Optimal. Leaf size=85 \[ -\frac{\sqrt{x} (A b-3 a B)}{a b^2}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{x^{3/2} (A b-a B)}{a b (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0353538, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 50, 63, 205} \[ -\frac{\sqrt{x} (A b-3 a B)}{a b^2}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{x^{3/2} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{(a+b x)^2} \, dx &=\frac{(A b-a B) x^{3/2}}{a b (a+b x)}-\frac{\left (\frac{A b}{2}-\frac{3 a B}{2}\right ) \int \frac{\sqrt{x}}{a+b x} \, dx}{a b}\\ &=-\frac{(A b-3 a B) \sqrt{x}}{a b^2}+\frac{(A b-a B) x^{3/2}}{a b (a+b x)}+\frac{(A b-3 a B) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 b^2}\\ &=-\frac{(A b-3 a B) \sqrt{x}}{a b^2}+\frac{(A b-a B) x^{3/2}}{a b (a+b x)}+\frac{(A b-3 a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{(A b-3 a B) \sqrt{x}}{a b^2}+\frac{(A b-a B) x^{3/2}}{a b (a+b x)}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0427333, size = 67, normalized size = 0.79 \[ \frac{\sqrt{x} (3 a B-A b+2 b B x)}{b^2 (a+b x)}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{B\sqrt{x}}{{b}^{2}}}-{\frac{A}{b \left ( bx+a \right ) }\sqrt{x}}+{\frac{Ba}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}+{\frac{A}{b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{Ba}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \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 = 2.34852, size = 441, normalized size = 5.19 \begin{align*} \left [\frac{{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (2 \, B a b^{2} x + 3 \, B a^{2} b - A a b^{2}\right )} \sqrt{x}}{2 \,{\left (a b^{4} x + a^{2} b^{3}\right )}}, \frac{{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (2 \, B a b^{2} x + 3 \, B a^{2} b - A a b^{2}\right )} \sqrt{x}}{a b^{4} x + a^{2} b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21492, size = 88, normalized size = 1.04 \begin{align*} \frac{2 \, B \sqrt{x}}{b^{2}} - \frac{{\left (3 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{B a \sqrt{x} - A b \sqrt{x}}{{\left (b x + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]