\(\int \frac {1}{x^7 (8 c-d x^3)^2 (c+d x^3)^{3/2}} \, dx\) [621]

Optimal result

Integrand size = 27, antiderivative size = 185 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {13 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{497664 c^{11/2}}-\frac {33 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{11/2}} \] Output:

665/41472*d^2/c^5/(d*x^3+c)^(1/2)-71/13824*d^2/c^4/(-d*x^3+8*c)/(d*x^3+c)^ 
(1/2)-1/48/c^2/x^6/(-d*x^3+8*c)/(d*x^3+c)^(1/2)+17/384*d/c^3/x^3/(-d*x^3+8 
*c)/(d*x^3+c)^(1/2)+13/497664*d^2*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^( 
11/2)-33/2048*d^2*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {\frac {12 \sqrt {c} \left (864 c^3-1836 c^2 d x^3-5107 c d^2 x^6+665 d^3 x^9\right )}{x^6 \left (-8 c+d x^3\right ) \sqrt {c+d x^3}}+13 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-8019 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{497664 c^{11/2}} \] Input:

Integrate[1/(x^7*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
 

Output:

((12*Sqrt[c]*(864*c^3 - 1836*c^2*d*x^3 - 5107*c*d^2*x^6 + 665*d^3*x^9))/(x 
^6*(-8*c + d*x^3)*Sqrt[c + d*x^3]) + 13*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqr 
t[c])] - 8019*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(497664*c^(11/2))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {948, 114, 27, 168, 27, 168, 27, 169, 27, 174, 73, 219, 221}

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.

\(\displaystyle \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {d \left (34 c-7 d x^3\right )}{2 x^6 \left (8 c-d x^3\right )^2 \left (d x^3+c\right )^{3/2}}dx^3}{16 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {d \int \frac {34 c-7 d x^3}{x^6 \left (8 c-d x^3\right )^2 \left (d x^3+c\right )^{3/2}}dx^3}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {\int \frac {c d \left (396 c-85 d x^3\right )}{x^3 \left (8 c-d x^3\right )^2 \left (d x^3+c\right )^{3/2}}dx^3}{8 c^2}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \int \frac {396 c-85 d x^3}{x^3 \left (8 c-d x^3\right )^2 \left (d x^3+c\right )^{3/2}}dx^3}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (-\frac {\int -\frac {6 c d \left (594 c-71 d x^3\right )}{x^3 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx^3}{72 c^2 d}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\int \frac {594 c-71 d x^3}{x^3 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx^3}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {2 \int \frac {c d \left (5346 c-665 d x^3\right )}{2 x^3 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{9 c^2 d}+\frac {1330}{9 c \sqrt {c+d x^3}}}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {\int \frac {5346 c-665 d x^3}{x^3 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{9 c}+\frac {1330}{9 c \sqrt {c+d x^3}}}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {\frac {2673}{4} \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3+\frac {13}{4} d \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{9 c}+\frac {1330}{9 c \sqrt {c+d x^3}}}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {\frac {13}{2} \int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}+\frac {2673 \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{2 d}}{9 c}+\frac {1330}{9 c \sqrt {c+d x^3}}}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {\frac {2673 \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{2 d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 \sqrt {c}}}{9 c}+\frac {1330}{9 c \sqrt {c+d x^3}}}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {\frac {13 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 \sqrt {c}}-\frac {2673 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{9 c}+\frac {1330}{9 c \sqrt {c+d x^3}}}{12 c}-\frac {71}{18 c \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{8 c}-\frac {17}{4 c x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )}{32 c^2}-\frac {1}{16 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right )\)

Input:

Int[1/(x^7*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
 

Output:

(-1/16*1/(c^2*x^6*(8*c - d*x^3)*Sqrt[c + d*x^3]) - (d*(-17/(4*c*x^3*(8*c - 
 d*x^3)*Sqrt[c + d*x^3]) - (d*(-71/(18*c*(8*c - d*x^3)*Sqrt[c + d*x^3]) + 
(1330/(9*c*Sqrt[c + d*x^3]) + ((13*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/( 
6*Sqrt[c]) - (2673*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(2*Sqrt[c]))/(9*c))/( 
12*c)))/(8*c)))/(32*c^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 114

