Optimal. Leaf size=16 \[ \frac{x}{a \left (a+b \sqrt{x}\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0050682, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 37} \[ \frac{x}{a \left (a+b \sqrt{x}\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 190
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{(a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=\frac{x}{a \left (a+b \sqrt{x}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0039473, size = 16, normalized size = 1. \[ \frac{x}{a \left (a+b \sqrt{x}\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.024, size = 131, normalized size = 8.2 \begin{align*} -{\frac{1}{{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-1}}+{\frac{a}{2\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-2}}-{\frac{1}{{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-1}}-{\frac{a}{2\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-2}}+{\frac{{a}^{3}}{2\, \left ({b}^{2}x-{a}^{2} \right ) ^{2}{b}^{2}}}-3\,{b}^{2}a \left ( -1/2\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{2}}}-{\frac{1}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) }} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.937931, size = 39, normalized size = 2.44 \begin{align*} -\frac{2}{{\left (b \sqrt{x} + a\right )} b^{2}} + \frac{a}{{\left (b \sqrt{x} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.2579, size = 95, normalized size = 5.94 \begin{align*} -\frac{2 \, b^{3} x^{\frac{3}{2}} - 3 \, a b^{2} x + a^{3}}{b^{6} x^{2} - 2 \, a^{2} b^{4} x + a^{4} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.941293, size = 34, normalized size = 2.12 \begin{align*} \begin{cases} \frac{x}{a^{3} + 2 a^{2} b \sqrt{x} + a b^{2} x} & \text{for}\: a \neq 0 \\- \frac{2}{b^{3} \sqrt{x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1001, size = 30, normalized size = 1.88 \begin{align*} -\frac{2 \, b \sqrt{x} + a}{{\left (b \sqrt{x} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]