\(\int \frac {1}{(a+b x^3)^{8/3} (c+d x^3)} \, dx\) [149]

Optimal result

Integrand size = 21, antiderivative size = 62 \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {8}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c \left (a+b x^3\right )^{2/3}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(429\) vs. \(2(62)=124\).

Time = 10.83 (sec) , antiderivative size = 429, normalized size of antiderivative = 6.92 \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )} \, dx=-\frac {x \left (\frac {b d (-4 b c+9 a d) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c}+\frac {4 \left (4 a c \left (10 a^3 d^2+4 b^3 c x^3 \left (2 c+d x^3\right )-a^2 b d \left (20 c+d x^3\right )+a b^2 \left (10 c^2-12 c d x^3-9 d^2 x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b x^3 \left (c+d x^3\right ) \left (11 a^2 d-4 b^2 c x^3+a b \left (-6 c+9 d x^3\right )\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (a+b x^3\right ) \left (c+d x^3\right ) \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}\right )}{40 a^2 (b c-a d)^2 \left (a+b x^3\right )^{2/3}} \] Input:

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

Output:

-1/40*(x*((b*d*(-4*b*c + 9*a*d)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/ 
3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c + (4*(4*a*c*(10*a^3*d^2 + 4*b^3* 
c*x^3*(2*c + d*x^3) - a^2*b*d*(20*c + d*x^3) + a*b^2*(10*c^2 - 12*c*d*x^3 
- 9*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + b*x 
^3*(c + d*x^3)*(11*a^2*d - 4*b^2*c*x^3 + a*b*(-6*c + 9*d*x^3))*(3*a*d*Appe 
llF1[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)])))/((a + b*x^3)*(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*Appe 
llF1[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)])))))/(a^2*(b*c - a*d)^2*(a + b*x^3 
)^(2/3))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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 [F]

\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {8}{3}} \left (d \,x^{3}+c \right )}d x\]

Input:

int(1/(b*x^3+a)^(8/3)/(d*x^3+c),x)
 

Output:

int(1/(b*x^3+a)^(8/3)/(d*x^3+c),x)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate(1/(b*x**3+a)**(8/3)/(d*x**3+c),x)
 

Output:

Integral(1/((a + b*x**3)**(8/3)*(c + d*x**3)), x)
 

Maxima [F]

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

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

Output:

integrate(1/((b*x^3 + a)^(8/3)*(d*x^3 + c)), x)
 

Giac [F]

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

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

Output:

integrate(1/((b*x^3 + a)^(8/3)*(d*x^3 + c)), x)
 

Mupad [F(-1)]

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

int(1/((a + b*x^3)^(8/3)*(c + d*x^3)),x)
 

Output:

int(1/((a + b*x^3)^(8/3)*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} c +\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} d \,x^{3}+2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b c \,x^{3}+2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b d \,x^{6}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} c \,x^{6}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} d \,x^{9}}d x \] Input:

int(1/(b*x^3+a)^(8/3)/(d*x^3+c),x)
 

Output:

int(1/((a + b*x**3)**(2/3)*a**2*c + (a + b*x**3)**(2/3)*a**2*d*x**3 + 2*(a 
 + b*x**3)**(2/3)*a*b*c*x**3 + 2*(a + b*x**3)**(2/3)*a*b*d*x**6 + (a + b*x 
**3)**(2/3)*b**2*c*x**6 + (a + b*x**3)**(2/3)*b**2*d*x**9),x)