Optimal. Leaf size=196 \[ -\frac{A b-a B}{2 a^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A b-a B}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\log (x) (a+b x) (3 A b-a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 A b-a B) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.135734, antiderivative size = 196, 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} \[ -\frac{A b-a B}{2 a^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A b-a B}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\log (x) (a+b x) (3 A b-a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 A b-a B) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^2 \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^2 \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^2}+\frac{-3 A b+a B}{a^4 b^3 x}+\frac{A b-a B}{a^2 b^2 (a+b x)^3}+\frac{2 A b-a B}{a^3 b^2 (a+b x)^2}+\frac{3 A b-a B}{a^4 b^2 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A b-a B}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{2 a^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(3 A b-a B) (a+b x) \log (x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-a B) (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.063217, size = 110, normalized size = 0.56 \[ \frac{a \left (a^2 (3 B x-2 A)+a b x (2 B x-9 A)-6 A b^2 x^2\right )+2 x \log (x) (a+b x)^2 (a B-3 A b)+2 x (a+b x)^2 (3 A b-a B) \log (a+b x)}{2 a^4 x (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 221, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 6\,A\ln \left ( x \right ){x}^{3}{b}^{3}-6\,A\ln \left ( bx+a \right ){x}^{3}{b}^{3}-2\,B\ln \left ( x \right ){x}^{3}a{b}^{2}+2\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{2}+12\,A\ln \left ( x \right ){x}^{2}a{b}^{2}-12\,A\ln \left ( bx+a \right ){x}^{2}a{b}^{2}-4\,B\ln \left ( x \right ){x}^{2}{a}^{2}b+4\,B\ln \left ( bx+a \right ){x}^{2}{a}^{2}b+6\,A\ln \left ( x \right ) x{a}^{2}b-6\,A\ln \left ( bx+a \right ) x{a}^{2}b+6\,A{x}^{2}a{b}^{2}-2\,B\ln \left ( x \right ) x{a}^{3}+2\,B\ln \left ( bx+a \right ) x{a}^{3}-2\,B{x}^{2}{a}^{2}b+9\,A{a}^{2}bx-3\,{a}^{3}Bx+2\,A{a}^{3} \right ) \left ( bx+a \right ) }{2\,x{a}^{4}} \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 = 1.85385, size = 400, normalized size = 2.04 \begin{align*} -\frac{2 \, A a^{3} - 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\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^{2} \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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