Optimal. Leaf size=69 \[ -\frac{2 \sqrt{x} (b B-A c)}{c^2}+\frac{2 \sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{5/2}}+\frac{2 B x^{3/2}}{3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0385251, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {781, 80, 50, 63, 205} \[ -\frac{2 \sqrt{x} (b B-A c)}{c^2}+\frac{2 \sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{5/2}}+\frac{2 B x^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 781
Rule 80
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{b x+c x^2} \, dx &=\int \frac{\sqrt{x} (A+B x)}{b+c x} \, dx\\ &=\frac{2 B x^{3/2}}{3 c}+\frac{\left (2 \left (-\frac{3 b B}{2}+\frac{3 A c}{2}\right )\right ) \int \frac{\sqrt{x}}{b+c x} \, dx}{3 c}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{3/2}}{3 c}+\frac{(b (b B-A c)) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{c^2}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{3/2}}{3 c}+\frac{(2 b (b B-A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{3/2}}{3 c}+\frac{2 \sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0327641, size = 63, normalized size = 0.91 \[ \frac{2 \sqrt{x} (3 A c-3 b B+B c x)}{3 c^2}+\frac{2 \sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 78, normalized size = 1.1 \begin{align*}{\frac{2\,B}{3\,c}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{c}}-2\,{\frac{bB\sqrt{x}}{{c}^{2}}}-2\,{\frac{Ab}{c\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) }+2\,{\frac{{b}^{2}B}{{c}^{2}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \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 = 2.18636, size = 305, normalized size = 4.42 \begin{align*} \left [-\frac{3 \,{\left (B b - A c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x - 2 \, c \sqrt{x} \sqrt{-\frac{b}{c}} - b}{c x + b}\right ) - 2 \,{\left (B c x - 3 \, B b + 3 \, A c\right )} \sqrt{x}}{3 \, c^{2}}, \frac{2 \,{\left (3 \,{\left (B b - A c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c \sqrt{x} \sqrt{\frac{b}{c}}}{b}\right ) +{\left (B c x - 3 \, B b + 3 \, A c\right )} \sqrt{x}\right )}}{3 \, c^{2}}\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 = 1.13373, size = 86, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (B b^{2} - A b c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c^{2}} + \frac{2 \,{\left (B c^{2} x^{\frac{3}{2}} - 3 \, B b c \sqrt{x} + 3 \, A c^{2} \sqrt{x}\right )}}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]