Optimal. Leaf size=66 \[ \frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0233972, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ \frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{x (c+d x)^{3/2}} \, dx &=\frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}+\frac{a \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{c}\\ &=\frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{c}\\ &=\frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0328611, size = 66, normalized size = 1. \[ \frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.018, size = 143, normalized size = 2.2 \begin{align*}{\frac{1}{c} \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xad-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) ac+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \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 [B] time = 3.09934, size = 566, normalized size = 8.58 \begin{align*} \left [\frac{{\left (d x + c\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (c d x + c^{2}\right )}}, \frac{{\left (d x + c\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{a}{c}}}{2 \,{\left (a b d x^{2} + a^{2} c +{\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \, \sqrt{b x + a} \sqrt{d x + c}}{c d x + c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x}}{x \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.34007, size = 176, normalized size = 2.67 \begin{align*} -\frac{2 \, \sqrt{b d} a b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c{\left | b \right |}} + \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} c{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]