\(\int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx\) [206]

Optimal result

Integrand size = 20, antiderivative size = 104 \[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=-\frac {b \sqrt {a+b x^3+c x^6}}{4 c^2}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c}+\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{24 c^{5/2}} \] Output:

-1/4*b*(c*x^6+b*x^3+a)^(1/2)/c^2+1/6*x^3*(c*x^6+b*x^3+a)^(1/2)/c+1/24*(-4* 
a*c+3*b^2)*arctanh(1/2*(2*c*x^3+b)/c^(1/2)/(c*x^6+b*x^3+a)^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {\left (-3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{12 c^2}+\frac {\left (-3 b^2+4 a c\right ) \log \left (b c^2+2 c^3 x^3-2 c^{5/2} \sqrt {a+b x^3+c x^6}\right )}{24 c^{5/2}} \] Input:

Integrate[x^8/Sqrt[a + b*x^3 + c*x^6],x]
 

Output:

((-3*b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*c^2) + ((-3*b^2 + 4*a*c)*Lo 
g[b*c^2 + 2*c^3*x^3 - 2*c^(5/2)*Sqrt[a + b*x^3 + c*x^6]])/(24*c^(5/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1693, 1166, 27, 1160, 1092, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {x^6}{\sqrt {c x^6+b x^3+a}}dx^3\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {1}{3} \left (\frac {\int -\frac {3 b x^3+2 a}{2 \sqrt {c x^6+b x^3+a}}dx^3}{2 c}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{2 c}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {x^3 \sqrt {a+b x^3+c x^6}}{2 c}-\frac {\int \frac {3 b x^3+2 a}{\sqrt {c x^6+b x^3+a}}dx^3}{4 c}\right )\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{3} \left (\frac {x^3 \sqrt {a+b x^3+c x^6}}{2 c}-\frac {\frac {3 b \sqrt {a+b x^3+c x^6}}{c}-\frac {\left (3 b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^6+b x^3+a}}dx^3}{2 c}}{4 c}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{3} \left (\frac {x^3 \sqrt {a+b x^3+c x^6}}{2 c}-\frac {\frac {3 b \sqrt {a+b x^3+c x^6}}{c}-\frac {\left (3 b^2-4 a c\right ) \int \frac {1}{4 c-x^6}d\frac {2 c x^3+b}{\sqrt {c x^6+b x^3+a}}}{c}}{4 c}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {x^3 \sqrt {a+b x^3+c x^6}}{2 c}-\frac {\frac {3 b \sqrt {a+b x^3+c x^6}}{c}-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{3/2}}}{4 c}\right )\)

Input:

Int[x^8/Sqrt[a + b*x^3 + c*x^6],x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

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

rule 1160

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

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(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] && IntegerQ 
[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {x^{8}}{\sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]

Input:

int(x^8/(c*x^6+b*x^3+a)^(1/2),x)
 

Output:

int(x^8/(c*x^6+b*x^3+a)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.95 \[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\left [-\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c^{2} x^{3} - 3 \, b c\right )}}{48 \, c^{3}}, -\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c^{2} x^{3} - 3 \, b c\right )}}{24 \, c^{3}}\right ] \] Input:

integrate(x^8/(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/48*((3*b^2 - 4*a*c)*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*sqrt(c*x^6 + b*x^3 + 
 a)*(2*c^2*x^3 - 3*b*c))/c^3, -1/24*((3*b^2 - 4*a*c)*sqrt(-c)*arctan(1/2*s 
qrt(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*sqrt(c*x^6 + b*x^3 + a)*(2*c^2*x^3 - 3*b*c))/c^3]
 

Sympy [F]

\[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\int \frac {x^{8}}{\sqrt {a + b x^{3} + c x^{6}}}\, dx \] Input:

integrate(x**8/(c*x**6+b*x**3+a)**(1/2),x)
 

Output:

Integral(x**8/sqrt(a + b*x**3 + c*x**6), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^8/(c*x^6+b*x^3+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {x^{8}}{\sqrt {c x^{6} + b x^{3} + a}} \,d x } \] Input:

integrate(x^8/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^8/sqrt(c*x^6 + b*x^3 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\int \frac {x^8}{\sqrt {c\,x^6+b\,x^3+a}} \,d x \] Input:

int(x^8/(a + b*x^3 + c*x^6)^(1/2),x)
 

Output:

int(x^8/(a + b*x^3 + c*x^6)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {8 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a \,c^{2}-12 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{2} c +4 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b \,c^{2} x^{3}+4 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) a b c -3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) b^{3}-4 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) a b c +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) b^{3}-24 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b \,c^{3}+18 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{3} c^{2}-24 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} c^{3}+6 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{2} c^{2}+9 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{4} c}{24 b \,c^{3}} \] Input:

int(x^8/(c*x^6+b*x^3+a)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

(8*sqrt(a + b*x**3 + c*x**6)*a*c**2 - 12*sqrt(a + b*x**3 + c*x**6)*b**2*c 
+ 4*sqrt(a + b*x**3 + c*x**6)*b*c**2*x**3 + 4*sqrt(c)*log(sqrt(a + b*x**3 
+ c*x**6) - sqrt(c)*x**3)*a*b*c - 3*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) 
- sqrt(c)*x**3)*b**3 - 4*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x 
**3)*a*b*c + 3*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*b**3 
- 24*int((sqrt(a + b*x**3 + c*x**6)*x**8)/(a**2 + 2*a*b*x**3 + a*c*x**6 + 
b**2*x**6 + b*c*x**9),x)*a*b*c**3 + 18*int((sqrt(a + b*x**3 + c*x**6)*x**8 
)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*b**3*c**2 - 24* 
int((sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2* 
x**6 + b*c*x**9),x)*a**2*c**3 + 6*int((sqrt(a + b*x**3 + c*x**6)*x**5)/(a* 
*2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*a*b**2*c**2 + 9*int( 
(sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 
 + b*c*x**9),x)*b**4*c)/(24*b*c**3)