Optimal. Leaf size=36 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2029, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{5/2}}{\sqrt {a x^2+b x^5}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 59, normalized size = 1.64 \begin {gather*} \frac {2 x \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}\right )}{3 \sqrt {b} \sqrt {x^2 \left (a+b x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.22, size = 53, normalized size = 1.47 \begin {gather*} \frac {2 \log \left (\sqrt {a x^2+b x^5}+\sqrt {b} x^{5/2}\right )}{3 \sqrt {b}}-\frac {4 \log \left (\sqrt {x}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 101, normalized size = 2.81 \begin {gather*} \left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} + a\right )} \sqrt {b} \sqrt {x} - a^{2}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {-b} \sqrt {x}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 41, normalized size = 1.14 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {b + \frac {a}{x^{3}}}}{\sqrt {-b}}\right )}{3 \, \sqrt {-b}} + \frac {2 \, \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{3 \, \sqrt {-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.24, size = 480, normalized size = 13.33 \begin {gather*} -\frac {4 \left (b \,x^{3}+a \right ) \left (i \sqrt {3}-1\right ) \sqrt {-\frac {\left (i \sqrt {3}-3\right ) b x}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )^{2} \sqrt {\frac {2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {-2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \left (\EllipticF \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) b x}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}-1\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )-\EllipticPi \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) b x}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \frac {i \sqrt {3}-1}{i \sqrt {3}-3}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}-1\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )\right ) x^{\frac {3}{2}}}{\sqrt {b \,x^{5}+a \,x^{2}}\, \sqrt {\left (b \,x^{3}+a \right ) x}\, \left (i \sqrt {3}-3\right ) \sqrt {\frac {\left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (-2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) x}{b^{2}}}\, b^{2}} \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 \frac {x^{\frac {3}{2}}}{\sqrt {b x^{5} + a x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {b\,x^5+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3}{2}}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________