\(\int \frac {x^5}{(a+b x^3+c x^6)^{3/2}} \, dx\) [237]

Optimal result

Integrand size = 20, antiderivative size = 39 \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]

[Out]

2/3*(b*x^3+2*a)/(-4*a*c+b^2)/(c*x^6+b*x^3+a)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1371, 650} \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]

[In]

Int[x^5/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {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] (verified)

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]

[In]

Integrate[x^5/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

Maple [A] (verified)

Time = 4.66 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97

method	result	size
gosper	\(-\frac {2 \left (b \,x^{3}+2 a \right )}{3 \sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )}\)	\(38\)
trager	\(-\frac {2 \left (b \,x^{3}+2 a \right )}{3 \sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )}\)	\(38\)

[In]

int(x^5/(c*x^6+b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/(c*x^6+b*x^3+a)^(1/2)*(b*x^3+2*a)/(4*a*c-b^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.74 \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )}}{3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )}} \]

[In]

integrate(x^5/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

2/3*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)/((b^2*c - 4*a*c^2)*x^6 + (b^3 - 4*a*b*c)*x^3 + a*b^2 - 4*a^2*c)

Sympy [F]

\[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(x**5/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**5/(a + b*x**3 + c*x**6)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^5/(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {b x^{3}}{b^{2} - 4 \, a c} + \frac {2 \, a}{b^{2} - 4 \, a c}\right )}}{3 \, \sqrt {c x^{6} + b x^{3} + a}} \]

[In]

integrate(x^5/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

2/3*(b*x^3/(b^2 - 4*a*c) + 2*a/(b^2 - 4*a*c))/sqrt(c*x^6 + b*x^3 + a)

Mupad [B] (verification not implemented)

Time = 8.65 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {2\,b\,x^3+4\,a}{\left (12\,a\,c-3\,b^2\right )\,\sqrt {c\,x^6+b\,x^3+a}} \]

[In]

int(x^5/(a + b*x^3 + c*x^6)^(3/2),x)

[Out]

-(4*a + 2*b*x^3)/((12*a*c - 3*b^2)*(a + b*x^3 + c*x^6)^(1/2))