\(\int \frac {(a+b x^3)^{4/3}}{x^6 (c+d x^3)} \, dx\) [728]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^6 \left (c+d x^3\right )} \, dx=-\frac {a \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (-\frac {5}{3},-\frac {4}{3},1,-\frac {2}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 c x^5 \sqrt [3]{1+\frac {b x^3}{a}}} \] Output:

-1/5*a*(b*x^3+a)^(1/3)*AppellF1(-5/3,-4/3,1,-2/3,-b*x^3/a,-d*x^3/c)/c/x^5/ 
(1+b*x^3/a)^(1/3)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(286\) vs. \(2(65)=130\).

Time = 10.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 4.40 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^6 \left (c+d x^3\right )} \, dx=\frac {-\frac {4 \left (a+b x^3\right ) \left (2 a c+6 b c x^3-5 a d x^3\right )}{c^2 x^5}+\frac {b d (-6 b c+5 a d) x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3}-\frac {16 a \left (4 b^2 c^2-15 a b c d+10 a^2 d^2\right ) x \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \left (c+d x^3\right ) \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}}{40 \left (a+b x^3\right )^{2/3}} \] Input:

Integrate[(a + b*x^3)^(4/3)/(x^6*(c + d*x^3)),x]
 

Output:

((-4*(a + b*x^3)*(2*a*c + 6*b*c*x^3 - 5*a*d*x^3))/(c^2*x^5) + (b*d*(-6*b*c 
 + 5*a*d)*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a 
), -((d*x^3)/c)])/c^3 - (16*a*(4*b^2*c^2 - 15*a*b*c*d + 10*a^2*d^2)*x*Appe 
llF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(c + d*x^3)*(-4*a*c 
*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*Appel 
lF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/ 
3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(40*(a + b*x^3)^(2/3))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 65, 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.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{4/3}}{x^6 \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {a \sqrt [3]{a+b x^3} \int \frac {\left (\frac {b x^3}{a}+1\right )^{4/3}}{x^6 \left (d x^3+c\right )}dx}{\sqrt [3]{\frac {b x^3}{a}+1}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {a \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (-\frac {5}{3},-\frac {4}{3},1,-\frac {2}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 c x^5 \sqrt [3]{\frac {b x^3}{a}+1}}\)

Input:

Int[(a + b*x^3)^(4/3)/(x^6*(c + d*x^3)),x]
 

Output:

-1/5*(a*(a + b*x^3)^(1/3)*AppellF1[-5/3, -4/3, 1, -2/3, -((b*x^3)/a), -((d 
*x^3)/c)])/(c*x^5*(1 + (b*x^3)/a)^(1/3))
 

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]

\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{x^{6} \left (d \,x^{3}+c \right )}d x\]

Input:

int((b*x^3+a)^(4/3)/x^6/(d*x^3+c),x)
 

Output:

int((b*x^3+a)^(4/3)/x^6/(d*x^3+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^6 \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^3+a)^(4/3)/x^6/(d*x^3+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**3+a)**(4/3)/x**6/(d*x**3+c),x)
 

Output:

Integral((a + b*x**3)**(4/3)/(x**6*(c + d*x**3)), x)
 

Maxima [F]

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

integrate((b*x^3+a)^(4/3)/x^6/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^6), x)
 

Giac [F]

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

integrate((b*x^3+a)^(4/3)/x^6/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^6 \left (c+d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{4/3}}{x^6\,\left (d\,x^3+c\right )} \,d x \] Input:

int((a + b*x^3)^(4/3)/(x^6*(c + d*x^3)),x)
 

