Integrand size = 21, antiderivative size = 61 \[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\frac {x \sqrt [3]{a+b x^n} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c \sqrt [3]{1+\frac {b x^n}{a}}} \] Output:
x*(a+b*x^n)^(1/3)*AppellF1(1/n,-1/3,1,1+1/n,-b*x^n/a,-d*x^n/c)/c/(1+b*x^n/ a)^(1/3)
Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(61)=122\).
Time = 0.37 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.97 \[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\frac {3 a c (1+n) x \sqrt [3]{a+b x^n} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{\left (c+d x^n\right ) \left (-3 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-\frac {1}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {2}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+3 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )} \] Input:
Integrate[(a + b*x^n)^(1/3)/(c + d*x^n),x]
Output:
(3*a*c*(1 + n)*x*(a + b*x^n)^(1/3)*AppellF1[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), 2/3, 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)]))
Time = 0.33 (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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt [3]{a+b x^n} \int \frac {\sqrt [3]{\frac {b x^n}{a}+1}}{d x^n+c}dx}{\sqrt [3]{\frac {b x^n}{a}+1}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt [3]{a+b x^n} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c \sqrt [3]{\frac {b x^n}{a}+1}}\) |
Input:
Int[(a + b*x^n)^(1/3)/(c + d*x^n),x]
Output:
(x*(a + b*x^n)^(1/3)*AppellF1[n^(-1), -1/3, 1, 1 + n^(-1), -((b*x^n)/a), - ((d*x^n)/c)])/(c*(1 + (b*x^n)/a)^(1/3))
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])
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])
\[\int \frac {\left (a +b \,x^{n}\right )^{\frac {1}{3}}}{c +d \,x^{n}}d x\]
Input:
int((a+b*x^n)^(1/3)/(c+d*x^n),x)
Output:
int((a+b*x^n)^(1/3)/(c+d*x^n),x)
\[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {1}{3}}}{d x^{n} + c} \,d x } \] Input:
integrate((a+b*x^n)^(1/3)/(c+d*x^n),x, algorithm="fricas")
Output:
integral((b*x^n + a)^(1/3)/(d*x^n + c), x)
\[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\int \frac {\sqrt [3]{a + b x^{n}}}{c + d x^{n}}\, dx \] Input:
integrate((a+b*x**n)**(1/3)/(c+d*x**n),x)
Output:
Integral((a + b*x**n)**(1/3)/(c + d*x**n), x)
\[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {1}{3}}}{d x^{n} + c} \,d x } \] Input:
integrate((a+b*x^n)^(1/3)/(c+d*x^n),x, algorithm="maxima")
Output:
integrate((b*x^n + a)^(1/3)/(d*x^n + c), x)
\[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {1}{3}}}{d x^{n} + c} \,d x } \] Input:
integrate((a+b*x^n)^(1/3)/(c+d*x^n),x, algorithm="giac")
Output:
integrate((b*x^n + a)^(1/3)/(d*x^n + c), x)
Timed out. \[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\int \frac {{\left (a+b\,x^n\right )}^{1/3}}{c+d\,x^n} \,d x \] Input:
int((a + b*x^n)^(1/3)/(c + d*x^n),x)
Output:
int((a + b*x^n)^(1/3)/(c + d*x^n), x)
\[ \int \frac {\sqrt [3]{a+b x^n}}{c+d x^n} \, dx=\int \frac {\left (x^{n} b +a \right )^{\frac {1}{3}}}{x^{n} d +c}d x \] Input:
int((a+b*x^n)^(1/3)/(c+d*x^n),x)
Output:
int((x**n*b + a)**(1/3)/(x**n*d + c),x)