\(\int \frac {1}{(a+b x^3)^{8/3} (c+d x^3)^3} \, dx\) [173]

Optimal result

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

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(515\) vs. \(2(62)=124\).

Time = 11.69 (sec) , antiderivative size = 515, normalized size of antiderivative = 8.31 \[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (b d \left (36 b^3 c^3-171 a b^2 c^2 d-110 a^2 b c d^2+25 a^3 d^3\right ) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 c \left (\frac {36 b^5 c^3 x^3 \left (c+d x^3\right )^2+9 a b^4 c^2 \left (6 c-19 d x^3\right ) \left (c+d x^3\right )^2+5 a^5 d^4 \left (8 c+5 d x^3\right )+5 a^3 b^2 d^3 x^3 \left (-50 c^2-36 c d x^3+5 d^2 x^6\right )+5 a^4 b d^3 \left (-25 c^2-6 c d x^3+10 d^2 x^6\right )-a^2 b^3 c d \left (189 c^3+378 c^2 d x^3+314 c d^2 x^6+110 d^3 x^9\right )}{a+b x^3}+\frac {4 a c \left (36 b^4 c^4-171 a b^3 c^3 d+540 a^2 b^2 c^2 d^2-235 a^3 b c d^3+50 a^4 d^4\right ) \left (c+d x^3\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{\left (c+d x^3\right )^2}\right )}{360 a^2 c^3 (b c-a d)^4 \left (a+b x^3\right )^{2/3}} \] Input:

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

Output:

(x*(b*d*(36*b^3*c^3 - 171*a*b^2*c^2*d - 110*a^2*b*c*d^2 + 25*a^3*d^3)*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 
)] + (4*c*((36*b^5*c^3*x^3*(c + d*x^3)^2 + 9*a*b^4*c^2*(6*c - 19*d*x^3)*(c 
 + d*x^3)^2 + 5*a^5*d^4*(8*c + 5*d*x^3) + 5*a^3*b^2*d^3*x^3*(-50*c^2 - 36* 
c*d*x^3 + 5*d^2*x^6) + 5*a^4*b*d^3*(-25*c^2 - 6*c*d*x^3 + 10*d^2*x^6) - a^ 
2*b^3*c*d*(189*c^3 + 378*c^2*d*x^3 + 314*c*d^2*x^6 + 110*d^3*x^9))/(a + b* 
x^3) + (4*a*c*(36*b^4*c^4 - 171*a*b^3*c^3*d + 540*a^2*b^2*c^2*d^2 - 235*a^ 
3*b*c*d^3 + 50*a^4*d^4)*(c + d*x^3)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a 
), -((d*x^3)/c)])/(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)]))))/(c + d* 
x^3)^2))/(360*a^2*c^3*(b*c - a*d)^4*(a + b*x^3)^(2/3))
 

Rubi [A] (verified)

Time = 0.29 (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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} c^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} c^{2} d \,x^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} c \,d^{2} x^{6}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} d^{3} x^{9}+2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b \,c^{3} x^{3}+6 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b \,c^{2} d \,x^{6}+6 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b c \,d^{2} x^{9}+2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b \,d^{3} x^{12}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} c^{3} x^{6}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} c^{2} d \,x^{9}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} c \,d^{2} x^{12}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} d^{3} x^{15}}d x \] Input:

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

Output:

int(1/((a + b*x**3)**(2/3)*a**2*c**3 + 3*(a + b*x**3)**(2/3)*a**2*c**2*d*x 
**3 + 3*(a + b*x**3)**(2/3)*a**2*c*d**2*x**6 + (a + b*x**3)**(2/3)*a**2*d* 
*3*x**9 + 2*(a + b*x**3)**(2/3)*a*b*c**3*x**3 + 6*(a + b*x**3)**(2/3)*a*b* 
c**2*d*x**6 + 6*(a + b*x**3)**(2/3)*a*b*c*d**2*x**9 + 2*(a + b*x**3)**(2/3 
)*a*b*d**3*x**12 + (a + b*x**3)**(2/3)*b**2*c**3*x**6 + 3*(a + b*x**3)**(2 
/3)*b**2*c**2*d*x**9 + 3*(a + b*x**3)**(2/3)*b**2*c*d**2*x**12 + (a + b*x* 
*3)**(2/3)*b**2*d**3*x**15),x)