Optimal. Leaf size=151 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) (d+e x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]
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Rubi [A] time = 0.097895, antiderivative size = 151, 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} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) (d+e x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]
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)^3} \, 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)^3} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e) (-B d+A e)}{e^2 (d+e x)^3}+\frac{b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^2}+\frac{b^2 B}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e) (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac{(2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}+\frac{b B \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 = 0.0507265, size = 89, normalized size = 0.59 \[ -\frac{\sqrt{(a+b x)^2} \left (a e (A e+B (d+2 e x))+b (A e (d+2 e x)-B d (3 d+4 e x))-2 b B (d+e x)^2 \log (d+e x)\right )}{2 e^3 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 117, normalized size = 0.8 \begin{align*} -{\frac{{\it csgn} \left ( bx+a \right ) \left ( -2\,B\ln \left ( bex+bd \right ){x}^{2}b{e}^{2}-4\,B\ln \left ( bex+bd \right ) xbde+2\,Axb{e}^{2}-2\,B\ln \left ( bex+bd \right ) b{d}^{2}+2\,aB{e}^{2}x-4\,Bxbde+aA{e}^{2}+Abde+aBde-3\,Bb{d}^{2} \right ) }{2\,{e}^{3} \left ( ex+d \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 = 1.60622, size = 227, normalized size = 1.5 \begin{align*} \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b e^{2} x^{2} + 2 \, B b d e x + B b d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14237, size = 94, normalized size = 0.62 \begin{align*} \frac{B b \log{\left (d + e x \right )}}{e^{3}} - \frac{A a e^{2} + A b d e + B a d e - 3 B b d^{2} + x \left (2 A b e^{2} + 2 B a e^{2} - 4 B b d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10246, size = 171, normalized size = 1.13 \begin{align*} B b e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (2 \,{\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 )} x +{\left (3 \, 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 )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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