\(\int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx\) [749]

Optimal result

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

-6*(d*x+c)^(11/6)*hypergeom([1, 5/3],[5/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b* 
c)/(b*x+a)^(1/6)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=-\frac {6 (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {5}{6},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \] Input:

Integrate[(c + d*x)^(5/6)/(a + b*x)^(7/6),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.29, 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.

\(\displaystyle \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {(c+d x)^{5/6} \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{5/6}}{(a+b x)^{7/6}}dx}{\left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}}\)

\(\Big \downarrow \) 79

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

Input:

Int[(c + d*x)^(5/6)/(a + b*x)^(7/6),x]
 

Output:

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

\[\int \frac {\left (x d +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]

Input:

int((d*x+c)^(5/6)/(b*x+a)^(7/6),x)
 

Output:

int((d*x+c)^(5/6)/(b*x+a)^(7/6),x)
 

Fricas [F]

\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:

integrate((d*x+c)^(5/6)/(b*x+a)^(7/6),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{6}}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \] Input:

integrate((d*x+c)**(5/6)/(b*x+a)**(7/6),x)
 

Output:

Integral((c + d*x)**(5/6)/(a + b*x)**(7/6), x)
 

Maxima [F]

\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:

integrate((d*x+c)^(5/6)/(b*x+a)^(7/6),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(5/6)/(b*x + a)^(7/6), x)
 

Giac [F]

\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:

integrate((d*x+c)^(5/6)/(b*x+a)^(7/6),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(5/6)/(b*x + a)^(7/6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \] Input:

int((c + d*x)^(5/6)/(a + b*x)^(7/6),x)
 

Output:

int((c + d*x)^(5/6)/(a + b*x)^(7/6), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\frac {-6 \left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}} \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) d +6 \left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}} \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) d -\left (\int \frac {\left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}}}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) a^{2} d^{2}+2 \left (\int \frac {\left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}}}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) a b c d -\left (\int \frac {\left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}}}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) b^{2} c^{2}}{b \left (a d -b c \right )} \] Input:

int((d*x+c)^(5/6)/(b*x+a)^(7/6),x)
 

Output:

( - 6*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*d + 6*(c + d 
*x)**(5/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*d - int(((c + d*x)**(5/6 
)*(a + b*x)**(5/6))/(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a*b*d*x**2 + b**2*c 
*x**2 + b**2*d*x**3),x)*a**2*d**2 + 2*int(((c + d*x)**(5/6)*(a + b*x)**(5/ 
6))/(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a*b*d*x**2 + b**2*c*x**2 + b**2*d*x 
**3),x)*a*b*c*d - int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a**2*c + a**2*d 
*x + 2*a*b*c*x + 2*a*b*d*x**2 + b**2*c*x**2 + b**2*d*x**3),x)*b**2*c**2)/( 
b*(a*d - b*c))