\(\int \frac {\sqrt [4]{a+b x^3}}{(c+d x^3)^{19/12}} \, dx\) [176]

Optimal result

Integrand size = 23, antiderivative size = 87 \[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\frac {x \sqrt [4]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{3},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \left (c+d x^3\right )^{7/12}} \] Output:

x*(b*x^3+a)^(1/4)*hypergeom([-1/4, 1/3],[4/3],-(-a*d+b*c)*x^3/a/(d*x^3+c)) 
/c/(c*(b*x^3+a)/a/(d*x^3+c))^(1/4)/(d*x^3+c)^(7/12)
 

Mathematica [A] (warning: unable to verify)

Time = 3.86 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\frac {x \sqrt [4]{a+b x^3} \sqrt [4]{1+\frac {d x^3}{c}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{3},\frac {4}{3},\frac {(-b c+a d) x^3}{a \left (c+d x^3\right )}\right )}{c \sqrt [4]{1+\frac {b x^3}{a}} \left (c+d x^3\right )^{7/12}} \] Input:

Integrate[(a + b*x^3)^(1/4)/(c + d*x^3)^(19/12),x]
 

Output:

(x*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(1/4)*Hypergeometric2F1[-1/4, 1/3, 4/ 
3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c*(1 + (b*x^3)/a)^(1/4)*(c + d* 
x^3)^(7/12))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {903, 905}

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.

\(\displaystyle \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx\)

\(\Big \downarrow \) 903

\(\displaystyle \frac {3 a \int \frac {1}{\left (b x^3+a\right )^{3/4} \left (d x^3+c\right )^{7/12}}dx}{7 c}+\frac {4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}}\)

\(\Big \downarrow \) 905

\(\displaystyle \frac {3 a x \left (c+d x^3\right )^{5/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{7 c^2 \left (a+b x^3\right )^{3/4}}+\frac {4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}}\)

Input:

Int[(a + b*x^3)^(1/4)/(c + d*x^3)^(19/12),x]
 

Output:

(4*x*(a + b*x^3)^(1/4))/(7*c*(c + d*x^3)^(7/12)) + (3*a*x*((c*(a + b*x^3)) 
/(a*(c + d*x^3)))^(3/4)*(c + d*x^3)^(5/12)*Hypergeometric2F1[1/3, 3/4, 4/3 
, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))])/(7*c^2*(a + b*x^3)^(3/4))
 

Defintions of rubi rules used

rule 903

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

rule 905

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[x*((a + b*x^n)^p/(c*(c*((a + b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n) 
^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c 
+ d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && 
EqQ[n*(p + q + 1) + 1, 0]
 

Maple [F]

\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{4}}}{\left (d \,x^{3}+c \right )^{\frac {19}{12}}}d x\]

Input:

int((b*x^3+a)^(1/4)/(d*x^3+c)^(19/12),x)
 

Output:

int((b*x^3+a)^(1/4)/(d*x^3+c)^(19/12),x)
 

Fricas [F]

\[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}}}{{\left (d x^{3} + c\right )}^{\frac {19}{12}}} \,d x } \] Input:

integrate((b*x^3+a)^(1/4)/(d*x^3+c)^(19/12),x, algorithm="fricas")
 

Output:

integral((b*x^3 + a)^(1/4)*(d*x^3 + c)^(5/12)/(d^2*x^6 + 2*c*d*x^3 + c^2), 
 x)
 

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\int \frac {\sqrt [4]{a + b x^{3}}}{\left (c + d x^{3}\right )^{\frac {19}{12}}}\, dx \] Input:

integrate((b*x**3+a)**(1/4)/(d*x**3+c)**(19/12),x)
 

Output:

Integral((a + b*x**3)**(1/4)/(c + d*x**3)**(19/12), x)
 

Maxima [F]

\[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}}}{{\left (d x^{3} + c\right )}^{\frac {19}{12}}} \,d x } \] Input:

integrate((b*x^3+a)^(1/4)/(d*x^3+c)^(19/12),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(1/4)/(d*x^3 + c)^(19/12), x)
 

Giac [F]

\[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}}}{{\left (d x^{3} + c\right )}^{\frac {19}{12}}} \,d x } \] Input:

integrate((b*x^3+a)^(1/4)/(d*x^3+c)^(19/12),x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(1/4)/(d*x^3 + c)^(19/12), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/4}}{{\left (d\,x^3+c\right )}^{19/12}} \,d x \] Input:

int((a + b*x^3)^(1/4)/(c + d*x^3)^(19/12),x)
 

Output:

int((a + b*x^3)^(1/4)/(c + d*x^3)^(19/12), x)
 

Reduce [F]

\[ \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx=\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{4}}}{\left (d \,x^{3}+c \right )^{\frac {7}{12}} c +\left (d \,x^{3}+c \right )^{\frac {7}{12}} d \,x^{3}}d x \] Input:

int((b*x^3+a)^(1/4)/(d*x^3+c)^(19/12),x)
 

Output:

int((a + b*x**3)**(1/4)/((c + d*x**3)**(7/12)*c + (c + d*x**3)**(7/12)*d*x 
**3),x)