Integrand size = 24, antiderivative size = 64 \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {x^5 \left (a+b x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 c \left (1+\frac {b x^3}{a}\right )^{2/3}} \] Output:
1/5*x^5*(b*x^3+a)^(2/3)*AppellF1(5/3,-2/3,1,8/3,-b*x^3/a,-d*x^3/c)/c/(1+b* x^3/a)^(2/3)
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(64)=128\).
Time = 6.99 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.20 \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {5 c x^2 \left (a+b x^3\right )-5 a c x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 (-2 b c+a d) x^5 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{20 c d \sqrt [3]{a+b x^3}} \] Input:
Integrate[(x^4*(a + b*x^3)^(2/3))/(c + d*x^3),x]
Output:
(5*c*x^2*(a + b*x^3) - 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)] + 2*(-2*b*c + a*d)*x^5*(1 + (b*x^3)/a) ^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*c*d*(a + b*x^3)^(1/3))
Time = 0.35 (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.
\(\displaystyle \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\left (a+b x^3\right )^{2/3} \int \frac {x^4 \left (\frac {b x^3}{a}+1\right )^{2/3}}{d x^3+c}dx}{\left (\frac {b x^3}{a}+1\right )^{2/3}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^5 \left (a+b x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 c \left (\frac {b x^3}{a}+1\right )^{2/3}}\) |
Input:
Int[(x^4*(a + b*x^3)^(2/3))/(c + d*x^3),x]
Output:
(x^5*(a + b*x^3)^(2/3)*AppellF1[5/3, -2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3) /c)])/(5*c*(1 + (b*x^3)/a)^(2/3))
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])
\[\int \frac {x^{4} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{d \,x^{3}+c}d x\]
Input:
int(x^4*(b*x^3+a)^(2/3)/(d*x^3+c),x)
Output:
int(x^4*(b*x^3+a)^(2/3)/(d*x^3+c),x)
Timed out. \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\text {Timed out} \] Input:
integrate(x^4*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int \frac {x^{4} \left (a + b x^{3}\right )^{\frac {2}{3}}}{c + d x^{3}}\, dx \] Input:
integrate(x**4*(b*x**3+a)**(2/3)/(d*x**3+c),x)
Output:
Integral(x**4*(a + b*x**3)**(2/3)/(c + d*x**3), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{4}}{d x^{3} + c} \,d x } \] Input:
integrate(x^4*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(2/3)*x^4/(d*x^3 + c), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{4}}{d x^{3} + c} \,d x } \] Input:
integrate(x^4*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(2/3)*x^4/(d*x^3 + c), x)
Timed out. \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int \frac {x^4\,{\left (b\,x^3+a\right )}^{2/3}}{d\,x^3+c} \,d x \] Input:
int((x^4*(a + b*x^3)^(2/3))/(c + d*x^3),x)
Output:
int((x^4*(a + b*x^3)^(2/3))/(c + d*x^3), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{2}+2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a d -4 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b c -2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a c}{4 d} \] Input:
int(x^4*(b*x^3+a)^(2/3)/(d*x^3+c),x)
Output:
((a + b*x**3)**(2/3)*x**2 + 2*int(((a + b*x**3)**(2/3)*x**4)/(a*c + a*d*x* *3 + b*c*x**3 + b*d*x**6),x)*a*d - 4*int(((a + b*x**3)**(2/3)*x**4)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b*c - 2*int(((a + b*x**3)**(2/3)*x)/(a *c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*c)/(4*d)