Optimal. Leaf size=33 \[ -\frac{2 (A b-x (b B-2 A c))}{b^2 \sqrt{b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0099222, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {636} \[ -\frac{2 (A b-x (b B-2 A c))}{b^2 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 636
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (A b-(b B-2 A c) x)}{b^2 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0127149, size = 30, normalized size = 0.91 \[ \frac{2 b B x-2 A (b+2 c x)}{b^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 37, normalized size = 1.1 \begin{align*} -2\,{\frac{x \left ( cx+b \right ) \left ( 2\,Acx-bBx+Ab \right ) }{{b}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01205, size = 74, normalized size = 2.24 \begin{align*} \frac{2 \, B x}{\sqrt{c x^{2} + b x} b} - \frac{4 \, A c x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, A}{\sqrt{c x^{2} + b x} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65545, size = 89, normalized size = 2.7 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (A b -{\left (B b - 2 \, A c\right )} x\right )}}{b^{2} c x^{2} + b^{3} 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{A + B 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 [A] time = 1.34254, size = 45, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (\frac{A}{b} - \frac{{\left (B b - 2 \, A c\right )} x}{b^{2}}\right )}}{\sqrt{c x^{2} + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]