Integrand size = 19, antiderivative size = 58 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\frac {6 (a+b x)^{11/6} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {17}{6},-\frac {d (a+b x)}{b c-a d}\right )}{11 (b c-a d) (c+d x)^{7/6}} \] Output:
6/11*(b*x+a)^(11/6)*hypergeom([2/3, 1],[17/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d +b*c)/(d*x+c)^(7/6)
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\frac {6 (a+b x)^{11/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{13/6} \operatorname {Hypergeometric2F1}\left (\frac {11}{6},\frac {13}{6},\frac {17}{6},\frac {d (a+b x)}{-b c+a d}\right )}{11 b (c+d x)^{13/6}} \] Input:
Integrate[(a + b*x)^(5/6)/(c + d*x)^(13/6),x]
Output:
(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(13/6)*Hypergeometric2F1[1 1/6, 13/6, 17/6, (d*(a + b*x))/(-(b*c) + a*d)])/(11*b*(c + d*x)^(13/6))
Time = 0.18 (sec) , antiderivative size = 82, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \int \frac {(a+b x)^{5/6}}{\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{13/6}}dx}{\sqrt [6]{c+d x} (b c-a d)^2}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {6 b (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {11}{6},\frac {13}{6},\frac {17}{6},-\frac {d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)^2}\) |
Input:
Int[(a + b*x)^(5/6)/(c + d*x)^(13/6),x]
Output:
(6*b*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[ 11/6, 13/6, 17/6, -((d*(a + b*x))/(b*c - a*d))])/(11*(b*c - a*d)^2*(c + d* x)^(1/6))
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]))
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])
\[\int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (x d +c \right )^{\frac {13}{6}}}d x\]
Input:
int((b*x+a)^(5/6)/(d*x+c)^(13/6),x)
Output:
int((b*x+a)^(5/6)/(d*x+c)^(13/6),x)
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:
integrate((b*x+a)^(5/6)/(d*x+c)^(13/6),x, algorithm="fricas")
Output:
integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d* x + c^3), x)
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{6}}}{\left (c + d x\right )^{\frac {13}{6}}}\, dx \] Input:
integrate((b*x+a)**(5/6)/(d*x+c)**(13/6),x)
Output:
Integral((a + b*x)**(5/6)/(c + d*x)**(13/6), x)
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:
integrate((b*x+a)^(5/6)/(d*x+c)^(13/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(5/6)/(d*x + c)^(13/6), x)
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:
integrate((b*x+a)^(5/6)/(d*x+c)^(13/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(5/6)/(d*x + c)^(13/6), x)
Timed out. \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/6}}{{\left (c+d\,x\right )}^{13/6}} \,d x \] Input:
int((a + b*x)^(5/6)/(c + d*x)^(13/6),x)
Output:
int((a + b*x)^(5/6)/(c + d*x)^(13/6), x)
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^(5/6)/(d*x+c)^(13/6),x)
Output:
(24*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*b**2*c + 24*(c + d*x)**(5/6)*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*b**2*d*x - 24*(c + d *x)**(5/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*b**2*c - 24*(c + d*x)**( 5/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*b**2*d*x - 2*(c + d*x)**(5/6)* (a + b*x)**(5/6)*a*b*d + 2*(c + d*x)**(5/6)*(a + b*x)**(5/6)*b**2*c + 3*in t(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2* x**2 + a*d**3*x**3 + b*c**3*x + 3*b*c**2*d*x**2 + 3*b*c*d**2*x**3 + b*d**3 *x**4),x)*a**3*c*d**3 + 3*int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*x + 3*b*c**2*d*x** 2 + 3*b*c*d**2*x**3 + b*d**3*x**4),x)*a**3*d**4*x - 9*int(((c + d*x)**(5/6 )*(a + b*x)**(5/6))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*x + 3*b*c**2*d*x**2 + 3*b*c*d**2*x**3 + b*d**3*x**4),x)*a**2*b*c **2*d**2 - 9*int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**3 + 3*a*c**2*d* x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*x + 3*b*c**2*d*x**2 + 3*b*c*d** 2*x**3 + b*d**3*x**4),x)*a**2*b*c*d**3*x + 9*int(((c + d*x)**(5/6)*(a + b* x)**(5/6))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3 *x + 3*b*c**2*d*x**2 + 3*b*c*d**2*x**3 + b*d**3*x**4),x)*a*b**2*c**3*d + 9 *int(((c + d*x)**(5/6)*(a + b*x)**(5/6))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d* *2*x**2 + a*d**3*x**3 + b*c**3*x + 3*b*c**2*d*x**2 + 3*b*c*d**2*x**3 + b*d **3*x**4),x)*a*b**2*c**2*d**2*x - 3*int(((c + d*x)**(5/6)*(a + b*x)**(5...