\(\int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx\) [732]

Optimal result

Integrand size = 19, antiderivative size = 56 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},1,\frac {7}{6},-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (c+d x)^{5/6}} \] Output:

6*(b*x+a)^(1/6)*hypergeom([-2/3, 1],[7/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c 
)/(d*x+c)^(5/6)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{11/6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{11/6}} \] Input:

Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(11/6)),x]
 

Output:

(6*(a + b*x)^(1/6)*((b*(c + d*x))/(b*c - a*d))^(11/6)*Hypergeometric2F1[1/ 
6, 11/6, 7/6, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(c + d*x)^(11/6))
 

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41, 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.

\(\displaystyle \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \int \frac {1}{(a+b x)^{5/6} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{11/6}}dx}{(c+d x)^{5/6} (b c-a d)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},-\frac {d (a+b x)}{b c-a d}\right )}{(c+d x)^{5/6} (b c-a d)}\)

Input:

Int[1/((a + b*x)^(5/6)*(c + d*x)^(11/6)),x]
 

Output:

(6*(a + b*x)^(1/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[1/6 
, 11/6, 7/6, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)*(c + d*x)^(5/6))
 

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]

\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{6}} \left (x d +c \right )^{\frac {11}{6}}}d x\]

Input:

int(1/(b*x+a)^(5/6)/(d*x+c)^(11/6),x)
 

Output:

int(1/(b*x+a)^(5/6)/(d*x+c)^(11/6),x)
 

Fricas [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(11/6),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \left (c + d x\right )^{\frac {11}{6}}}\, dx \] Input:

integrate(1/(b*x+a)**(5/6)/(d*x+c)**(11/6),x)
 

Output:

Integral(1/((a + b*x)**(5/6)*(c + d*x)**(11/6)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(11/6),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(11/6)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(11/6),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(11/6)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{11/6}} \,d x \] Input:

int(1/((a + b*x)^(5/6)*(c + d*x)^(11/6)),x)
 

Output:

int(1/((a + b*x)^(5/6)*(c + d*x)^(11/6)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int \frac {\left (d x +c \right )^{\frac {2}{3}} \left (b x +a \right )^{\frac {2}{3}}}{\sqrt {d x +c}\, \sqrt {b x +a}\, a \,c^{2}+2 \sqrt {d x +c}\, \sqrt {b x +a}\, a c d x +\sqrt {d x +c}\, \sqrt {b x +a}\, a \,d^{2} x^{2}+\sqrt {d x +c}\, \sqrt {b x +a}\, b \,c^{2} x +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b c d \,x^{2}+\sqrt {d x +c}\, \sqrt {b x +a}\, b \,d^{2} x^{3}}d x \] Input:

int(1/(b*x+a)^(5/6)/(d*x+c)^(11/6),x)
 

Output:

int(((c + d*x)**(2/3)*(a + b*x)**(2/3))/(sqrt(c + d*x)*sqrt(a + b*x)*a*c** 
2 + 2*sqrt(c + d*x)*sqrt(a + b*x)*a*c*d*x + sqrt(c + d*x)*sqrt(a + b*x)*a* 
d**2*x**2 + sqrt(c + d*x)*sqrt(a + b*x)*b*c**2*x + 2*sqrt(c + d*x)*sqrt(a 
+ b*x)*b*c*d*x**2 + sqrt(c + d*x)*sqrt(a + b*x)*b*d**2*x**3),x)