Optimal. Leaf size=50 \[ \frac{2 \sqrt{x}}{b \sqrt{a-b x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0162758, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {47, 63, 217, 203} \[ \frac{2 \sqrt{x}}{b \sqrt{a-b x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{(a-b x)^{3/2}} \, dx &=\frac{2 \sqrt{x}}{b \sqrt{a-b x}}-\frac{\int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{a-b x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{a-b x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{a-b x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0737895, size = 66, normalized size = 1.32 \[ \frac{2 \sqrt{b} \sqrt{x}-2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2} \sqrt{a-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{x} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \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 = 1.7616, size = 312, normalized size = 6.24 \begin{align*} \left [-\frac{{\left (b x - a\right )} \sqrt{-b} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \, \sqrt{-b x + a} b \sqrt{x}}{b^{3} x - a b^{2}}, \frac{2 \,{\left ({\left (b x - a\right )} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) - \sqrt{-b x + a} b \sqrt{x}\right )}}{b^{3} x - a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.11018, size = 104, normalized size = 2.08 \begin{align*} \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{2 i \sqrt{x}}{\sqrt{a} b \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x}}{\sqrt{a} b \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 59.564, size = 138, normalized size = 2.76 \begin{align*} -\frac{{\left (\frac{4 \, a \sqrt{-b}}{{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b} - \frac{\sqrt{-b} \log \left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{b}\right )}{\left | b \right |}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]