\(\int \frac {1}{(a+b x^n)^{4/3} (c+d x^n)} \, dx\) [117]

Optimal result

Integrand size = 21, antiderivative size = 64 \[ \int \frac {1}{\left (a+b x^n\right )^{4/3} \left (c+d x^n\right )} \, dx=\frac {x \sqrt [3]{1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {4}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{a c \sqrt [3]{a+b x^n}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(382\) vs. \(2(64)=128\).

Time = 0.85 (sec) , antiderivative size = 382, normalized size of antiderivative = 5.97 \[ \int \frac {1}{\left (a+b x^n\right )^{4/3} \left (c+d x^n\right )} \, dx=-\frac {x \left (\frac {b d (-3+n) x^n \sqrt [3]{1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c (1+n)}+\frac {3 \left (3 a b d n x^n \left (c+d x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b^2 c n x^n \left (c+d x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {4}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \left (a d n-b \left (c n+3 d x^n\right )\right ) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{\left (c+d x^n\right ) \left (3 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {4}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}\right )}{a (-b c+a d) n \sqrt [3]{a+b x^n}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x^n + a)^(2/3)/(b^2*d*x^(3*n) + a^2*c + (b^2*c + 2*a*b*d)*x^(2 
*n) + (2*a*b*c + a^2*d)*x^n), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^n\right )^{4/3} \left (c+d x^n\right )} \, dx=\int \frac {1}{x^{2 n} \left (x^{n} b +a \right )^{\frac {1}{3}} b d +x^{n} \left (x^{n} b +a \right )^{\frac {1}{3}} a d +x^{n} \left (x^{n} b +a \right )^{\frac {1}{3}} b c +\left (x^{n} b +a \right )^{\frac {1}{3}} a c}d x \] Input:

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

Output:

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