\(\int \frac {1}{(8 c-d x^3) \sqrt {c+d x^3}} \, dx\) [497]

Optimal result

Integrand size = 24, antiderivative size = 64 \[ \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {x \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c \sqrt {c+d x^3}} \] Output:

1/8*x*(1+d*x^3/c)^(1/2)*AppellF1(1/3,1/2,1,4/3,-d*x^3/c,1/8*d*x^3/c)/c/(d* 
x^3+c)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(166\) vs. \(2(64)=128\).

Time = 10.17 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.59 \[ \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {32 c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3} \left (32 c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+3 d x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-4 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )\right )} \] Input:

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

Output:

(32*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d 
*x^3)*Sqrt[c + d*x^3]*(32*c*AppellF1[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)])))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {937, 936}

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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

\(\displaystyle \frac {x \sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c \sqrt {c+d x^3}}\)

Input:

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

Output:

(x*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1, 1/2, 4/3, (d*x^3)/(8*c), -((d*x^3) 
/c)])/(8*c*Sqrt[c + d*x^3])
 

Defintions of rubi rules used

rule 936

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

rule 937

Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[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 = 1.07 (sec) , antiderivative size = 416, normalized size of antiderivative = 6.50

method	result	size
default	\(-\frac {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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{27 d^{3} c}\)	\(416\)
elliptic	\(-\frac {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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{27 d^{3} c}\)	\(416\)

Input:

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

Output:

-1/27*I/d^3/c*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)),_alpha=RootOf(_Z^3*d-8*c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2319 vs. \(2 (50) = 100\).

Time = 0.58 (sec) , antiderivative size = 2319, normalized size of antiderivative = 36.23 \[ \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/432*(2*c*d*(1/(c^7*d^2))^(1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d 
*x^3 + 640*c^3 + 18*(c^5*d^4*x^8 + 38*c^6*d^3*x^5 + 64*c^7*d^2*x^2)*(1/(c^ 
7*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^4*x^7 + 80*c^7*d^3*x^4 + 160*c^8 
*d^2*x)*(1/(c^7*d^2))^(5/6) + (7*c^4*d^3*x^6 + 152*c^5*d^2*x^3 + 64*c^6*d) 
*sqrt(1/(c^7*d^2)) + 6*(5*c^2*d^2*x^5 + 32*c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) 
 + 18*(5*c^3*d^3*x^7 + 64*c^4*d^2*x^4 + 32*c^5*d*x)*(1/(c^7*d^2))^(1/3))/( 
d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 2*c*d*(1/(c^7*d^2))^( 
1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3 + 640*c^3 + 18*(c^5*d^4 
*x^8 + 38*c^6*d^3*x^5 + 64*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) - 6*sqrt(d*x^3 
 + c)*((c^6*d^4*x^7 + 80*c^7*d^3*x^4 + 160*c^8*d^2*x)*(1/(c^7*d^2))^(5/6) 
+ (7*c^4*d^3*x^6 + 152*c^5*d^2*x^3 + 64*c^6*d)*sqrt(1/(c^7*d^2)) + 6*(5*c^ 
2*d^2*x^5 + 32*c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) + 18*(5*c^3*d^3*x^7 + 64*c^ 
4*d^2*x^4 + 32*c^5*d*x)*(1/(c^7*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192 
*c^2*d*x^3 - 512*c^3)) + (sqrt(-3)*c*d + c*d)*(1/(c^7*d^2))^(1/6)*log((d^3 
*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3 + 640*c^3 - 9*(c^5*d^4*x^8 + 38*c^6* 
d^3*x^5 + 64*c^7*d^2*x^2 + sqrt(-3)*(c^5*d^4*x^8 + 38*c^6*d^3*x^5 + 64*c^7 
*d^2*x^2))*(1/(c^7*d^2))^(2/3) + 3*sqrt(d*x^3 + c)*((c^6*d^4*x^7 + 80*c^7* 
d^3*x^4 + 160*c^8*d^2*x - sqrt(-3)*(c^6*d^4*x^7 + 80*c^7*d^3*x^4 + 160*c^8 
*d^2*x))*(1/(c^7*d^2))^(5/6) - 2*(7*c^4*d^3*x^6 + 152*c^5*d^2*x^3 + 64*c^6 
*d)*sqrt(1/(c^7*d^2)) + 6*(5*c^2*d^2*x^5 + 32*c^3*d*x^2 + sqrt(-3)*(5*c...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \] Input:

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

Output:

int(sqrt(c + d*x**3)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)