Integrand size = 27, antiderivative size = 66 \[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (-\frac {5}{3},2,\frac {3}{2},-\frac {2}{3},\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{320 c^3 x^5 \sqrt {c+d x^3}} \] Output:
-1/320*(1+d*x^3/c)^(1/2)*AppellF1(-5/3,3/2,2,-2/3,-d*x^3/c,1/8*d*x^3/c)/c^ 3/x^5/(d*x^3+c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(66)=132\).
Time = 10.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 4.29 \[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {\frac {64 \left (2592 c^3-7128 c^2 d x^3-15373 c d^2 x^6+2027 d^3 x^9\right )}{c^5 x^5 \left (-8 c+d x^3\right )}-\frac {2027 d^3 x^4 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{c^6}+\frac {16789504 d^2 x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{c^3 \left (8 c-d x^3\right ) \left (32 c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+3 d x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-4 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )\right )}}{6635520 \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^6*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
((64*(2592*c^3 - 7128*c^2*d*x^3 - 15373*c*d^2*x^6 + 2027*d^3*x^9))/(c^5*x^ 5*(-8*c + d*x^3)) - (2027*d^3*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1 , 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^6 + (16789504*d^2*x*AppellF1[1/3, 1 /2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(c^3*(8*c - d*x^3)*(32*c*AppellF 1[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)]))))/(6635520*Sqrt[c + d*x^3])
Time = 0.35 (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.
| \(\displaystyle \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle \frac {\sqrt {\frac {d x^3}{c}+1} \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (\frac {d x^3}{c}+1\right )^{3/2}}dx}{c \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle -\frac {\sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (-\frac {5}{3},2,\frac {3}{2},-\frac {2}{3},\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{320 c^3 x^5 \sqrt {c+d x^3}}\) |
Input:
Int[1/(x^6*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
-1/320*(Sqrt[1 + (d*x^3)/c]*AppellF1[-5/3, 2, 3/2, -2/3, (d*x^3)/(8*c), -( (d*x^3)/c)])/(c^3*x^5*Sqrt[c + d*x^3])
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])
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])
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 4.78 (sec) , antiderivative size = 787, normalized size of antiderivative = 11.92
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(787\) |
| risch | \(\text {Expression too large to display}\) | \(1772\) |
| default | \(\text {Expression too large to display}\) | \(2157\) |
Input:
int(1/x^6/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/320*(d*x^3+c)^(1/2)/c^4/x^5+29/2560*d*(d*x^3+c)^(1/2)/c^5/x^2+2/243*d^2 *x/c^5/((x^3+c/d)*d)^(1/2)+1/124416*d^2*x/c^5*(d*x^3+c)^(1/2)/(-d*x^3+8*c) -2027/311040*I*d/c^5*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))-1/31104*I/c^5/d*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)*_alp ha^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)),_alpha=RootOf(_Z^3*d-8*c))
Leaf count of result is larger than twice the leaf count of optimal. 2698 vs. \(2 (52) = 104\).
Time = 5.78 (sec) , antiderivative size = 2698, normalized size of antiderivative = 40.88 \[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^6/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
Output:
1/2488320*(49008*(d^3*x^11 - 7*c*d^2*x^8 - 8*c^2*d*x^5)*sqrt(d)*weierstras sPInverse(0, -4*c/d, x) + 5*(c^5*d^2*x^11 - 7*c^6*d*x^8 - 8*c^7*x^5 + sqrt (-3)*(c^5*d^2*x^11 - 7*c^6*d*x^8 - 8*c^7*x^5))*(d^10/c^31)^(1/6)*log((d^11 *x^9 + 318*c*d^10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8 - 9*(c^21*d^4*x^8 + 38*c^22*d^3*x^5 + 64*c^23*d^2*x^2 + sqrt(-3)*(c^21*d^4*x^8 + 38*c^22*d^3* x^5 + 64*c^23*d^2*x^2))*(d^10/c^31)^(2/3) + 3*sqrt(d*x^3 + c)*((c^26*d^2*x ^7 + 80*c^27*d*x^4 + 160*c^28*x - sqrt(-3)*(c^26*d^2*x^7 + 80*c^27*d*x^4 + 160*c^28*x))*(d^10/c^31)^(5/6) - 2*(7*c^16*d^5*x^6 + 152*c^17*d^4*x^3 + 6 4*c^18*d^3)*sqrt(d^10/c^31) + 6*(5*c^6*d^8*x^5 + 32*c^7*d^7*x^2 + sqrt(-3) *(5*c^6*d^8*x^5 + 32*c^7*d^7*x^2))*(d^10/c^31)^(1/6)) - 9*(5*c^11*d^7*x^7 + 64*c^12*d^6*x^4 + 32*c^13*d^5*x - sqrt(-3)*(5*c^11*d^7*x^7 + 64*c^12*d^6 *x^4 + 32*c^13*d^5*x))*(d^10/c^31)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^ 2*d*x^3 - 512*c^3)) - 5*(c^5*d^2*x^11 - 7*c^6*d*x^8 - 8*c^7*x^5 + sqrt(-3) *(c^5*d^2*x^11 - 7*c^6*d*x^8 - 8*c^7*x^5))*(d^10/c^31)^(1/6)*log((d^11*x^9 + 318*c*d^10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8 - 9*(c^21*d^4*x^8 + 38* c^22*d^3*x^5 + 64*c^23*d^2*x^2 + sqrt(-3)*(c^21*d^4*x^8 + 38*c^22*d^3*x^5 + 64*c^23*d^2*x^2))*(d^10/c^31)^(2/3) - 3*sqrt(d*x^3 + c)*((c^26*d^2*x^7 + 80*c^27*d*x^4 + 160*c^28*x - sqrt(-3)*(c^26*d^2*x^7 + 80*c^27*d*x^4 + 160 *c^28*x))*(d^10/c^31)^(5/6) - 2*(7*c^16*d^5*x^6 + 152*c^17*d^4*x^3 + 64*c^ 18*d^3)*sqrt(d^10/c^31) + 6*(5*c^6*d^8*x^5 + 32*c^7*d^7*x^2 + sqrt(-3)*...
\[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^{6} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**6/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
Output:
Integral(1/(x**6*(-8*c + d*x**3)**2*(c + d*x**3)**(3/2)), x)
\[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-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^6), x)
\[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-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^6), x)
Timed out. \[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^6\,{\left (d\,x^3+c\right )}^{3/2}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int(1/(x^6*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2),x)
Output:
int(1/(x^6*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2), x)
\[ \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-8 \sqrt {d \,x^{3}+c}\, c +22 \sqrt {d \,x^{3}+c}\, d \,x^{3}+4656 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{3} d^{2} x^{5}+4074 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{3} x^{8}-582 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{4} x^{11}-1144 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{3} x^{5}-1001 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{4} x^{8}+143 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) d^{5} x^{11}}{320 c^{3} x^{5} \left (-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}\right )} \] Input:
int(1/x^6/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
Output:
( - 8*sqrt(c + d*x**3)*c + 22*sqrt(c + d*x**3)*d*x**3 + 4656*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**2*x**5 + 4074*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**3 *x**8 - 582*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**4*x**11 - 1144*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**3*x**5 - 1001*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**4*x**8 + 143*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**5*x**11)/(320* c**3*x**5*(8*c**2 + 7*c*d*x**3 - d**2*x**6))