Optimal. Leaf size=54 \[ \frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0363245, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0368101, size = 50, normalized size = 0.93 \[ \frac{\frac{2 a^3}{a+b \sqrt{x}}+6 a^2 \log \left (a+b \sqrt{x}\right )-4 a b \sqrt{x}+b^2 x}{b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 49, normalized size = 0.9 \begin{align*}{\frac{x}{{b}^{2}}}+6\,{\frac{{a}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}-4\,{\frac{a\sqrt{x}}{{b}^{3}}}+2\,{\frac{{a}^{3}}{{b}^{4} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.978247, size = 81, normalized size = 1.5 \begin{align*} \frac{6 \, a^{2} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{{\left (b \sqrt{x} + a\right )}^{2}}{b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )} a}{b^{4}} + \frac{2 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.27179, size = 167, normalized size = 3.09 \begin{align*} \frac{b^{4} x^{2} - a^{2} b^{2} x - 2 \, a^{4} + 6 \,{\left (a^{2} b^{2} x - a^{4}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (2 \, a b^{3} x - 3 \, a^{3} b\right )} \sqrt{x}}{b^{6} x - a^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.618201, size = 134, normalized size = 2.48 \begin{align*} \begin{cases} \frac{6 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{4} + b^{5} \sqrt{x}} + \frac{6 a^{3}}{a b^{4} + b^{5} \sqrt{x}} + \frac{6 a^{2} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{4} + b^{5} \sqrt{x}} - \frac{3 a b^{2} x}{a b^{4} + b^{5} \sqrt{x}} + \frac{b^{3} x^{\frac{3}{2}}}{a b^{4} + b^{5} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12539, size = 70, normalized size = 1.3 \begin{align*} \frac{6 \, a^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{b^{2} x - 4 \, a b \sqrt{x}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]