\(\int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx\) [701]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\frac {6 (a+b x)^{11/6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {17}{6},-\frac {d (a+b x)}{b c-a d}\right )}{11 (b c-a d) (c+d x)^{13/6}} \] Output:

6/11*(b*x+a)^(11/6)*hypergeom([-1/3, 1],[17/6],-d*(b*x+a)/(-a*d+b*c))/(-a* 
d+b*c)/(d*x+c)^(13/6)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\frac {6 b (a+b x)^{11/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \operatorname {Hypergeometric2F1}\left (\frac {11}{6},\frac {19}{6},\frac {17}{6},\frac {d (a+b x)}{-b c+a d}\right )}{11 (b c-a d)^2 (c+d x)^{7/6}} \] Input:

Integrate[(a + b*x)^(5/6)/(c + d*x)^(19/6),x]
 

Output:

(6*b*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[ 
11/6, 19/6, 17/6, (d*(a + b*x))/(-(b*c) + a*d)])/(11*(b*c - a*d)^2*(c + d* 
x)^(7/6))
 

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {80, 79}

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.

\(\displaystyle \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \int \frac {(a+b x)^{5/6}}{\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{19/6}}dx}{\sqrt [6]{c+d x} (b c-a d)^3}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {6 b^2 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {11}{6},\frac {19}{6},\frac {17}{6},-\frac {d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)^3}\)

Input:

Int[(a + b*x)^(5/6)/(c + d*x)^(19/6),x]
 

Output:

(6*b^2*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F 
1[11/6, 19/6, 17/6, -((d*(a + b*x))/(b*c - a*d))])/(11*(b*c - a*d)^3*(c + 
d*x)^(1/6))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (x d +c \right )^{\frac {19}{6}}}d x\]

Input:

int((b*x+a)^(5/6)/(d*x+c)^(19/6),x)
 

Output:

int((b*x+a)^(5/6)/(d*x+c)^(19/6),x)
 

Fricas [F]

\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:

integrate((b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(5/6)/(d*x+c)**(19/6),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:

integrate((b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(5/6)/(d*x + c)^(19/6), x)
 

Giac [F]

\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:

integrate((b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(5/6)/(d*x + c)^(19/6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/6}}{{\left (c+d\,x\right )}^{19/6}} \,d x \] Input:

int((a + b*x)^(5/6)/(c + d*x)^(19/6),x)
 

Output:

int((a + b*x)^(5/6)/(c + d*x)^(19/6), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx=\text {too large to display} \] Input:

int((b*x+a)^(5/6)/(d*x+c)^(19/6),x)
 

Output:

( - 48*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*b**3*c**2 - 
 96*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*b**3*c*d*x - 4 
8*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*b**3*d**2*x**2 + 
 48*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*b**3*c**2 + 96 
*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*b**3*c*d*x + 48*( 
c + d*x)**(5/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*b**3*d**2*x**2 - 2* 
(c + d*x)**(5/6)*(a + b*x)**(5/6)*a**2*b*d**2 + 8*(c + d*x)**(5/6)*(a + b* 
x)**(5/6)*a*b**2*c*d + 4*(c + d*x)**(5/6)*(a + b*x)**(5/6)*a*b**2*d**2*x - 
 6*(c + d*x)**(5/6)*(a + b*x)**(5/6)*b**3*c**2 - 4*(c + d*x)**(5/6)*(a + b 
*x)**(5/6)*b**3*c*d*x + 5*int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**4 
+ 4*a*c**3*d*x + 6*a*c**2*d**2*x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 + b*c* 
*4*x + 4*b*c**3*d*x**2 + 6*b*c**2*d**2*x**3 + 4*b*c*d**3*x**4 + b*d**4*x** 
5),x)*a**4*c**2*d**4 + 10*int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**4 
+ 4*a*c**3*d*x + 6*a*c**2*d**2*x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 + b*c* 
*4*x + 4*b*c**3*d*x**2 + 6*b*c**2*d**2*x**3 + 4*b*c*d**3*x**4 + b*d**4*x** 
5),x)*a**4*c*d**5*x + 5*int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**4 + 
4*a*c**3*d*x + 6*a*c**2*d**2*x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 + b*c**4 
*x + 4*b*c**3*d*x**2 + 6*b*c**2*d**2*x**3 + 4*b*c*d**3*x**4 + b*d**4*x**5) 
,x)*a**4*d**6*x**2 - 20*int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**4 + 
4*a*c**3*d*x + 6*a*c**2*d**2*x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 + b*c...