\(\int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx\) [1677]

Optimal result

Integrand size = 28, antiderivative size = 206 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\frac {2 d (a+b x)^{4/3} \sqrt [6]{e+f x}}{3 (b c-a d) (d e-c f) (c+d x)^{3/2}}+\frac {(b d e-9 b c f+8 a d f) (a+b x)^{4/3} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{3/2} \sqrt [6]{e+f x} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {7}{3},-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{12 (b c-a d) (b e-a f) (d e-c f) (c+d x)^{3/2}} \] Output:

2/3*d*(b*x+a)^(4/3)*(f*x+e)^(1/6)/(-a*d+b*c)/(-c*f+d*e)/(d*x+c)^(3/2)+1/12 
*(8*a*d*f-9*b*c*f+b*d*e)*(b*x+a)^(4/3)*((-a*f+b*e)*(d*x+c)/(-a*d+b*c)/(f*x 
+e))^(3/2)*(f*x+e)^(1/6)*hypergeom([4/3, 3/2],[7/3],-(-c*f+d*e)*(b*x+a)/(- 
a*d+b*c)/(f*x+e))/(-a*d+b*c)/(-a*f+b*e)/(-c*f+d*e)/(d*x+c)^(3/2)
 

Mathematica [A] (verified)

Time = 10.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\frac {2 \sqrt [3]{a+b x} \sqrt [6]{e+f x} \left (-d (a+b x)-\frac {(b d e-9 b c f+8 a d f) (c+d x) \left (-1+\sqrt {\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )\right )}{3 d e-3 c f}\right )}{3 (b c-a d) (-d e+c f) (c+d x)^{3/2}} \] Input:

Integrate[(a + b*x)^(1/3)/((c + d*x)^(5/2)*(e + f*x)^(5/6)),x]
 

Output:

(2*(a + b*x)^(1/3)*(e + f*x)^(1/6)*(-(d*(a + b*x)) - ((b*d*e - 9*b*c*f + 8 
*a*d*f)*(c + d*x)*(-1 + Sqrt[((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x 
))]*Hypergeometric2F1[1/3, 1/2, 4/3, ((-(d*e) + c*f)*(a + b*x))/((b*c - a* 
d)*(e + f*x))]))/(3*d*e - 3*c*f)))/(3*(b*c - a*d)*(-(d*e) + c*f)*(c + d*x) 
^(3/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {107, 105, 142}

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.

\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx\)

\(\Big \downarrow \) 107

\(\displaystyle \frac {(8 a d f-9 b c f+b d e) \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{3/2} (e+f x)^{5/6}}dx}{9 (b c-a d) (d e-c f)}+\frac {2 d (a+b x)^{4/3} \sqrt [6]{e+f x}}{3 (c+d x)^{3/2} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {(8 a d f-9 b c f+b d e) \left (\frac {2 (b e-a f) \int \frac {1}{(a+b x)^{2/3} \sqrt {c+d x} (e+f x)^{5/6}}dx}{3 (d e-c f)}-\frac {2 \sqrt [3]{a+b x} \sqrt [6]{e+f x}}{\sqrt {c+d x} (d e-c f)}\right )}{9 (b c-a d) (d e-c f)}+\frac {2 d (a+b x)^{4/3} \sqrt [6]{e+f x}}{3 (c+d x)^{3/2} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 142

\(\displaystyle \frac {(8 a d f-9 b c f+b d e) \left (\frac {2 \sqrt [3]{a+b x} \sqrt [6]{e+f x} \sqrt {\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{\sqrt {c+d x} (d e-c f)}-\frac {2 \sqrt [3]{a+b x} \sqrt [6]{e+f x}}{\sqrt {c+d x} (d e-c f)}\right )}{9 (b c-a d) (d e-c f)}+\frac {2 d (a+b x)^{4/3} \sqrt [6]{e+f x}}{3 (c+d x)^{3/2} (b c-a d) (d e-c f)}\)

Input:

Int[(a + b*x)^(1/3)/((c + d*x)^(5/2)*(e + f*x)^(5/6)),x]
 

Output:

(2*d*(a + b*x)^(4/3)*(e + f*x)^(1/6))/(3*(b*c - a*d)*(d*e - c*f)*(c + d*x) 
^(3/2)) + ((b*d*e - 9*b*c*f + 8*a*d*f)*((-2*(a + b*x)^(1/3)*(e + f*x)^(1/6 
))/((d*e - c*f)*Sqrt[c + d*x]) + (2*(a + b*x)^(1/3)*Sqrt[((b*e - a*f)*(c + 
 d*x))/((b*c - a*d)*(e + f*x))]*(e + f*x)^(1/6)*Hypergeometric2F1[1/3, 1/2 
, 4/3, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((d*e - c*f)*S 
qrt[c + d*x])))/(9*(b*c - a*d)*(d*e - c*f))
 

Defintions of rubi rules used

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (x d +c \right )^{\frac {5}{2}} \left (f x +e \right )^{\frac {5}{6}}}d x\]

Input:

int((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x)
 

Output:

int((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x)
 

Fricas [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {5}{2}} {\left (f x + e\right )}^{\frac {5}{6}}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(1/3)*sqrt(d*x + c)*(f*x + e)^(1/6)/(d^3*f*x^4 + c^3*e 
+ (d^3*e + 3*c*d^2*f)*x^3 + 3*(c*d^2*e + c^2*d*f)*x^2 + (3*c^2*d*e + c^3*f 
)*x), x)
 

Sympy [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\int \frac {\sqrt [3]{a + b x}}{\left (c + d x\right )^{\frac {5}{2}} \left (e + f x\right )^{\frac {5}{6}}}\, dx \] Input:

integrate((b*x+a)**(1/3)/(d*x+c)**(5/2)/(f*x+e)**(5/6),x)
 

Output:

Integral((a + b*x)**(1/3)/((c + d*x)**(5/2)*(e + f*x)**(5/6)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {5}{2}} {\left (f x + e\right )}^{\frac {5}{6}}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/3)/((d*x + c)^(5/2)*(f*x + e)^(5/6)), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {5}{2}} {\left (f x + e\right )}^{\frac {5}{6}}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/3)/((d*x + c)^(5/2)*(f*x + e)^(5/6)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (e+f\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((a + b*x)^(1/3)/((e + f*x)^(5/6)*(c + d*x)^(5/2)),x)
 

Output:

int((a + b*x)^(1/3)/((e + f*x)^(5/6)*(c + d*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{5/2} (e+f x)^{5/6}} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {5}{2}} \left (f x +e \right )^{\frac {5}{6}}}d x \] Input:

int((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x)
 

Output:

int((b*x+a)^(1/3)/(d*x+c)^(5/2)/(f*x+e)^(5/6),x)