Optimal. Leaf size=144 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x) (-a B e-A b e+2 b B d)}{e^3 (a+b x)}+\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]
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Rubi [A] time = 0.100658, antiderivative size = 144, 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} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x) (-a B e-A b e+2 b B d)}{e^3 (a+b x)}+\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]
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)^2} \, 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)^2} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{b^2 B}{e^2}-\frac{b (b d-a e) (-B d+A e)}{e^2 (d+e x)^2}+\frac{b (-2 b B d+A b e+a B e)}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}-\frac{(b d-a e) (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}-\frac{(2 b B d-A b e-a B 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 = 0.0497593, size = 96, normalized size = 0.67 \[ \frac{\sqrt{(a+b x)^2} \left (-(d+e x) \log (d+e x) (-a B e-A b e+2 b B d)+a e (B d-A e)+b \left (A d e-B d^2+B d e x+B e^2 x^2\right )\right )}{e^3 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 158, normalized size = 1.1 \begin{align*}{\frac{{\it csgn} \left ( bx+a \right ) \left ( A\ln \left ( bex+bd \right ) xb{e}^{2}+B\ln \left ( bex+bd \right ) xa{e}^{2}-2\,B\ln \left ( bex+bd \right ) xbde+B{x}^{2}b{e}^{2}+A\ln \left ( bex+bd \right ) bde+B\ln \left ( bex+bd \right ) ade-2\,B\ln \left ( bex+bd \right ) b{d}^{2}+aB{e}^{2}x+Bxbde-aA{e}^{2}+Abde+2\,aBde-Bb{d}^{2} \right ) }{{e}^{3} \left ( ex+d \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 = 1.65317, size = 216, normalized size = 1.5 \begin{align*} \frac{B b e^{2} x^{2} + B b d e x - B b d^{2} - A a e^{2} +{\left (B a + A b\right )} d e -{\left (2 \, B b d^{2} -{\left (B a + A b\right )} d e +{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.691983, size = 71, normalized size = 0.49 \begin{align*} \frac{B b x}{e^{2}} + \frac{- A a e^{2} + A b d e + B a d e - B b d^{2}}{d e^{3} + e^{4} x} + \frac{\left (A b e + B a e - 2 B 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 = 1.12119, size = 166, normalized size = 1.15 \begin{align*} B b x e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) -{\left (2 \, B b d \mathrm{sgn}\left (b x + a\right ) - B a e \mathrm{sgn}\left (b x + a\right ) - A b e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\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 )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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