Optimal. Leaf size=51 \[ \frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0134114, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {652, 613} \[ \frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 652
Rule 613
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a x+b x^2\right )^{5/2}} \, dx &=-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{\int \frac{1}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}}+\frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0110211, size = 29, normalized size = 0.57 \[ \frac{2 x^2 (3 a+2 b x)}{3 a^2 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 33, normalized size = 0.7 \begin{align*}{\frac{2\,{x}^{3} \left ( bx+a \right ) \left ( 2\,bx+3\,a \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.1577, size = 73, normalized size = 1.43 \begin{align*} \frac{4 \, x}{3 \, \sqrt{b x^{2} + a x} a^{2}} - \frac{2 \, x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b} + \frac{2}{3 \, \sqrt{b x^{2} + a x} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.07175, size = 93, normalized size = 1.82 \begin{align*} \frac{2 \, \sqrt{b x^{2} + a x}{\left (2 \, b x + 3 \, a\right )}}{3 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\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^{2}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21426, size = 82, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} b + 2 \, a \sqrt{b}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} + a\right )}^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]