\(\int \frac {x^8}{(8 c-d x^3)^2 \sqrt {c+d x^3}} \, dx\) [598]

Optimal result

Integrand size = 27, antiderivative size = 83 \[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {224 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3} \] Output:

2/3*(d*x^3+c)^(1/2)/d^3+64/27*c*(d*x^3+c)^(1/2)/d^3/(-d*x^3+8*c)-224/81*c^ 
(1/2)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^3
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84 \[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2 \left (\frac {3 \sqrt {c+d x^3} \left (-104 c+9 d x^3\right )}{-8 c+d x^3}-112 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{81 d^3} \] Input:

Integrate[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
 

Output:

(2*((3*Sqrt[c + d*x^3]*(-104*c + 9*d*x^3))/(-8*c + d*x^3) - 112*Sqrt[c]*Ar 
cTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(81*d^3)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {948, 100, 27, 90, 73, 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.

\(\displaystyle \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {x^6}{\left (8 c-d x^3\right )^2 \sqrt {d x^3+c}}dx^3\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{3} \left (\frac {64 c \sqrt {c+d x^3}}{9 d^3 \left (8 c-d x^3\right )}-\frac {\int \frac {c d \left (9 d x^3+40 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{9 c d^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {64 c \sqrt {c+d x^3}}{9 d^3 \left (8 c-d x^3\right )}-\frac {\int \frac {9 d x^3+40 c}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{9 d^2}\right )\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{3} \left (\frac {64 c \sqrt {c+d x^3}}{9 d^3 \left (8 c-d x^3\right )}-\frac {112 c \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3-\frac {18 \sqrt {c+d x^3}}{d}}{9 d^2}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{3} \left (\frac {64 c \sqrt {c+d x^3}}{9 d^3 \left (8 c-d x^3\right )}-\frac {\frac {224 c \int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}}{d}-\frac {18 \sqrt {c+d x^3}}{d}}{9 d^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {64 c \sqrt {c+d x^3}}{9 d^3 \left (8 c-d x^3\right )}-\frac {\frac {224 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{3 d}-\frac {18 \sqrt {c+d x^3}}{d}}{9 d^2}\right )\)

Input:

Int[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
 

Output:

((64*c*Sqrt[c + d*x^3])/(9*d^3*(8*c - d*x^3)) - ((-18*Sqrt[c + d*x^3])/d + 
 (224*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(3*d))/(9*d^2))/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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {\frac {2 \sqrt {d \,x^{3}+c}}{3}+\frac {32 c \left (\frac {2 \sqrt {d \,x^{3}+c}}{-d \,x^{3}+8 c}-\frac {7 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 \sqrt {c}}\right )}{27}}{d^{3}}\)	\(63\)
default	\(\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{3}}-\frac {32 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{9 d^{3}}+\frac {64 c^{2} \left (\frac {\sqrt {d \,x^{3}+c}}{c \left (-d \,x^{3}+8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{27 d^{3}}\)	\(93\)
risch	\(\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{3}}+\frac {16 c \left (-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{9 \sqrt {c}\, d}+\frac {4 c \left (-\frac {\sqrt {d \,x^{3}+c}}{c \left (d \,x^{3}-8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{27 d}\right )}{d^{2}}\)	\(98\)
elliptic	\(\frac {64 c \sqrt {d \,x^{3}+c}}{27 d^{3} \left (-d \,x^{3}+8 c \right )}+\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{3}}+\frac {112 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{243 d^{5}}\)	\(452\)

Input:

int(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*((d*x^3+c)^(1/2)+16/9*c*(2*(d*x^3+c)^(1/2)/(-d*x^3+8*c)-7/3*arctanh(1/ 
3*(d*x^3+c)^(1/2)/c^(1/2))/c^(1/2)))/d^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.98 \[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\left [\frac {2 \, {\left (56 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \, {\left (9 \, d x^{3} - 104 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, \frac {2 \, {\left (112 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {3 \, \sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + 3 \, {\left (9 \, d x^{3} - 104 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \] Input:

integrate(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="fricas")
 

Output:

[2/81*(56*(d*x^3 - 8*c)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c) + 1 
0*c)/(d*x^3 - 8*c)) + 3*(9*d*x^3 - 104*c)*sqrt(d*x^3 + c))/(d^4*x^3 - 8*c* 
d^3), 2/81*(112*(d*x^3 - 8*c)*sqrt(-c)*arctan(3*sqrt(-c)/sqrt(d*x^3 + c)) 
+ 3*(9*d*x^3 - 104*c)*sqrt(d*x^3 + c))/(d^4*x^3 - 8*c*d^3)]
 

Sympy [F]

\[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {x^{8}}{\left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \] Input:

integrate(x**8/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)
 

Output:

Integral(x**8/((-8*c + d*x**3)**2*sqrt(c + d*x**3)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2 \, {\left (56 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 27 \, \sqrt {d x^{3} + c} - \frac {96 \, \sqrt {d x^{3} + c} c}{d x^{3} - 8 \, c}\right )}}{81 \, d^{3}} \] Input:

integrate(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")
 

Output:

2/81*(56*sqrt(c)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3*sq 
rt(c))) + 27*sqrt(d*x^3 + c) - 96*sqrt(d*x^3 + c)*c/(d*x^3 - 8*c))/d^3
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2 \, {\left (\frac {112 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d} + \frac {27 \, \sqrt {d x^{3} + c}}{d} - \frac {96 \, \sqrt {d x^{3} + c} c}{{\left (d x^{3} - 8 \, c\right )} d}\right )}}{81 \, d^{2}} \] Input:

integrate(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="giac")
 

Output:

2/81*(112*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d) + 27*sqrt(d* 
x^3 + c)/d - 96*sqrt(d*x^3 + c)*c/((d*x^3 - 8*c)*d))/d^2
 

Mupad [B] (verification not implemented)

Time = 2.60 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05 \[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2\,\sqrt {d\,x^3+c}}{3\,d^3}+\frac {112\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,d^3}+\frac {64\,c\,\sqrt {d\,x^3+c}}{27\,d^3\,\left (8\,c-d\,x^3\right )} \] Input:

int(x^8/((c + d*x^3)^(1/2)*(8*c - d*x^3)^2),x)
 

Output:

(2*(c + d*x^3)^(1/2))/(3*d^3) + (112*c^(1/2)*log((10*c + d*x^3 - 6*c^(1/2) 
*(c + d*x^3)^(1/2))/(8*c - d*x^3)))/(81*d^3) + (64*c*(c + d*x^3)^(1/2))/(2 
7*d^3*(8*c - d*x^3))
 

Reduce [F]

\[ \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {\frac {16 \sqrt {d \,x^{3}+c}\, c}{15}-\frac {2 \sqrt {d \,x^{3}+c}\, d \,x^{3}}{3}+\frac {896 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d^{2}}{5}-\frac {112 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{3} x^{3}}{5}}{d^{3} \left (-d \,x^{3}+8 c \right )} \] Input:

int(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)
 

Output:

(2*(8*sqrt(c + d*x**3)*c - 5*sqrt(c + d*x**3)*d*x**3 + 1344*int((sqrt(c + 
d*x**3)*x**5)/(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c 
**2*d**2 - 168*int((sqrt(c + d*x**3)*x**5)/(64*c**3 + 48*c**2*d*x**3 - 15* 
c*d**2*x**6 + d**3*x**9),x)*c*d**3*x**3))/(15*d**3*(8*c - d*x**3))