Optimal. Leaf size=51 \[ \frac {2}{3} x \sqrt {\frac {a}{x^2}+b x}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1979, 2007, 2029, 206} \begin {gather*} \frac {2}{3} x \sqrt {\frac {a}{x^2}+b x}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 1979
Rule 2007
Rule 2029
Rubi steps
\begin {align*} \int \sqrt {\frac {a+b x^3}{x^2}} \, dx &=\int \sqrt {\frac {a}{x^2}+b x} \, dx\\ &=\frac {2}{3} x \sqrt {\frac {a}{x^2}+b x}+a \int \frac {1}{x^2 \sqrt {\frac {a}{x^2}+b x}} \, dx\\ &=\frac {2}{3} x \sqrt {\frac {a}{x^2}+b x}-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x \sqrt {\frac {a}{x^2}+b x}}\right )\\ &=\frac {2}{3} x \sqrt {\frac {a}{x^2}+b x}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 66, normalized size = 1.29 \begin {gather*} \frac {2 x \sqrt {\frac {a}{x^2}+b x} \left (\sqrt {a+b x^3}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )}{3 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 4.03, size = 69, normalized size = 1.35 \begin {gather*} \frac {x \sqrt {\frac {a}{x^2}+b x} \left (\frac {2}{3} \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )}{\sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 104, normalized size = 2.04 \begin {gather*} \left [\frac {2}{3} \, x \sqrt {\frac {b x^{3} + a}{x^{2}}} + \frac {1}{3} \, \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {a} x \sqrt {\frac {b x^{3} + a}{x^{2}}} + 2 \, a}{x^{3}}\right ), \frac {2}{3} \, x \sqrt {\frac {b x^{3} + a}{x^{2}}} + \frac {2}{3} \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {b x^{3} + a}{x^{2}}}}{a}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 71, normalized size = 1.39 \begin {gather*} \frac {2 \, a \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{3 \, \sqrt {-a}} + \frac {2}{3} \, \sqrt {b x^{3} + a} \mathrm {sgn}\relax (x) - \frac {2 \, {\left (a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\relax (x)}{3 \, \sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 55, normalized size = 1.08 \begin {gather*} \frac {2 \sqrt {\frac {b \,x^{3}+a}{x^{2}}}\, \left (-\sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )+\sqrt {b \,x^{3}+a}\right ) x}{3 \sqrt {b \,x^{3}+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {b x^{3} + a}{x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.34, size = 63, normalized size = 1.24 \begin {gather*} \frac {2\,x\,\sqrt {b\,x+\frac {a}{x^2}}}{3}+\frac {\sqrt {a}\,\mathrm {asin}\left (\frac {\sqrt {a}\,1{}\mathrm {i}}{\sqrt {b}\,x^{3/2}}\right )\,\sqrt {b\,x+\frac {a}{x^2}}\,2{}\mathrm {i}}{3\,\sqrt {b}\,\sqrt {x}\,\sqrt {\frac {a}{b\,x^3}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________