Optimal. Leaf size=148 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{(d+e x) (b d-a e)^2}+\frac{(a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rubi [A] time = 0.0808597, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {769, 646, 36, 31} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{(d+e x) (b d-a e)^2}+\frac{(a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 769
Rule 646
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{(B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{(b d-a e)^2 (d+e x)}+\frac{(A b-a B) \int \frac{1}{(d+e x) \sqrt{a^2+2 a b x+b^2 x^2}} \, dx}{b d-a e}\\ &=\frac{(B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{(b d-a e)^2 (d+e x)}+\frac{\left ((A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)} \, dx}{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{(b d-a e)^2 (d+e x)}+\frac{\left (b (A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{a b+b^2 x} \, dx}{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left ((A b-a B) e \left (a b+b^2 x\right )\right ) \int \frac{1}{d+e x} \, dx}{b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{(b d-a e)^2 (d+e x)}+\frac{(A b-a 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{(A b-a 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 = 0.128731, size = 97, normalized size = 0.66 \[ \frac{(a+b x) \left (\frac{B d-A e}{e (d+e x) (a e-b d)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}+\frac{(a B-A b) \log (d+e x)}{(b d-a e)^2}\right )}{\sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 162, normalized size = 1.1 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( A\ln \left ( ex+d \right ) xb{e}^{2}-A\ln \left ( bx+a \right ) xb{e}^{2}-B\ln \left ( ex+d \right ) xa{e}^{2}+B\ln \left ( bx+a \right ) xa{e}^{2}+A\ln \left ( ex+d \right ) bde-A\ln \left ( bx+a \right ) bde-B\ln \left ( ex+d \right ) ade+B\ln \left ( bx+a \right ) ade+aA{e}^{2}-Abde-aBde+Bb{d}^{2} \right ) }{ \left ( ae-bd \right ) ^{2}e \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 = 1.26387, size = 308, normalized size = 2.08 \begin{align*} -\frac{B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e +{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \log \left (b x + a\right ) -{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \log \left (e x + d\right )}{b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.54498, size = 355, normalized size = 2.4 \begin{align*} \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b e - A b^{2} d + B a^{2} e + B a b d - \frac{a^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b d e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{3 a b^{2} d^{2} e \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{b^{3} d^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}}}{- 2 A b^{2} e + 2 B a b e} \right )}}{\left (a e - b d\right )^{2}} - \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b e - A b^{2} d + B a^{2} e + B a b d + \frac{a^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b d e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{3 a b^{2} d^{2} e \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{b^{3} d^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}}}{- 2 A b^{2} e + 2 B a b e} \right )}}{\left (a e - b d\right )^{2}} + \frac{- A e + B d}{a d e^{2} - b d^{2} e + x \left (a e^{3} - b d e^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11963, size = 257, normalized size = 1.74 \begin{align*} -\frac{{\left (B a b \mathrm{sgn}\left (b x + a\right ) - A b^{2} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} + \frac{{\left (B a e \mathrm{sgn}\left (b x + a\right ) - A b e \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} - \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 (-1\right )}}{{\left (b d - a e\right )}^{2}{\left (x e + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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