\(\int \frac {1}{(a+b x^3)^{2/3} (c+d x^3)^2} \, dx\) [158]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \left (a+b x^3\right )^{2/3}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(393\) vs. \(2(59)=118\).

Time = 10.33 (sec) , antiderivative size = 393, normalized size of antiderivative = 6.66 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\frac {4 a c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) \left (4 c \left (-3 b c+3 a d+b d x^3\right )+b d x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )-d x^4 \left (4 c \left (a+b x^3\right )+b x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}{12 c^2 (b c-a d) \left (a+b x^3\right )^{2/3} \left (c+d x^3\right ) \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \] Input:

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

Output:

(4*a*c*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)]*(4*c*(-3*b 
*c + 3*a*d + b*d*x^3) + b*d*x^3*(1 + (b*x^3)/a)^(2/3)*(c + d*x^3)*AppellF1 
[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]) - d*x^4*(4*c*(a + b*x^3) + 
 b*x^3*(1 + (b*x^3)/a)^(2/3)*(c + d*x^3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x 
^3)/a), -((d*x^3)/c)])*(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)]) 
)/(12*c^2*(b*c - a*d)*(a + b*x^3)^(2/3)*(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*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)])))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 59, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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