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 221

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

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 = 1.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {d^{2} \left (-\frac {\sqrt {d \,x^{3}+c}\, \left (-3 d \,x^{3}+c \right )}{128 d^{2} x^{6}}-\frac {99 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{2048 \sqrt {c}}+\frac {\sqrt {d \,x^{3}+c}}{-41472 d \,x^{3}+331776 c}+\frac {13 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{165888 \sqrt {c}}+\frac {2}{81 \sqrt {d \,x^{3}+c}}\right )}{3 c^{5}}\)	\(108\)
risch	\(-\frac {\sqrt {d \,x^{3}+c}\, \left (-3 d \,x^{3}+c \right )}{384 c^{5} x^{6}}+\frac {d^{2} \left (-\frac {33 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{8 \sqrt {c}}+\frac {512}{243 \sqrt {d \,x^{3}+c}}+\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{5832 \sqrt {c}}+\frac {c \left (-\frac {\sqrt {d \,x^{3}+c}}{c \left (d \,x^{3}-8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{486}\right )}{256 c^{5}}\)	\(135\)
default	\(\frac {-\frac {\sqrt {d \,x^{3}+c}}{6 c^{2} x^{6}}+\frac {7 d \sqrt {d \,x^{3}+c}}{12 c^{3} x^{3}}+\frac {2 d^{2}}{3 c^{3} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {5 d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{4 c^{\frac {7}{2}}}}{64 c^{2}}+\frac {d \left (-\frac {\sqrt {d \,x^{3}+c}}{3 c^{2} x^{3}}-\frac {2 d}{3 c^{2} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}+\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )}{256 c^{3}}+\frac {3 d^{2} \left (\frac {2}{3 c \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{4096 c^{4}}+\frac {d^{2} \left (-\frac {2}{\sqrt {d \,x^{3}+c}}+\frac {\sqrt {d \,x^{3}+c}}{-d \,x^{3}+8 c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{\sqrt {c}}\right )}{124416 c^{5}}+\frac {d^{2} \left (-\frac {1}{c \sqrt {d \,x^{3}+c}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{18432 c^{4}}\)	\(298\)
elliptic	\(\text {Expression too large to display}\)	\(1601\)

Input:

int(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*d^2/c^5*(-1/128*(d*x^3+c)^(1/2)*(-3*d*x^3+c)/d^2/x^6-99/2048*arctanh(( 
d*x^3+c)^(1/2)/c^(1/2))/c^(1/2)+1/41472*(d*x^3+c)^(1/2)/(-d*x^3+8*c)+13/16 
5888*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(1/2)+2/81/(d*x^3+c)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.12 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\left [\frac {13 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 8019 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 24 \, {\left (665 \, c d^{3} x^{9} - 5107 \, c^{2} d^{2} x^{6} - 1836 \, c^{3} d x^{3} + 864 \, c^{4}\right )} \sqrt {d x^{3} + c}}{995328 \, {\left (c^{6} d^{2} x^{12} - 7 \, c^{7} d x^{9} - 8 \, c^{8} x^{6}\right )}}, -\frac {13 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {3 \, \sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) - 8019 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) - 12 \, {\left (665 \, c d^{3} x^{9} - 5107 \, c^{2} d^{2} x^{6} - 1836 \, c^{3} d x^{3} + 864 \, c^{4}\right )} \sqrt {d x^{3} + c}}{497664 \, {\left (c^{6} d^{2} x^{12} - 7 \, c^{7} d x^{9} - 8 \, c^{8} x^{6}\right )}}\right ] \] Input:

integrate(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
 

Output:

[1/995328*(13*(d^4*x^12 - 7*c*d^3*x^9 - 8*c^2*d^2*x^6)*sqrt(c)*log((d*x^3 
+ 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) + 8019*(d^4*x^12 - 7*c* 
d^3*x^9 - 8*c^2*d^2*x^6)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 
2*c)/x^3) + 24*(665*c*d^3*x^9 - 5107*c^2*d^2*x^6 - 1836*c^3*d*x^3 + 864*c^ 
4)*sqrt(d*x^3 + c))/(c^6*d^2*x^12 - 7*c^7*d*x^9 - 8*c^8*x^6), -1/497664*(1 
3*(d^4*x^12 - 7*c*d^3*x^9 - 8*c^2*d^2*x^6)*sqrt(-c)*arctan(3*sqrt(-c)/sqrt 
(d*x^3 + c)) - 8019*(d^4*x^12 - 7*c*d^3*x^9 - 8*c^2*d^2*x^6)*sqrt(-c)*arct 
an(sqrt(-c)/sqrt(d*x^3 + c)) - 12*(665*c*d^3*x^9 - 5107*c^2*d^2*x^6 - 1836 
*c^3*d*x^3 + 864*c^4)*sqrt(d*x^3 + c))/(c^6*d^2*x^12 - 7*c^7*d*x^9 - 8*c^8 
*x^6)]
 

Sympy [F]

\[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^{7} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/x**7/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
 

Output:

Integral(1/(x**7*(-8*c + d*x**3)**2*(c + d*x**3)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{7}} \,d x } \] Input:

integrate(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^7), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {33 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{2048 \, \sqrt {-c} c^{5}} - \frac {13 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{497664 \, \sqrt {-c} c^{5}} + \frac {341 \, {\left (d x^{3} + c\right )} d^{2} - 3072 \, c d^{2}}{41472 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c\right )} c^{5}} + \frac {3 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 4 \, \sqrt {d x^{3} + c} c d^{2}}{384 \, c^{5} d^{2} x^{6}} \] Input:

integrate(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="giac")
 

Output:

33/2048*d^2*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^5) - 13/497664*d^ 
2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^5) + 1/41472*(341*(d*x^ 
3 + c)*d^2 - 3072*c*d^2)/(((d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c)*c^5) + 
 1/384*(3*(d*x^3 + c)^(3/2)*d^2 - 4*sqrt(d*x^3 + c)*c*d^2)/(c^5*d^2*x^6)
 

