Optimal. Leaf size=60 \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0204526, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {78, 63, 217, 206} \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{x} (a+b x)^{3/2}} \, dx &=\frac{2 (A b-a B) \sqrt{x}}{a b \sqrt{a+b x}}+\frac{B \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{b}\\ &=\frac{2 (A b-a B) \sqrt{x}}{a b \sqrt{a+b x}}+\frac{(2 B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{2 (A b-a B) \sqrt{x}}{a b \sqrt{a+b x}}+\frac{(2 B) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{b}\\ &=\frac{2 (A b-a B) \sqrt{x}}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0581439, size = 76, normalized size = 1.27 \[ \frac{2 a^{3/2} B \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )+2 \sqrt{b} \sqrt{x} (A b-a B)}{a b^{3/2} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 121, normalized size = 2. \begin{align*}{\frac{1}{a} \left ( B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) xab+2\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){a}^{2}-2\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ) \sqrt{x}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \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 = 2.60359, size = 378, normalized size = 6.3 \begin{align*} \left [\frac{{\left (B a b x + B a^{2}\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (B a b - A b^{2}\right )} \sqrt{b x + a} \sqrt{x}}{a b^{3} x + a^{2} b^{2}}, -\frac{2 \,{\left ({\left (B a b x + B a^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (B a b - A b^{2}\right )} \sqrt{b x + a} \sqrt{x}\right )}}{a b^{3} x + a^{2} b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 15.7603, size = 68, normalized size = 1.13 \begin{align*} \frac{2 A}{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}} + B \left (\frac{2 \operatorname{asinh}{\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}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 87.5327, size = 131, normalized size = 2.18 \begin{align*} -\frac{B \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}{\left | b \right |}} - \frac{4 \,{\left (B a \sqrt{b} - A b^{\frac{3}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]