Integrand size = 24, antiderivative size = 64 \[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=-\frac {\sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (-\frac {4}{3},\frac {1}{3},1,-\frac {1}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}} \] Output:
-1/4*(1+b*x^3/a)^(1/3)*AppellF1(-4/3,1/3,1,-1/3,-b*x^3/a,-d*x^3/c)/c/x^4/( b*x^3+a)^(1/3)
Leaf count is larger than twice the leaf count of optimal. \(183\) vs. \(2(64)=128\).
Time = 10.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.86 \[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\frac {5 c \left (a+b x^3\right ) \left (-a c+2 b c x^3+4 a d x^3\right )+5 \left (-b^2 c^2-2 a b c d+2 a^2 d^2\right ) x^6 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 b d (b c+2 a d) x^9 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{20 a^2 c^3 x^4 \sqrt [3]{a+b x^3}} \] Input:
Integrate[1/(x^5*(a + b*x^3)^(1/3)*(c + d*x^3)),x]
Output:
(5*c*(a + b*x^3)*(-(a*c) + 2*b*c*x^3 + 4*a*d*x^3) + 5*(-(b^2*c^2) - 2*a*b* c*d + 2*a^2*d^2)*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b *x^3)/a), -((d*x^3)/c)] - 2*b*d*(b*c + 2*a*d)*x^9*(1 + (b*x^3)/a)^(1/3)*Ap pellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a^2*c^3*x^4*(a + b*x^3)^(1/3))
Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1013, 1012}
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.
\(\displaystyle \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt [3]{\frac {b x^3}{a}+1} \int \frac {1}{x^5 \sqrt [3]{\frac {b x^3}{a}+1} \left (d x^3+c\right )}dx}{\sqrt [3]{a+b x^3}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {\sqrt [3]{\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {4}{3},\frac {1}{3},1,-\frac {1}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}}\) |
Input:
Int[1/(x^5*(a + b*x^3)^(1/3)*(c + d*x^3)),x]
Output:
-1/4*((1 + (b*x^3)/a)^(1/3)*AppellF1[-4/3, 1/3, 1, -1/3, -((b*x^3)/a), -(( d*x^3)/c)])/(c*x^4*(a + b*x^3)^(1/3))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {1}{x^{5} \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )}d x\]
Input:
int(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x)
Output:
int(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x)
Timed out. \[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{x^{5} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \] Input:
integrate(1/x**5/(b*x**3+a)**(1/3)/(d*x**3+c),x)
Output:
Integral(1/(x**5*(a + b*x**3)**(1/3)*(c + d*x**3)), x)
\[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{5}} \,d x } \] Input:
integrate(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate(1/((b*x^3 + a)^(1/3)*(d*x^3 + c)*x^5), x)
\[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{5}} \,d x } \] Input:
integrate(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")
Output:
integrate(1/((b*x^3 + a)^(1/3)*(d*x^3 + c)*x^5), x)
Timed out. \[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{x^5\,{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \] Input:
int(1/(x^5*(a + b*x^3)^(1/3)*(c + d*x^3)),x)
Output:
int(1/(x^5*(a + b*x^3)^(1/3)*(c + d*x^3)), x)
\[ \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} c \,x^{5}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} d \,x^{8}}d x \] Input:
int(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x)
Output:
int(1/((a + b*x**3)**(1/3)*c*x**5 + (a + b*x**3)**(1/3)*d*x**8),x)