Output:

int((a + b*x^3)^(4/3)/(x^6*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^6 \left (c+d x^3\right )} \, dx=\frac {-\left (b \,x^{3}+a \right )^{\frac {1}{3}} a -25 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{5 a b \,d^{2} x^{9}+4 b^{2} c d \,x^{9}+5 a^{2} d^{2} x^{6}+9 a b c d \,x^{6}+4 b^{2} c^{2} x^{6}+5 a^{2} c d \,x^{3}+4 a b \,c^{2} x^{3}}d x \right ) a^{3} d^{2} x^{5}+10 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{5 a b \,d^{2} x^{9}+4 b^{2} c d \,x^{9}+5 a^{2} d^{2} x^{6}+9 a b c d \,x^{6}+4 b^{2} c^{2} x^{6}+5 a^{2} c d \,x^{3}+4 a b \,c^{2} x^{3}}d x \right ) a^{2} b c d \,x^{5}+24 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{5 a b \,d^{2} x^{9}+4 b^{2} c d \,x^{9}+5 a^{2} d^{2} x^{6}+9 a b c d \,x^{6}+4 b^{2} c^{2} x^{6}+5 a^{2} c d \,x^{3}+4 a b \,c^{2} x^{3}}d x \right ) a \,b^{2} c^{2} x^{5}-20 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{5 a b \,d^{2} x^{6}+4 b^{2} c d \,x^{6}+5 a^{2} d^{2} x^{3}+9 a b c d \,x^{3}+4 b^{2} c^{2} x^{3}+5 a^{2} c d +4 a b \,c^{2}}d x \right ) a^{2} b \,d^{2} x^{5}+9 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{5 a b \,d^{2} x^{6}+4 b^{2} c d \,x^{6}+5 a^{2} d^{2} x^{3}+9 a b c d \,x^{3}+4 b^{2} c^{2} x^{3}+5 a^{2} c d +4 a b \,c^{2}}d x \right ) a \,b^{2} c d \,x^{5}+20 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{5 a b \,d^{2} x^{6}+4 b^{2} c d \,x^{6}+5 a^{2} d^{2} x^{3}+9 a b c d \,x^{3}+4 b^{2} c^{2} x^{3}+5 a^{2} c d +4 a b \,c^{2}}d x \right ) b^{3} c^{2} x^{5}}{5 c \,x^{5}} \] Input:

int((b*x^3+a)^(4/3)/x^6/(d*x^3+c),x)
 

Output:

( - (a + b*x**3)**(1/3)*a - 25*int((a + b*x**3)**(1/3)/(5*a**2*c*d*x**3 + 
5*a**2*d**2*x**6 + 4*a*b*c**2*x**3 + 9*a*b*c*d*x**6 + 5*a*b*d**2*x**9 + 4* 
b**2*c**2*x**6 + 4*b**2*c*d*x**9),x)*a**3*d**2*x**5 + 10*int((a + b*x**3)* 
*(1/3)/(5*a**2*c*d*x**3 + 5*a**2*d**2*x**6 + 4*a*b*c**2*x**3 + 9*a*b*c*d*x 
**6 + 5*a*b*d**2*x**9 + 4*b**2*c**2*x**6 + 4*b**2*c*d*x**9),x)*a**2*b*c*d* 
x**5 + 24*int((a + b*x**3)**(1/3)/(5*a**2*c*d*x**3 + 5*a**2*d**2*x**6 + 4* 
a*b*c**2*x**3 + 9*a*b*c*d*x**6 + 5*a*b*d**2*x**9 + 4*b**2*c**2*x**6 + 4*b* 
*2*c*d*x**9),x)*a*b**2*c**2*x**5 - 20*int((a + b*x**3)**(1/3)/(5*a**2*c*d 
+ 5*a**2*d**2*x**3 + 4*a*b*c**2 + 9*a*b*c*d*x**3 + 5*a*b*d**2*x**6 + 4*b** 
2*c**2*x**3 + 4*b**2*c*d*x**6),x)*a**2*b*d**2*x**5 + 9*int((a + b*x**3)**( 
1/3)/(5*a**2*c*d + 5*a**2*d**2*x**3 + 4*a*b*c**2 + 9*a*b*c*d*x**3 + 5*a*b* 
d**2*x**6 + 4*b**2*c**2*x**3 + 4*b**2*c*d*x**6),x)*a*b**2*c*d*x**5 + 20*in 
t((a + b*x**3)**(1/3)/(5*a**2*c*d + 5*a**2*d**2*x**3 + 4*a*b*c**2 + 9*a*b* 
c*d*x**3 + 5*a*b*d**2*x**6 + 4*b**2*c**2*x**3 + 4*b**2*c*d*x**6),x)*b**3*c 
**2*x**5)/(5*c*x**5)