Mupad [B] (verification not implemented)

Time = 3.63 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {\frac {2\,d^2}{9\,c^2}-\frac {10373\,d^2\,\left (d\,x^3+c\right )}{13824\,c^3}+\frac {3551\,d^2\,{\left (d\,x^3+c\right )}^2}{6912\,c^4}-\frac {665\,d^2\,{\left (d\,x^3+c\right )}^3}{13824\,c^5}}{33\,c\,{\left (d\,x^3+c\right )}^{5/2}-3\,{\left (d\,x^3+c\right )}^{7/2}+27\,c^3\,\sqrt {d\,x^3+c}-57\,c^2\,{\left (d\,x^3+c\right )}^{3/2}}+\frac {d^2\,\left (\mathrm {atanh}\left (\frac {c^5\,\sqrt {d\,x^3+c}}{\sqrt {c^{11}}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c^5\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^{11}}}\right )\,13{}\mathrm {i}}{8019}\right )\,33{}\mathrm {i}}{2048\,\sqrt {c^{11}}} \] Input:

int(1/(x^7*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2),x)
 

Output:

((2*d^2)/(9*c^2) - (10373*d^2*(c + d*x^3))/(13824*c^3) + (3551*d^2*(c + d* 
x^3)^2)/(6912*c^4) - (665*d^2*(c + d*x^3)^3)/(13824*c^5))/(33*c*(c + d*x^3 
)^(5/2) - 3*(c + d*x^3)^(7/2) + 27*c^3*(c + d*x^3)^(1/2) - 57*c^2*(c + d*x 
^3)^(3/2)) + (d^2*(atanh((c^5*(c + d*x^3)^(1/2))/(c^11)^(1/2))*1i - (atanh 
((c^5*(c + d*x^3)^(1/2))/(3*(c^11)^(1/2)))*13i)/8019)*33i)/(2048*(c^11)^(1 
/2))
 

Reduce [F]

\[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
 

Output:

( - 4930464*sqrt(c + d*x**3)*c**4 + 10477236*sqrt(c + d*x**3)*c**3*d*x**3 
+ 29076630*sqrt(c + d*x**3)*c**2*d**2*x**6 - 3814800*sqrt(c + d*x**3)*c*d* 
*3*x**9 + 15259200*sqrt(c)*log(sqrt(c + d*x**3) - sqrt(c))*c**2*d**2*x**6 
+ 13351800*sqrt(c)*log(sqrt(c + d*x**3) - sqrt(c))*c*d**3*x**9 - 1907400*s 
qrt(c)*log(sqrt(c + d*x**3) - sqrt(c))*d**4*x**12 - 15259200*sqrt(c)*log(s 
qrt(c + d*x**3) + sqrt(c))*c**2*d**2*x**6 - 13351800*sqrt(c)*log(sqrt(c + 
d*x**3) + sqrt(c))*c*d**3*x**9 + 1907400*sqrt(c)*log(sqrt(c + d*x**3) + sq 
rt(c))*d**4*x**12 + 692224*int(sqrt(c + d*x**3)/(64*c**4*x**7 + 112*c**3*d 
*x**10 + 33*c**2*d**2*x**13 - 14*c*d**3*x**16 + d**4*x**19),x)*c**8*x**6 + 
 605696*int(sqrt(c + d*x**3)/(64*c**4*x**7 + 112*c**3*d*x**10 + 33*c**2*d* 
*2*x**13 - 14*c*d**3*x**16 + d**4*x**19),x)*c**7*d*x**9 - 86528*int(sqrt(c 
 + d*x**3)/(64*c**4*x**7 + 112*c**3*d*x**10 + 33*c**2*d**2*x**13 - 14*c*d* 
*3*x**16 + d**4*x**19),x)*c**6*d**2*x**12 - 1070784*int(sqrt(c + d*x**3)/( 
64*c**4*x + 112*c**3*d*x**4 + 33*c**2*d**2*x**7 - 14*c*d**3*x**10 + d**4*x 
**13),x)*c**6*d**2*x**6 - 936936*int(sqrt(c + d*x**3)/(64*c**4*x + 112*c** 
3*d*x**4 + 33*c**2*d**2*x**7 - 14*c*d**3*x**10 + d**4*x**13),x)*c**5*d**3* 
x**9 + 133848*int(sqrt(c + d*x**3)/(64*c**4*x + 112*c**3*d*x**4 + 33*c**2* 
d**2*x**7 - 14*c*d**3*x**10 + d**4*x**13),x)*c**4*d**4*x**12 + 6126120*int 
((sqrt(c + d*x**3)*x**5)/(64*c**4 + 112*c**3*d*x**3 + 33*c**2*d**2*x**6 - 
14*c*d**3*x**9 + d**4*x**12),x)*c**4*d**4*x**6 + 5360355*int((sqrt(c + ...