Optimal. Leaf size=25 \[ -\frac{4 \sqrt{a x+b \sqrt{x}}}{b \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0379828, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2014} \[ -\frac{4 \sqrt{a x+b \sqrt{x}}}{b \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{b \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0080369, size = 25, normalized size = 1. \[ -\frac{4 \sqrt{a x+b \sqrt{x}}}{b \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.011, size = 160, normalized size = 6.4 \begin{align*} -{\frac{1}{{b}^{2}x}\sqrt{b\sqrt{x}+ax} \left ( 4\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}-2\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}x-\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) xab-2\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }x+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ) xab \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b \sqrt{x}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.43133, size = 51, normalized size = 2.04 \begin{align*} -\frac{4 \, \sqrt{a x + b \sqrt{x}}}{b \sqrt{x}} \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{1}{x \sqrt{a x + b \sqrt{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22164, size = 34, normalized size = 1.36 \begin{align*} \frac{4}{\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]