\(\int \frac {1}{x^3 (8 c-d x^3)^2 (c+d x^3)^{3/2}} \, dx\) [631]

Optimal result

Integrand size = 27, antiderivative size = 66 \[ \int \frac {1}{x^3 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (-\frac {2}{3},2,\frac {3}{2},\frac {1}{3},\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{128 c^3 x^2 \sqrt {c+d x^3}} \] Output:

-1/128*(1+d*x^3/c)^(1/2)*AppellF1(-2/3,3/2,2,1/3,-d*x^3/c,1/8*d*x^3/c)/c^3 
/x^2/(d*x^3+c)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(66)=132\).

Time = 10.28 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.92 \[ \int \frac {1}{x^3 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {167 d^2 x^6 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+\frac {64 c \left (-648 c^2-1249 c d x^3+167 d^2 x^6-\frac {19648 c^2 d x^3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{32 c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+3 d x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-4 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}\right )}{8 c-d x^3}}{663552 c^5 x^2 \sqrt {c+d x^3}} \] Input:

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

Output:

(167*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), 
(d*x^3)/(8*c)] + (64*c*(-648*c^2 - 1249*c*d*x^3 + 167*d^2*x^6 - (19648*c^2 
*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*Appe 
llF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/ 
3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3 
, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(8*c - d*x^3))/(663552*c^5*x^2*Sqrt[c + 
 d*x^3])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1013, 1012}

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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (-\frac {2}{3},2,\frac {3}{2},\frac {1}{3},\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{128 c^3 x^2 \sqrt {c+d x^3}}\)

Input:

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

Output:

-1/128*(Sqrt[1 + (d*x^3)/c]*AppellF1[-2/3, 2, 3/2, 1/3, (d*x^3)/(8*c), -(( 
d*x^3)/c)])/(c^3*x^2*Sqrt[c + d*x^3])
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 6.

Time = 4.06 (sec) , antiderivative size = 764, normalized size of antiderivative = 11.58

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(764\)
risch	\(\text {Expression too large to display}\)	\(1760\)
default	\(\text {Expression too large to display}\)	\(1806\)

Input:

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

Output:

-1/128*(d*x^3+c)^(1/2)/c^4/x^2-2/243*d*x/c^4/((x^3+c/d)*d)^(1/2)+1/15552*d 
*x/c^4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)+167/31104*I/c^4*3^(1/2)*(-c*d^2)^(1/3) 
*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c* 
d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^( 
1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d* 
(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*EllipticF( 
1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^( 
1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^ 
(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))-1/5184*I/c^4/d^2*2^(1/2)*sum 
(1/_alpha^2*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(- 
c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2 
)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c 
*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c* 
d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2-(-c*d^ 
2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d 
^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2), 
-1/18/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(-c*d^2)^(2/3)*_alpha*3^( 
1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^ 
(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alph 
a=RootOf(_Z^3*d-8*c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2650 vs. \(2 (52) = 104\).

Time = 1.88 (sec) , antiderivative size = 2650, normalized size of antiderivative = 40.15 \[ \int \frac {1}{x^3 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/82944*(1264*(d^2*x^8 - 7*c*d*x^5 - 8*c^2*x^2)*sqrt(d)*weierstrassPInver 
se(0, -4*c/d, x) - (c^4*d^2*x^8 - 7*c^5*d*x^5 - 8*c^6*x^2 + sqrt(-3)*(c^4* 
d^2*x^8 - 7*c^5*d*x^5 - 8*c^6*x^2))*(d^4/c^25)^(1/6)*log((d^6*x^9 + 318*c* 
d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 - 9*(c^17*d^3*x^8 + 38*c^18*d^2*x 
^5 + 64*c^19*d*x^2 + sqrt(-3)*(c^17*d^3*x^8 + 38*c^18*d^2*x^5 + 64*c^19*d* 
x^2))*(d^4/c^25)^(2/3) + 3*sqrt(d*x^3 + c)*((c^21*d^2*x^7 + 80*c^22*d*x^4 
+ 160*c^23*x - sqrt(-3)*(c^21*d^2*x^7 + 80*c^22*d*x^4 + 160*c^23*x))*(d^4/ 
c^25)^(5/6) - 2*(7*c^13*d^3*x^6 + 152*c^14*d^2*x^3 + 64*c^15*d)*sqrt(d^4/c 
^25) + 6*(5*c^5*d^4*x^5 + 32*c^6*d^3*x^2 + sqrt(-3)*(5*c^5*d^4*x^5 + 32*c^ 
6*d^3*x^2))*(d^4/c^25)^(1/6)) - 9*(5*c^9*d^4*x^7 + 64*c^10*d^3*x^4 + 32*c^ 
11*d^2*x - sqrt(-3)*(5*c^9*d^4*x^7 + 64*c^10*d^3*x^4 + 32*c^11*d^2*x))*(d^ 
4/c^25)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + (c^4* 
d^2*x^8 - 7*c^5*d*x^5 - 8*c^6*x^2 + sqrt(-3)*(c^4*d^2*x^8 - 7*c^5*d*x^5 - 
8*c^6*x^2))*(d^4/c^25)^(1/6)*log((d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x 
^3 + 640*c^3*d^3 - 9*(c^17*d^3*x^8 + 38*c^18*d^2*x^5 + 64*c^19*d*x^2 + sqr 
t(-3)*(c^17*d^3*x^8 + 38*c^18*d^2*x^5 + 64*c^19*d*x^2))*(d^4/c^25)^(2/3) - 
 3*sqrt(d*x^3 + c)*((c^21*d^2*x^7 + 80*c^22*d*x^4 + 160*c^23*x - sqrt(-3)* 
(c^21*d^2*x^7 + 80*c^22*d*x^4 + 160*c^23*x))*(d^4/c^25)^(5/6) - 2*(7*c^13* 
d^3*x^6 + 152*c^14*d^2*x^3 + 64*c^15*d)*sqrt(d^4/c^25) + 6*(5*c^5*d^4*x^5 
+ 32*c^6*d^3*x^2 + sqrt(-3)*(5*c^5*d^4*x^5 + 32*c^6*d^3*x^2))*(d^4/c^25...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x^3/(-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^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^3 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}-368 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{3} d \,x^{2}-322 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{2} x^{5}+46 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{3} x^{8}+104 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{2} x^{2}+91 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{3} x^{5}-13 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) d^{4} x^{8}}{32 c^{2} x^{2} \left (-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}\right )} \] Input:

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

Output:

( - 2*sqrt(c + d*x**3) - 368*int(sqrt(c + d*x**3)/(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**3*d*x**2 - 322 
*int(sqrt(c + d*x**3)/(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**2*d**2*x**5 + 46*int(sqrt(c + d*x**3)/(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*d**3*x**8 + 104*int((sqrt(c + d*x**3)*x**3)/(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**2*d**2*x**2 + 9 
1*int((sqrt(c + d*x**3)*x**3)/(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*d**3*x**5 - 13*int((sqrt(c + d*x**3 
)*x**3)/(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)*d**4*x**8)/(32*c**2*x**2*(8*c**2 + 7*c*d*x**3 - d**2*x**6))