\(\int \frac {x^7 (a+b x^3)^{2/3}}{c+d x^3} \, dx\) [706]

Optimal result

Integrand size = 24, antiderivative size = 64 \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {x^8 \left (a+b x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {8}{3},-\frac {2}{3},1,\frac {11}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 c \left (1+\frac {b x^3}{a}\right )^{2/3}} \] Output:

1/8*x^8*(b*x^3+a)^(2/3)*AppellF1(8/3,-2/3,1,11/3,-b*x^3/a,-d*x^3/c)/c/(1+b 
*x^3/a)^(2/3)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(64)=128\).

Time = 7.62 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.83 \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {x^2 \left (5 c \left (a+b x^3\right ) \left (-7 b c+2 a d+4 b d x^3\right )+5 a c (7 b c-2 a d) \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 \left (-14 b^2 c^2+7 a b c d+2 a^2 d^2\right ) x^3 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}{140 b c d^2 \sqrt [3]{a+b x^3}} \] Input:

Integrate[(x^7*(a + b*x^3)^(2/3))/(c + d*x^3),x]
 

Output:

(x^2*(5*c*(a + b*x^3)*(-7*b*c + 2*a*d + 4*b*d*x^3) + 5*a*c*(7*b*c - 2*a*d) 
*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/ 
c)] - 2*(-14*b^2*c^2 + 7*a*b*c*d + 2*a^2*d^2)*x^3*(1 + (b*x^3)/a)^(1/3)*Ap 
pellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]))/(140*b*c*d^2*(a + b 
*x^3)^(1/3))
 

Rubi [A] (verified)

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.

\(\displaystyle \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {\left (a+b x^3\right )^{2/3} \int \frac {x^7 \left (\frac {b x^3}{a}+1\right )^{2/3}}{d x^3+c}dx}{\left (\frac {b x^3}{a}+1\right )^{2/3}}\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^8 \left (a+b x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {8}{3},-\frac {2}{3},1,\frac {11}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 c \left (\frac {b x^3}{a}+1\right )^{2/3}}\)

Input:

Int[(x^7*(a + b*x^3)^(2/3))/(c + d*x^3),x]
 

Output:

(x^8*(a + b*x^3)^(2/3)*AppellF1[8/3, -2/3, 1, 11/3, -((b*x^3)/a), -((d*x^3 
)/c)])/(8*c*(1 + (b*x^3)/a)^(2/3))
 

Defintions of rubi rules used

rule 1012

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])
 

rule 1013

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])
 

Maple [F]

\[\int \frac {x^{7} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{d \,x^{3}+c}d x\]

Input:

int(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x)
 

Output:

int(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\text {Timed out} \] Input:

integrate(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int \frac {x^{7} \left (a + b x^{3}\right )^{\frac {2}{3}}}{c + d x^{3}}\, dx \] Input:

integrate(x**7*(b*x**3+a)**(2/3)/(d*x**3+c),x)
 

Output:

Integral(x**7*(a + b*x**3)**(2/3)/(c + d*x**3), x)
 

Maxima [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{7}}{d x^{3} + c} \,d x } \] Input:

integrate(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(2/3)*x^7/(d*x^3 + c), x)
 

Giac [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{7}}{d x^{3} + c} \,d x } \] Input:

integrate(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(2/3)*x^7/(d*x^3 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int \frac {x^7\,{\left (b\,x^3+a\right )}^{2/3}}{d\,x^3+c} \,d x \] Input:

int((x^7*(a + b*x^3)^(2/3))/(c + d*x^3),x)
 

Output:

int((x^7*(a + b*x^3)^(2/3))/(c + d*x^3), x)
 

Reduce [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a d \,x^{2}-7 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b c \,x^{2}+4 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b d \,x^{5}-4 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} d^{2}-14 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b c d +28 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{2} c^{2}-4 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} c d +14 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b \,c^{2}}{28 b \,d^{2}} \] Input:

int(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x)
 

Output:

(2*(a + b*x**3)**(2/3)*a*d*x**2 - 7*(a + b*x**3)**(2/3)*b*c*x**2 + 4*(a + 
b*x**3)**(2/3)*b*d*x**5 - 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)*a**2*d**2 - 14*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*b*c*d + 28*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**2*c**2 - 4*int(((a 
 + b*x**3)**(2/3)*x)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a**2*c*d + 
14*int(((a + b*x**3)**(2/3)*x)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a 
*b*c**2)/(28*b*d**2)