Optimal. Leaf size=127 \[ -\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{-2 a B e+A b e+b B d}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0988835, antiderivative size = 127, 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} \[ -\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{-2 a B e+A b e+b B d}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{(A+B x) (d+e x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac{(A b-a B) (b d-a e)}{b^5 (a+b x)^3}+\frac{b B d+A b e-2 a B e}{b^5 (a+b x)^2}+\frac{B e}{b^5 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{b B d+A b e-2 a B e}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B 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 = 0.0483492, size = 86, normalized size = 0.68 \[ \frac{B \left (3 a^2 e-a b d+4 a b e x-2 b^2 d x\right )-A b (a e+b d+2 b e x)+2 B e (a+b x)^2 \log (a+b x)}{2 b^3 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 109, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -2\,B\ln \left ( bx+a \right ){x}^{2}{b}^{2}e-4\,B\ln \left ( bx+a \right ) xabe+2\,Ax{b}^{2}e-2\,B\ln \left ( bx+a \right ){a}^{2}e-4\,Bxabe+2\,Bx{b}^{2}d+aAeb+Ad{b}^{2}-3\,Be{a}^{2}+Bdab \right ) \left ( bx+a \right ) }{2\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995631, size = 185, normalized size = 1.46 \begin{align*} \frac{B e \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, B a^{2} b^{2} e}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, B a b e x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{B d + A e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{A d}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{{\left (B d + A e\right )} a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29137, size = 234, normalized size = 1.84 \begin{align*} -\frac{{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} - A a b\right )} e + 2 \,{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x - 2 \,{\left (B b^{2} e x^{2} + 2 \, B a b e x + B a^{2} e\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (d + e x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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