\(\int \frac {(a+b x^3)^{4/3}}{c+d x^3} \, dx\) [726]

Optimal result

Integrand size = 21, antiderivative size = 60 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {a x \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {4}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt [3]{1+\frac {b x^3}{a}}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(346\) vs. \(2(60)=120\).

Time = 0.21 (sec) , antiderivative size = 346, normalized size of antiderivative = 5.77 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {x \left (\frac {b (-2 b c+3 a d) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c}+\frac {4 \left (-4 a c \left (2 a^2 d+a b d x^3+b^2 x^3 \left (c+d x^3\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right ) \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}\right )}{8 d \left (a+b x^3\right )^{2/3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {937, 936}

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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((a + b*x**3)**(1/3)*b*x + 2*int((a + b*x**3)**(1/3)/(a*c + a*d*x**3 + b*c 
*x**3 + b*d*x**6),x)*a**2*d - int((a + b*x**3)**(1/3)/(a*c + a*d*x**3 + b* 
c*x**3 + b*d*x**6),x)*a*b*c + 3*int(((a + b*x**3)**(1/3)*x**3)/(a*c + a*d* 
x**3 + b*c*x**3 + b*d*x**6),x)*a*b*d - 2*int(((a + b*x**3)**(1/3)*x**3)/(a 
*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b**2*c)/(2*d)