\(\int \frac {\sqrt [3]{a+b x^3}}{(c+d x^3)^3} \, dx\) [170]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int \frac {\sqrt [3]{a+b x^3}}{\left (c+d x^3\right )^3} \, dx=\frac {x \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \sqrt [3]{1+\frac {b x^3}{a}}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(407\) vs. \(2(59)=118\).

Time = 10.65 (sec) , antiderivative size = 407, normalized size of antiderivative = 6.90 \[ \int \frac {\sqrt [3]{a+b x^3}}{\left (c+d x^3\right )^3} \, dx=\frac {-b (-4 b c+5 a d) x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {c \left (16 a c x \left (-b^2 c x^3 \left (7 c+4 d x^3\right )+3 a^2 d \left (6 c+5 d x^3\right )+a b \left (-18 c^2-7 c d x^3+5 d^2 x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-4 x^4 \left (a+b x^3\right ) \left (-b c \left (7 c+4 d x^3\right )+a d \left (8 c+5 d x^3\right )\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right )^2 \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}}{72 c^3 (b c-a d) \left (a+b x^3\right )^{2/3}} \] Input:

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

Output:

(-(b*(-4*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*(16*a*c*x*(-(b^2*c*x^3*(7*c + 4*d*x^3)) 
 + 3*a^2*d*(6*c + 5*d*x^3) + a*b*(-18*c^2 - 7*c*d*x^3 + 5*d^2*x^6))*Appell 
F1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 4*x^4*(a + b*x^3)*(-(b* 
c*(7*c + 4*d*x^3)) + a*d*(8*c + 5*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)^2*(-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)]))))/(72*c^3*(b*c - a*d)*(a + b*x^3)^(2/3))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 59, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (d \,x^{3}+c \right )^{3}}d x\]

Input:

int((b*x^3+a)^(1/3)/(d*x^3+c)^3,x)
 

Output:

int((b*x^3+a)^(1/3)/(d*x^3+c)^3,x)
 

Fricas [F(-1)]

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

integrate((b*x^3+a)^(1/3)/(d*x^3+c)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**3+a)**(1/3)/(d*x**3+c)**3,x)
 

Output:

Integral((a + b*x**3)**(1/3)/(c + d*x**3)**3, x)
 

Maxima [F]

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

integrate((b*x^3+a)^(1/3)/(d*x^3+c)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(1/3)/(d*x^3 + c)^3, x)
 

Giac [F]

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

integrate((b*x^3+a)^(1/3)/(d*x^3+c)^3,x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(1/3)/(d*x^3 + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/3}}{{\left (d\,x^3+c\right )}^3} \,d x \] Input:

int((a + b*x^3)^(1/3)/(c + d*x^3)^3,x)
 

Output:

int((a + b*x^3)^(1/3)/(c + d*x^3)^3, x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{d^{3} x^{9}+3 c \,d^{2} x^{6}+3 c^{2} d \,x^{3}+c^{3}}d x \] Input:

int((b*x^3+a)^(1/3)/(d*x^3+c)^3,x)
 

Output:

int((a + b*x**3)**(1/3)/(c**3 + 3*c**2*d*x**3 + 3*c*d**2*x**6 + d**3*x**9) 
,x)