Optimal. Leaf size=140 \[ \frac{A b-a B}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0885879, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {770, 77} \[ \frac{A b-a B}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{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}{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}{a^3 b^3 x}+\frac{-A b+a B}{a b^3 (a+b x)^3}-\frac{A}{a^2 b^2 (a+b x)^2}-\frac{A}{a^3 b^2 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{A}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A (a+b x) \log (x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0362897, size = 80, normalized size = 0.57 \[ \frac{a \left (a^2 (-B)+3 a A b+2 A b^2 x\right )+2 A b \log (x) (a+b x)^2-2 A b (a+b x)^2 \log (a+b x)}{2 a^3 b (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 117, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,A\ln \left ( x \right ){x}^{2}{b}^{3}-2\,A\ln \left ( bx+a \right ){x}^{2}{b}^{3}+4\,A\ln \left ( x \right ) xa{b}^{2}-4\,A\ln \left ( bx+a \right ) xa{b}^{2}+2\,A\ln \left ( x \right ){a}^{2}b-2\,A\ln \left ( bx+a \right ){a}^{2}b+2\,Aa{b}^{2}x+3\,Ab{a}^{2}-B{a}^{3} \right ) \left ( bx+a \right ) }{2\,b{a}^{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 [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 = 1.52404, size = 236, normalized size = 1.69 \begin{align*} \frac{2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\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{A + B x}{x \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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