\(\int \frac {\sqrt [3]{a+b x^n}}{(c+d x^n)^2} \, dx\) [110]

Optimal result

Integrand size = 21, antiderivative size = 61 \[ \int \frac {\sqrt [3]{a+b x^n}}{\left (c+d x^n\right )^2} \, dx=\frac {x \sqrt [3]{a+b x^n} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{3},2,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 \sqrt [3]{1+\frac {b x^n}{a}}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(364\) vs. \(2(61)=122\).

Time = 0.93 (sec) , antiderivative size = 364, normalized size of antiderivative = 5.97 \[ \int \frac {\sqrt [3]{a+b x^n}}{\left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {b (-3+2 n) x^n \left (1+\frac {b x^n}{a}\right )^{2/3} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {2}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+n}+\frac {3 c \left (3 a d n x^n \left (a+b x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {2}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+2 b c n x^n \left (a+b x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {5}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a c (1+n) \left (a n+b x^n\right ) \operatorname {AppellF1}\left (\frac {1}{n},\frac {2}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{\left (c+d x^n\right ) \left (3 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {2}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+2 b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {5}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {2}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}\right )}{3 c^2 n \left (a+b x^n\right )^{2/3}} \] Input:

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

Output:

(x*((b*(-3 + 2*n)*x^n*(1 + (b*x^n)/a)^(2/3)*AppellF1[1 + n^(-1), 2/3, 1, 2 
 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/(1 + n) + (3*c*(3*a*d*n*x^n*(a + b 
*x^n)*AppellF1[1 + n^(-1), 2/3, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] 
 + 2*b*c*n*x^n*(a + b*x^n)*AppellF1[1 + n^(-1), 5/3, 1, 2 + n^(-1), -((b*x 
^n)/a), -((d*x^n)/c)] - 3*a*c*(1 + n)*(a*n + b*x^n)*AppellF1[n^(-1), 2/3, 
1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))/((c + d*x^n)*(3*a*d*n*x^n*App 
ellF1[1 + n^(-1), 2/3, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + 2*b*c* 
n*x^n*AppellF1[1 + n^(-1), 5/3, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] 
 - 3*a*c*(1 + n)*AppellF1[n^(-1), 2/3, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x 
^n)/c)]))))/(3*c^2*n*(a + b*x^n)^(2/3))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 61, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

\(\displaystyle \frac {x \sqrt [3]{a+b x^n} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{3},2,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 \sqrt [3]{\frac {b x^n}{a}+1}}\)

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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