\(\int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx\) [753]

Optimal result

Integrand size = 19, antiderivative size = 56 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=-\frac {6 \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},1,\frac {5}{6},-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{13/6}} \] Output:

-6*hypergeom([-7/3, 1],[5/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c)/(b*x+a)^(1/ 
6)/(d*x+c)^(13/6)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=-\frac {6 b \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {19}{6},\frac {5}{6},\frac {d (a+b x)}{-b c+a d}\right )}{(b c-a d)^2 \sqrt [6]{a+b x} (c+d x)^{7/6}} \] Input:

Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(19/6)),x]
 

Output:

(-6*b*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[-1/6, 19/6, 5/6, 
 (d*(a + b*x))/(-(b*c) + a*d)])/((b*c - a*d)^2*(a + b*x)^(1/6)*(c + d*x)^( 
7/6))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.46, 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.

\(\displaystyle \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \int \frac {1}{(a+b x)^{7/6} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{19/6}}dx}{\sqrt [6]{c+d x} (b c-a d)^3}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {6 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {19}{6},\frac {5}{6},-\frac {d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}\)

Input:

Int[1/((a + b*x)^(7/6)*(c + d*x)^(19/6)),x]
 

Output:

(-6*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-1/6, 19/6, 5/ 
6, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*(a + b*x)^(1/6)*(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]

\[\int \frac {1}{\left (b x +a \right )^{\frac {7}{6}} \left (x d +c \right )^{\frac {19}{6}}}d x\]

Input:

int(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x)
 

Output:

int(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x)
 

Fricas [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(b^2*d^4*x^6 + a^2*c^4 + 2*(2*b^2 
*c*d^3 + a*b*d^4)*x^5 + (6*b^2*c^2*d^2 + 8*a*b*c*d^3 + a^2*d^4)*x^4 + 4*(b 
^2*c^3*d + 3*a*b*c^2*d^2 + a^2*c*d^3)*x^3 + (b^2*c^4 + 8*a*b*c^3*d + 6*a^2 
*c^2*d^2)*x^2 + 2*(a*b*c^4 + 2*a^2*c^3*d)*x), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x+a)**(7/6)/(d*x+c)**(19/6),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{7/6}\,{\left (c+d\,x\right )}^{19/6}} \,d x \] Input:

int(1/((a + b*x)^(7/6)*(c + d*x)^(19/6)),x)
 

Output:

int(1/((a + b*x)^(7/6)*(c + d*x)^(19/6)), x)
 

Reduce [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 420, normalized size of antiderivative = 7.50 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx=\frac {2 \left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}} \left (-24 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c^{2}-48 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c d x -24 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} d^{2} x^{2}+24 \,\mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c^{2}+48 \,\mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c d x +24 \,\mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} d^{2} x^{2}-a^{2} d^{2}+4 a b c d +2 a b \,d^{2} x -3 b^{2} c^{2}-2 b^{2} c d x \right )}{5 a^{3} b \,d^{6} x^{4}-15 a^{2} b^{2} c \,d^{5} x^{4}+15 a \,b^{3} c^{2} d^{4} x^{4}-5 b^{4} c^{3} d^{3} x^{4}+5 a^{4} d^{6} x^{3}-30 a^{2} b^{2} c^{2} d^{4} x^{3}+40 a \,b^{3} c^{3} d^{3} x^{3}-15 b^{4} c^{4} d^{2} x^{3}+15 a^{4} c \,d^{5} x^{2}-30 a^{3} b \,c^{2} d^{4} x^{2}+30 a \,b^{3} c^{4} d^{2} x^{2}-15 b^{4} c^{5} d \,x^{2}+15 a^{4} c^{2} d^{4} x -40 a^{3} b \,c^{3} d^{3} x +30 a^{2} b^{2} c^{4} d^{2} x -5 b^{4} c^{6} x +5 a^{4} c^{3} d^{3}-15 a^{3} b \,c^{4} d^{2}+15 a^{2} b^{2} c^{5} d -5 a \,b^{3} c^{6}} \] Input:

int(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x)
 

Output:

(2*(c + d*x)**(5/6)*(a + b*x)**(5/6)*( - 24*log((a + b*x)**(1/6))*b**2*c** 
2 - 48*log((a + b*x)**(1/6))*b**2*c*d*x - 24*log((a + b*x)**(1/6))*b**2*d* 
*2*x**2 + 24*log((c + d*x)**(1/6))*b**2*c**2 + 48*log((c + d*x)**(1/6))*b* 
*2*c*d*x + 24*log((c + d*x)**(1/6))*b**2*d**2*x**2 - a**2*d**2 + 4*a*b*c*d 
 + 2*a*b*d**2*x - 3*b**2*c**2 - 2*b**2*c*d*x))/(5*(a**4*c**3*d**3 + 3*a**4 
*c**2*d**4*x + 3*a**4*c*d**5*x**2 + a**4*d**6*x**3 - 3*a**3*b*c**4*d**2 - 
8*a**3*b*c**3*d**3*x - 6*a**3*b*c**2*d**4*x**2 + a**3*b*d**6*x**4 + 3*a**2 
*b**2*c**5*d + 6*a**2*b**2*c**4*d**2*x - 6*a**2*b**2*c**2*d**4*x**3 - 3*a* 
*2*b**2*c*d**5*x**4 - a*b**3*c**6 + 6*a*b**3*c**4*d**2*x**2 + 8*a*b**3*c** 
3*d**3*x**3 + 3*a*b**3*c**2*d**4*x**4 - b**4*c**6*x - 3*b**4*c**5*d*x**2 - 
 3*b**4*c**4*d**2*x**3 - b**4*c**3*d**3*x**4))