Optimal. Leaf size=38 \[ \frac{2 B \sqrt{a+b x}}{b^2}-\frac{2 (A b-a B)}{b^2 \sqrt{a+b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0133618, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{2 B \sqrt{a+b x}}{b^2}-\frac{2 (A b-a B)}{b^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^{3/2}} \, dx &=\int \left (\frac{A b-a B}{b (a+b x)^{3/2}}+\frac{B}{b \sqrt{a+b x}}\right ) \, dx\\ &=-\frac{2 (A b-a B)}{b^2 \sqrt{a+b x}}+\frac{2 B \sqrt{a+b x}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0159248, size = 27, normalized size = 0.71 \[ \frac{2 (2 a B-A b+b B x)}{b^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 26, normalized size = 0.7 \begin{align*} -2\,{\frac{-bBx+Ab-2\,Ba}{\sqrt{bx+a}{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.79799, size = 50, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{b x + a} B}{b} + \frac{B a - A b}{\sqrt{b x + a} b}\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.22688, size = 74, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (B b x + 2 \, B a - A b\right )} \sqrt{b x + a}}{b^{3} x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.21379, size = 60, normalized size = 1.58 \begin{align*} \begin{cases} - \frac{2 A}{b \sqrt{a + b x}} + \frac{4 B a}{b^{2} \sqrt{a + b x}} + \frac{2 B x}{b \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{A x + \frac{B x^{2}}{2}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19465, size = 46, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{b x + a} B}{b^{2}} + \frac{2 \,{\left (B a - A b\right )}}{\sqrt{b x + a} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]