Optimal. Leaf size=51 \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0129913, 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 = {658, 613} \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 658
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{3 b x \sqrt{b x+c x^2}}-\frac{(4 c) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2}{3 b x \sqrt{b x+c x^2}}+\frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.011478, size = 40, normalized size = 0.78 \[ \frac{2 \left (-b^2+4 b c x+8 c^2 x^2\right )}{3 b^3 x \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 39, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -8\,{c}^{2}{x}^{2}-4\,bcx+{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.90221, size = 99, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (8 \, c^{2} x^{2} + 4 \, b c x - b^{2}\right )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{3} c x^{3} + b^{4} x^{2}\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{1}{x \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]