Optimal. Leaf size=154 \[ -\frac{a^2 (A b-a B)}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-3 a B) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.105934, antiderivative size = 154, 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} \[ -\frac{a^2 (A b-a B)}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-3 a B) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{x^2 (A+B 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{x^2 (A+B 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{B}{b^6}-\frac{a^2 (-A b+a B)}{b^6 (a+b x)^3}+\frac{a (-2 A b+3 a B)}{b^6 (a+b x)^2}+\frac{A b-3 a B}{b^6 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{a (2 A b-3 a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^2 (A b-a B)}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-3 a B) (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0418458, size = 89, normalized size = 0.58 \[ \frac{a^2 b (3 A-4 B x)-5 a^3 B+4 a b^2 x (A+B x)+2 (a+b x)^2 (A b-3 a B) \log (a+b x)+2 b^3 B x^3}{2 b^4 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 153, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,A\ln \left ( bx+a \right ){x}^{2}{b}^{3}-6\,B\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+2\,{b}^{3}B{x}^{3}+4\,A\ln \left ( bx+a \right ) xa{b}^{2}-12\,B\ln \left ( bx+a \right ) x{a}^{2}b+4\,B{x}^{2}a{b}^{2}+2\,A\ln \left ( bx+a \right ){a}^{2}b+4\,Aa{b}^{2}x-6\,B\ln \left ( bx+a \right ){a}^{3}-4\,B{a}^{2}bx+3\,Ab{a}^{2}-5\,B{a}^{3} \right ) \left ( bx+a \right ) }{2\,{b}^{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 [A] time = 0.992126, size = 266, normalized size = 1.73 \begin{align*} \frac{B x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{A \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{3 \, B a \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} - \frac{9 \, B a^{3} b}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{3 \, A a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, B a^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, A a b x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, B a^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{B a^{3}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}{\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 = 1.55436, size = 278, normalized size = 1.81 \begin{align*} \frac{2 \, B b^{3} x^{3} + 4 \, B a b^{2} x^{2} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \,{\left (B a^{2} b - A a b^{2}\right )} x - 2 \,{\left (3 \, B a^{3} - A a^{2} b +{\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\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{x^{2} \left (A + B 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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