Optimal. Leaf size=54 \[ \frac{4 a \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0441508, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac{4 a \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 369
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x^2} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{c} \sqrt{a+b \sqrt{c} x}}+\frac{\sqrt{a+b \sqrt{c} x}}{b \sqrt{c}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.033207, size = 42, normalized size = 0.78 \[ -\frac{4 \left (b \sqrt{\frac{c}{x}}-2 a\right ) \sqrt{a+b \sqrt{\frac{c}{x}}}}{3 b^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.046, size = 274, normalized size = 5.1 \begin{align*} -{\frac{1}{3\,{b}^{3}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 6\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{5/2}{x}^{3/2}+6\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{5/2}{x}^{3/2}+3\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{2}{a}^{2}b-3\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{2}{a}^{2}b+4\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}\sqrt{a}\sqrt{x}b-12\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{3/2}\sqrt{x} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.925253, size = 57, normalized size = 1.06 \begin{align*} -\frac{4 \,{\left (\frac{{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{b^{2}}\right )}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.4377, size = 76, normalized size = 1.41 \begin{align*} -\frac{4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}{\left (b \sqrt{\frac{c}{x}} - 2 \, a\right )}}{3 \, b^{2} c} \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^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15511, size = 51, normalized size = 0.94 \begin{align*} -\frac{4 \,{\left ({\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a\right )}}{3 \, b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]