\(\int \frac {x^{11}}{(a+b x^3+c x^6)^{3/2}} \, dx\) [235]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{5/2}} \]

[Out]

-1/2*b*arctanh(1/2*(2*c*x^3+b)/c^(1/2)/(c*x^6+b*x^3+a)^(1/2))/c^(5/2)+2/3*x^6*(b*x^3+2*a)/(-4*a*c+b^2)/(c*x^6+
b*x^3+a)^(1/2)+1/3*(-2*b*c*x^3-8*a*c+3*b^2)*(c*x^6+b*x^3+a)^(1/2)/c^2/(-4*a*c+b^2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 752, 793, 635, 212} \[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {b \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{5/2}}+\frac {\left (-8 a c+3 b^2-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]

[In]

Int[x^11/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*x^6*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) + ((3*b^2 - 8*a*c - 2*b*c*x^3)*Sqrt[a + b*x^3
+ c*x^6])/(3*c^2*(b^2 - 4*a*c)) - (b*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(2*c^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

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^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right ) \\ & = \frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {2 \text {Subst}\left (\int \frac {x (4 a+2 b x)}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3 \left (b^2-4 a c\right )} \\ & = \frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{2 c^2} \\ & = \frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right )}{c^2} \\ & = \frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {-3 a b^2+8 a^2 c-3 b^3 x^3+10 a b c x^3-b^2 c x^6+4 a c^2 x^6}{3 c^2 \left (-b^2+4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {b \log \left (b c^2+2 c^3 x^3-2 c^{5/2} \sqrt {a+b x^3+c x^6}\right )}{2 c^{5/2}} \]

[In]

Integrate[x^11/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-3*a*b^2 + 8*a^2*c - 3*b^3*x^3 + 10*a*b*c*x^3 - b^2*c*x^6 + 4*a*c^2*x^6)/(3*c^2*(-b^2 + 4*a*c)*Sqrt[a + b*x^3
+ c*x^6]) + (b*Log[b*c^2 + 2*c^3*x^3 - 2*c^(5/2)*Sqrt[a + b*x^3 + c*x^6]])/(2*c^(5/2))

Maple [F]

\[\int \frac {x^{11}}{\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}d x\]

[In]

int(x^11/(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^11/(c*x^6+b*x^3+a)^(3/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.35 \[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{12 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{6 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3}\right )}}\right ] \]

[In]

integrate(x^11/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(3*((b^3*c - 4*a*b*c^2)*x^6 + a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*x^3)*sqrt(c)*log(-8*c^2*x^6 - 8*b*c*
x^3 - b^2 + 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) + 4*((b^2*c^2 - 4*a*c^3)*x^6 + 3*a*b^2*c
- 8*a^2*c^2 + (3*b^3*c - 10*a*b*c^2)*x^3)*sqrt(c*x^6 + b*x^3 + a))/((b^2*c^4 - 4*a*c^5)*x^6 + a*b^2*c^3 - 4*a^
2*c^4 + (b^3*c^3 - 4*a*b*c^4)*x^3), 1/6*(3*((b^3*c - 4*a*b*c^2)*x^6 + a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*x^
3)*sqrt(-c)*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(-c)/(c^2*x^6 + b*c*x^3 + a*c)) + 2*((b^2*c^2
- 4*a*c^3)*x^6 + 3*a*b^2*c - 8*a^2*c^2 + (3*b^3*c - 10*a*b*c^2)*x^3)*sqrt(c*x^6 + b*x^3 + a))/((b^2*c^4 - 4*a
*c^5)*x^6 + a*b^2*c^3 - 4*a^2*c^4 + (b^3*c^3 - 4*a*b*c^4)*x^3)]

Sympy [F]

\[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {x^{11}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(x**11/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**11/(a + b*x**3 + c*x**6)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^11/(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 [F]

\[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int { \frac {x^{11}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^11/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^11/(c*x^6 + b*x^3 + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {x^{11}}{{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \]

[In]

int(x^11/(a + b*x^3 + c*x^6)^(3/2),x)

[Out]

int(x^11/(a + b*x^3 + c*x^6)^(3/2), x)