d+A e)}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{b (B d-A e) x \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}+\frac{B (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b e}+\frac{(b d-a e) (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0393942, size = 74, normalized size = 0.56 \[ \frac{\sqrt{(a+b x)^2} (e x (2 a B e+b (2 A e-2 B d+B e x))+2 (b d-a e) (B d-A e) \log (d+e x))}{2 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 146, normalized size = 1.1 \begin{align*}{\frac{{\it csgn} \left ( bx+a \right ) \left ( B{x}^{2}{b}^{2}{e}^{2}+2\,A\ln \left ( bex+bd \right ) ab{e}^{2}-2\,A\ln \left ( bex+bd \right ){b}^{2}de+2\,Ax{b}^{2}{e}^{2}-2\,B\ln \left ( bex+bd \right ) abde+2\,B\ln \left ( bex+bd \right ){b}^{2}{d}^{2}+2\,Bxab{e}^{2}-2\,Bx{b}^{2}de+2\,Aab{e}^{2}+{a}^{2}B{e}^{2}-2\,Bdabe \right ) }{2\,b{e}^{3}}} \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 = 1.62862, size = 151, normalized size = 1.14 \begin{align*} \frac{B b e^{2} x^{2} - 2 \,{\left (B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.513379, size = 53, normalized size = 0.4 \begin{align*} \frac{B b x^{2}}{2 e} + \frac{x \left (A b e + B a e - B b d\right )}{e^{2}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10322, size = 161, normalized size = 1.22 \begin{align*}{\left (B b d^{2} \mathrm{sgn}\left (b x + a\right ) - B a d e \mathrm{sgn}\left (b x + a\right ) - A b d e \mathrm{sgn}\left (b x + a\right ) + A a e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (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 )\right )} e^{\left (-2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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