Optimal. Leaf size=68 \[ \frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0232231, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{a+\frac{b}{x}} \, dx &=\int \frac{x^{5/2}}{b+a x} \, dx\\ &=\frac{2 x^{5/2}}{5 a}-\frac{b \int \frac{x^{3/2}}{b+a x} \, dx}{a}\\ &=-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a}+\frac{b^2 \int \frac{\sqrt{x}}{b+a x} \, dx}{a^2}\\ &=\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a}-\frac{b^3 \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{a^3}\\ &=\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a}-\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0216179, size = 61, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (3 a^2 x^2-5 a b x+15 b^2\right )}{15 a^3}-\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 54, normalized size = 0.8 \begin{align*}{\frac{2}{5\,a}{x}^{{\frac{5}{2}}}}-{\frac{2\,b}{3\,{a}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{{b}^{2}\sqrt{x}}{{a}^{3}}}-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{a\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 = 1.68958, size = 308, normalized size = 4.53 \begin{align*} \left [\frac{15 \, b^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt{x}}{15 \, a^{3}}, -\frac{2 \,{\left (15 \, b^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) -{\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt{x}\right )}}{15 \, a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.4348, size = 121, normalized size = 1.78 \begin{align*} \begin{cases} \frac{2 x^{\frac{5}{2}}}{5 a} - \frac{2 b x^{\frac{3}{2}}}{3 a^{2}} + \frac{2 b^{2} \sqrt{x}}{a^{3}} + \frac{i b^{\frac{5}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{4} \sqrt{\frac{1}{a}}} - \frac{i b^{\frac{5}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{4} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{7}{2}}}{7 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11601, size = 80, normalized size = 1.18 \begin{align*} -\frac{2 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{2 \,{\left (3 \, a^{4} x^{\frac{5}{2}} - 5 \, a^{3} b x^{\frac{3}{2}} + 15 \, a^{2} b^{2} \sqrt{x}\right )}}{15 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]