Integrand size = 24, antiderivative size = 67 \[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {x^8 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {8}{3},\frac {4}{3},1,\frac {11}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 a c \sqrt [3]{a+b x^3}} \] Output:
1/8*x^8*(1+b*x^3/a)^(1/3)*AppellF1(8/3,4/3,1,11/3,-b*x^3/a,-d*x^3/c)/a/c/( b*x^3+a)^(1/3)
Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(67)=134\).
Time = 10.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.15 \[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {5 a c x^2-5 a c x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+(b c-2 a d) x^5 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 b c (b c-a d) \sqrt [3]{a+b x^3}} \] Input:
Integrate[x^7/((a + b*x^3)^(4/3)*(c + d*x^3)),x]
Output:
(5*a*c*x^2 - 5*a*c*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)] + (b*c - 2*a*d)*x^5*(1 + (b*x^3)/a)^(1/3)*Appell F1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*b*c*(b*c - a*d)*(a + b*x^3)^(1/3))
Time = 0.37 (sec) , antiderivative size = 67, 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.
\(\displaystyle \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt [3]{\frac {b x^3}{a}+1} \int \frac {x^7}{\left (\frac {b x^3}{a}+1\right )^{4/3} \left (d x^3+c\right )}dx}{a \sqrt [3]{a+b x^3}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^8 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {8}{3},\frac {4}{3},1,\frac {11}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 a c \sqrt [3]{a+b x^3}}\) |
Input:
Int[x^7/((a + b*x^3)^(4/3)*(c + d*x^3)),x]
Output:
(x^8*(1 + (b*x^3)/a)^(1/3)*AppellF1[8/3, 4/3, 1, 11/3, -((b*x^3)/a), -((d* x^3)/c)])/(8*a*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])
\[\int \frac {x^{7}}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )}d x\]
Input:
int(x^7/(b*x^3+a)^(4/3)/(d*x^3+c),x)
Output:
int(x^7/(b*x^3+a)^(4/3)/(d*x^3+c),x)
Timed out. \[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x^7/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {x^{7}}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \] Input:
integrate(x**7/(b*x**3+a)**(4/3)/(d*x**3+c),x)
Output:
Integral(x**7/((a + b*x**3)**(4/3)*(c + d*x**3)), x)
\[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}} \,d x } \] Input:
integrate(x^7/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate(x^7/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)
\[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}} \,d x } \] Input:
integrate(x^7/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")
Output:
integrate(x^7/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)
Timed out. \[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {x^7}{{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \] Input:
int(x^7/((a + b*x^3)^(4/3)*(c + d*x^3)),x)
Output:
int(x^7/((a + b*x^3)^(4/3)*(c + d*x^3)), x)
\[ \int \frac {x^7}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {x^{7}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a c +\left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b c \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b d \,x^{6}}d x \] Input:
int(x^7/(b*x^3+a)^(4/3)/(d*x^3+c),x)
Output:
int(x**7/((a + b*x**3)**(1/3)*a*c + (a + b*x**3)**(1/3)*a*d*x**3 + (a + b* x**3)**(1/3)*b*c*x**3 + (a + b*x**3)**(1/3)*b*d*x**6),x)