Integrand size = 21, antiderivative size = 61 \[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\frac {x \sqrt [3]{1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{3},2,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 \sqrt [3]{a+b x^n}} \] Output:
x*(1+b*x^n/a)^(1/3)*AppellF1(1/n,1/3,2,1+1/n,-b*x^n/a,-d*x^n/c)/c^2/(a+b*x ^n)^(1/3)
Leaf count is larger than twice the leaf count of optimal. \(391\) vs. \(2(61)=122\).
Time = 0.87 (sec) , antiderivative size = 391, normalized size of antiderivative = 6.41 \[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {b d (-3+n) x^n \sqrt [3]{1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{(-b c+a d) (1+n)}-\frac {3 c \left (3 a d^2 n x^n \left (a+b x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c d n x^n \left (a+b x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {4}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a c (1+n) \left (-b c n+a d n+b d x^n\right ) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{(b c-a d) \left (c+d x^n\right ) \left (3 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{3},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {4}{3},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{3},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}\right )}{3 c^2 n \sqrt [3]{a+b x^n}} \] Input:
Integrate[1/((a + b*x^n)^(1/3)*(c + d*x^n)^2),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)])/((-(b*c) + a*d)*(1 + n)) - (3*c*(3 *a*d^2*n*x^n*(a + b*x^n)*AppellF1[1 + n^(-1), 1/3, 2, 2 + n^(-1), -((b*x^n )/a), -((d*x^n)/c)] + b*c*d*n*x^n*(a + b*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)*(-(b*c*n) + a*d*n + b*d*x^n)*AppellF1[n^(-1), 1/3, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c )]))/((b*c - a*d)*(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)]))))/(3*c^2*n*(a + b*x^ n)^(1/3))
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt [3]{\frac {b x^n}{a}+1} \int \frac {1}{\sqrt [3]{\frac {b x^n}{a}+1} \left (d x^n+c\right )^2}dx}{\sqrt [3]{a+b x^n}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt [3]{\frac {b x^n}{a}+1} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{3},2,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 \sqrt [3]{a+b x^n}}\) |
Input:
Int[1/((a + b*x^n)^(1/3)*(c + d*x^n)^2),x]
Output:
(x*(1 + (b*x^n)/a)^(1/3)*AppellF1[n^(-1), 1/3, 2, 1 + n^(-1), -((b*x^n)/a) , -((d*x^n)/c)])/(c^2*(a + b*x^n)^(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])
\[\int \frac {1}{\left (a +b \,x^{n}\right )^{\frac {1}{3}} \left (c +d \,x^{n}\right )^{2}}d x\]
Input:
int(1/(a+b*x^n)^(1/3)/(c+d*x^n)^2,x)
Output:
int(1/(a+b*x^n)^(1/3)/(c+d*x^n)^2,x)
\[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {1}{3}} {\left (d x^{n} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*x^n)^(1/3)/(c+d*x^n)^2,x, algorithm="fricas")
Output:
integral((b*x^n + a)^(2/3)/(b*d^2*x^(3*n) + a*c^2 + (2*b*c*d + a*d^2)*x^(2 *n) + (b*c^2 + 2*a*c*d)*x^n), x)
\[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\int \frac {1}{\sqrt [3]{a + b x^{n}} \left (c + d x^{n}\right )^{2}}\, dx \] Input:
integrate(1/(a+b*x**n)**(1/3)/(c+d*x**n)**2,x)
Output:
Integral(1/((a + b*x**n)**(1/3)*(c + d*x**n)**2), x)
\[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {1}{3}} {\left (d x^{n} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*x^n)^(1/3)/(c+d*x^n)^2,x, algorithm="maxima")
Output:
integrate(1/((b*x^n + a)^(1/3)*(d*x^n + c)^2), x)
\[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {1}{3}} {\left (d x^{n} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*x^n)^(1/3)/(c+d*x^n)^2,x, algorithm="giac")
Output:
integrate(1/((b*x^n + a)^(1/3)*(d*x^n + c)^2), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\int \frac {1}{{\left (a+b\,x^n\right )}^{1/3}\,{\left (c+d\,x^n\right )}^2} \,d x \] Input:
int(1/((a + b*x^n)^(1/3)*(c + d*x^n)^2),x)
Output:
int(1/((a + b*x^n)^(1/3)*(c + d*x^n)^2), x)
\[ \int \frac {1}{\sqrt [3]{a+b x^n} \left (c+d x^n\right )^2} \, dx=\int \frac {1}{x^{2 n} \left (x^{n} b +a \right )^{\frac {1}{3}} d^{2}+2 x^{n} \left (x^{n} b +a \right )^{\frac {1}{3}} c d +\left (x^{n} b +a \right )^{\frac {1}{3}} c^{2}}d x \] Input:
int(1/(a+b*x^n)^(1/3)/(c+d*x^n)^2,x)
Output:
int(1/(x**(2*n)*(x**n*b + a)**(1/3)*d**2 + 2*x**n*(x**n*b + a)**(1/3)*c*d + (x**n*b + a)**(1/3)*c**2),x)