Optimal. Leaf size=113 \[ -\frac{A b-2 a B}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0735174, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {770, 77} \[ -\frac{A b-2 a B}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{x (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 (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{a (-A b+a B)}{b^5 (a+b x)^3}+\frac{A b-2 a B}{b^5 (a+b x)^2}+\frac{B}{b^5 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-2 a B}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (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 = 0.0278377, size = 65, normalized size = 0.58 \[ \frac{3 a^2 B-a b (A-4 B x)+2 B (a+b x)^2 \log (a+b x)-2 A b^2 x}{2 b^3 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 83, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2\,B\ln \left ( bx+a \right ){x}^{2}{b}^{2}-4\,B\ln \left ( bx+a \right ) xab+2\,A{b}^{2}x-2\,B\ln \left ( bx+a \right ){a}^{2}-4\,abBx+Aab-3\,B{a}^{2} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.999024, size = 140, normalized size = 1.24 \begin{align*} \frac{B \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, B a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, B a b x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{A}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{A 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58908, size = 173, normalized size = 1.53 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x + 2 \,{\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]