Optimal. Leaf size=67 \[ -\frac{3 \sqrt{x} \sqrt{b x+2}}{2 b^2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{x^{3/2} \sqrt{b x+2}}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0135392, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ -\frac{3 \sqrt{x} \sqrt{b x+2}}{2 b^2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{x^{3/2} \sqrt{b x+2}}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{2+b x}} \, dx &=\frac{x^{3/2} \sqrt{2+b x}}{2 b}-\frac{3 \int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx}{2 b}\\ &=-\frac{3 \sqrt{x} \sqrt{2+b x}}{2 b^2}+\frac{x^{3/2} \sqrt{2+b x}}{2 b}+\frac{3 \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{2 b^2}\\ &=-\frac{3 \sqrt{x} \sqrt{2+b x}}{2 b^2}+\frac{x^{3/2} \sqrt{2+b x}}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{3 \sqrt{x} \sqrt{2+b x}}{2 b^2}+\frac{x^{3/2} \sqrt{2+b x}}{2 b}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0283173, size = 51, normalized size = 0.76 \[ \frac{\sqrt{x} \sqrt{b x+2} (b x-3)}{2 b^2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 78, normalized size = 1.2 \begin{align*}{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}-{\frac{3}{2\,{b}^{2}}\sqrt{x}\sqrt{bx+2}}+{\frac{3}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \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 = 1.92437, size = 285, normalized size = 4.25 \begin{align*} \left [\frac{{\left (b^{2} x - 3 \, b\right )} \sqrt{b x + 2} \sqrt{x} + 3 \, \sqrt{b} \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right )}{2 \, b^{3}}, \frac{{\left (b^{2} x - 3 \, b\right )} \sqrt{b x + 2} \sqrt{x} - 6 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{2 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.22213, size = 75, normalized size = 1.12 \begin{align*} \frac{x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} - \frac{x^{\frac{3}{2}}}{2 b \sqrt{b x + 2}} - \frac{3 \sqrt{x}}{b^{2} \sqrt{b x + 2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]