\(\int \frac {1}{x^3 (a+b x^3)^{2/3} (c+d x^3)} \, dx\) [761]

Optimal result

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

-1/2*(1+b*x^3/a)^(2/3)*AppellF1(-2/3,2/3,1,1/3,-b*x^3/a,-d*x^3/c)/c/x^2/(b 
*x^3+a)^(2/3)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(338\) vs. \(2(64)=128\).

Time = 10.21 (sec) , antiderivative size = 338, normalized size of antiderivative = 5.28 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx=\frac {-b d x^6 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 c \left (-4 a c \left (a c+2 b c x^3+3 a d x^3+b d x^6\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right ) \left (4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}}{8 a c^2 x^2 \left (a+b x^3\right )^{2/3}} \] Input:

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

Output:

(-(b*d*x^6*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), 
-((d*x^3)/c)]) + (4*c*(-4*a*c*(a*c + 2*b*c*x^3 + 3*a*d*x^3 + b*d*x^6)*Appe 
llF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(a + b*x^3)*(c + 
d*x^3)*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b 
*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/((c + d*x^3)* 
(4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - x^3*(3*a*d 
*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4 
/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(8*a*c^2*x^2*(a + b*x^3)^ 
(2/3))
 

Rubi [A] (verified)

Time = 0.36 (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 = {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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {2}{3},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c x^2 \left (a+b x^3\right )^{2/3}}\)

Input:

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

Output:

-1/2*((1 + (b*x^3)/a)^(2/3)*AppellF1[-2/3, 2/3, 1, 1/3, -((b*x^3)/a), -((d 
*x^3)/c)])/(c*x^2*(a + b*x^3)^(2/3))
 

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 [F]

\[\int \frac {1}{x^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )}d x\]

Input:

int(1/x^3/(b*x^3+a)^(2/3)/(d*x^3+c),x)
 

Output:

int(1/x^3/(b*x^3+a)^(2/3)/(d*x^3+c),x)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate(1/x**3/(b*x**3+a)**(2/3)/(d*x**3+c),x)
 

Output:

Integral(1/(x**3*(a + b*x**3)**(2/3)*(c + d*x**3)), x)
 

Maxima [F]

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

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

Output:

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)*x^3), x)
 

Giac [F]

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

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

Output:

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)*x^3), x)
 

Mupad [F(-1)]

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

int(1/(x^3*(a + b*x^3)^(2/3)*(c + d*x^3)),x)
 

Output:

int(1/(x^3*(a + b*x^3)^(2/3)*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {1}{x^3 \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} c \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} d \,x^{6}}d x \] Input:

int(1/x^3/(b*x^3+a)^(2/3)/(d*x^3+c),x)
 

Output:

int(1/((a + b*x**3)**(2/3)*c*x**3 + (a + b*x**3)**(2/3)*d*x**6),x)