\(\int \frac {1}{(a+b x^4)^{4/3} (c+d x^4)} \, dx\) [73]

Optimal result

Integrand size = 21, antiderivative size = 62 \[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\frac {x \sqrt [3]{1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {4}{3},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a c \sqrt [3]{a+b x^4}} \] Output:

x*(1+b*x^4/a)^(1/3)*AppellF1(1/4,4/3,1,5/4,-b*x^4/a,-d*x^4/c)/a/c/(b*x^4+a 
)^(1/3)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(329\) vs. \(2(62)=124\).

Time = 10.32 (sec) , antiderivative size = 329, normalized size of antiderivative = 5.31 \[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\frac {x \left (-\frac {b d x^4 \sqrt [3]{1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{3},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}+\frac {75 a c \left (-4 b c+4 a d-3 b d x^4\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{3},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+60 b x^4 \left (c+d x^4\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{3},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {4}{3},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (15 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{3},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )-4 x^4 \left (3 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{3},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {4}{3},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{20 a (-b c+a d) \sqrt [3]{a+b x^4}} \] Input:

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

Output:

(x*(-((b*d*x^4*(1 + (b*x^4)/a)^(1/3)*AppellF1[5/4, 1/3, 1, 9/4, -((b*x^4)/ 
a), -((d*x^4)/c)])/c) + (75*a*c*(-4*b*c + 4*a*d - 3*b*d*x^4)*AppellF1[1/4, 
 1/3, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 60*b*x^4*(c + d*x^4)*(3*a*d*Ap 
pellF1[5/4, 1/3, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 4 
/3, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))/((c + d*x^4)*(15*a*c*AppellF1[1/ 
4, 1/3, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - 4*x^4*(3*a*d*AppellF1[5/4, 1 
/3, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 4/3, 1, 9/4, - 
((b*x^4)/a), -((d*x^4)/c)])))))/(20*a*(-(b*c) + a*d)*(a + b*x^4)^(1/3))
 

Rubi [A] (verified)

Time = 0.31 (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.

\(\displaystyle \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx\)

\(\Big \downarrow \) 937

\(\displaystyle \frac {\sqrt [3]{\frac {b x^4}{a}+1} \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{4/3} \left (d x^4+c\right )}dx}{a \sqrt [3]{a+b x^4}}\)

\(\Big \downarrow \) 936

\(\displaystyle \frac {x \sqrt [3]{\frac {b x^4}{a}+1} \operatorname {AppellF1}\left (\frac {1}{4},\frac {4}{3},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a c \sqrt [3]{a+b x^4}}\)

Input:

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

Output:

(x*(1 + (b*x^4)/a)^(1/3)*AppellF1[1/4, 4/3, 1, 5/4, -((b*x^4)/a), -((d*x^4 
)/c)])/(a*c*(a + b*x^4)^(1/3))
 

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]

\[\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {4}{3}} \left (d \,x^{4}+c \right )}d x\]

Input:

int(1/(b*x^4+a)^(4/3)/(d*x^4+c),x)
 

Output:

int(1/(b*x^4+a)^(4/3)/(d*x^4+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^4+a)^(4/3)/(d*x^4+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\int \frac {1}{\left (a + b x^{4}\right )^{\frac {4}{3}} \left (c + d x^{4}\right )}\, dx \] Input:

integrate(1/(b*x**4+a)**(4/3)/(d*x**4+c),x)
 

Output:

Integral(1/((a + b*x**4)**(4/3)*(c + d*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {4}{3}} {\left (d x^{4} + c\right )}} \,d x } \] Input:

integrate(1/(b*x^4+a)^(4/3)/(d*x^4+c),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^4 + a)^(4/3)*(d*x^4 + c)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {4}{3}} {\left (d x^{4} + c\right )}} \,d x } \] Input:

integrate(1/(b*x^4+a)^(4/3)/(d*x^4+c),x, algorithm="giac")
 

Output:

integrate(1/((b*x^4 + a)^(4/3)*(d*x^4 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{4/3}\,\left (d\,x^4+c\right )} \,d x \] Input:

int(1/((a + b*x^4)^(4/3)*(c + d*x^4)),x)
 

Output:

int(1/((a + b*x^4)^(4/3)*(c + d*x^4)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{4/3} \left (c+d x^4\right )} \, dx=\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{3}} a c +\left (b \,x^{4}+a \right )^{\frac {1}{3}} a d \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {1}{3}} b c \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {1}{3}} b d \,x^{8}}d x \] Input:

int(1/(b*x^4+a)^(4/3)/(d*x^4+c),x)
 

Output:

int(1/((a + b*x**4)**(1/3)*a*c + (a + b*x**4)**(1/3)*a*d*x**4 + (a + b*x** 
4)**(1/3)*b*c*x**4 + (a + b*x**4)**(1/3)*b*d*x**8),x)