Optimal. Leaf size=39 \[ \frac {2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1357, 636} \[ \frac {2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 636
Rule 1357
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 41, normalized size = 1.05 \[ -\frac {2 \left (2 a+b x^3\right )}{3 \left (4 a c-b^2\right ) \sqrt {a+b x^3+c x^6}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.97, size = 68, normalized size = 1.74 \[ \frac {2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )}}{3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.32, size = 45, normalized size = 1.15 \[ \frac {2 \, {\left (\frac {b x^{3}}{b^{2} - 4 \, a c} + \frac {2 \, a}{b^{2} - 4 \, a c}\right )}}{3 \, \sqrt {c x^{6} + b x^{3} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 38, normalized size = 0.97 \[ -\frac {2 \left (b \,x^{3}+2 a \right )}{3 \sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.43, size = 38, normalized size = 0.97 \[ -\frac {2\,b\,x^3+4\,a}{\left (12\,a\,c-3\,b^2\right )\,\sqrt {c\,x^6+b\,x^3+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________