Optimal. Leaf size=48 \[ \frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0181612, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}}-\frac{(2 b) \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0129798, size = 28, normalized size = 0.58 \[ \frac{2 \sqrt{x} (2 b+c x)}{c^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 32, normalized size = 0.7 \begin{align*} 2\,{\frac{ \left ( cx+b \right ) \left ( cx+2\,b \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, \sqrt{c x + b} x}{c^{2} x + b c} - \int \frac{2 \,{\left (b c x + b^{2}\right )} x}{{\left (c^{3} x^{3} + 2 \, b c^{2} x^{2} + b^{2} c x\right )} \sqrt{c x + b}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02542, size = 82, normalized size = 1.71 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (c x + 2 \, b\right )} \sqrt{x}}{c^{3} x^{2} + b c^{2} x} \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^{\frac{5}{2}}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2398, size = 42, normalized size = 0.88 \begin{align*} \frac{2 \,{\left (\sqrt{c x + b} + \frac{b}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{4 \, \sqrt{b}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]