Integrand size = 22, antiderivative size = 64 \[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\frac {x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{a+b x^3}} \] Output:
1/2*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,1,5/3,-b*x^3/a,-d*x^3/c)/c/(b*x ^3+a)^(1/3)
Time = 9.75 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\frac {x^2 \sqrt [3]{\frac {a+b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{a+b x^3}} \] Input:
Integrate[x/((a + b*x^3)^(1/3)*(c + d*x^3)),x]
Output:
(x^2*((a + b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x ^3)/c)])/(2*c*(a + b*x^3)^(1/3))
Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt [3]{\frac {b x^3}{a}+1} \int \frac {x}{\sqrt [3]{\frac {b x^3}{a}+1} \left (d x^3+c\right )}dx}{\sqrt [3]{a+b x^3}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{a+b x^3}}\) |
Input:
Int[x/((a + b*x^3)^(1/3)*(c + d*x^3)),x]
Output:
(x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x ^3)/c)])/(2*c*(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])
\[\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )}d x\]
Input:
int(x/(b*x^3+a)^(1/3)/(d*x^3+c),x)
Output:
int(x/(b*x^3+a)^(1/3)/(d*x^3+c),x)
Timed out. \[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {x}{\sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \] Input:
integrate(x/(b*x**3+a)**(1/3)/(d*x**3+c),x)
Output:
Integral(x/((a + b*x**3)**(1/3)*(c + d*x**3)), x)
\[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}} \,d x } \] Input:
integrate(x/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate(x/((b*x^3 + a)^(1/3)*(d*x^3 + c)), x)
\[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}} \,d x } \] Input:
integrate(x/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")
Output:
integrate(x/((b*x^3 + a)^(1/3)*(d*x^3 + c)), x)
Timed out. \[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {x}{{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \] Input:
int(x/((a + b*x^3)^(1/3)*(c + d*x^3)),x)
Output:
int(x/((a + b*x^3)^(1/3)*(c + d*x^3)), x)
\[ \int \frac {x}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} c +\left (b \,x^{3}+a \right )^{\frac {1}{3}} d \,x^{3}}d x \] Input:
int(x/(b*x^3+a)^(1/3)/(d*x^3+c),x)
Output:
int(x/((a + b*x**3)**(1/3)*c + (a + b*x**3)**(1/3)*d*x**3),x)