Integrand size = 19, antiderivative size = 58 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\frac {6 (a+b x)^{13/6} \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {19}{6},-\frac {d (a+b x)}{b c-a d}\right )}{13 (b c-a d) (c+d x)^{5/6}} \] Output:
6/13*(b*x+a)^(13/6)*hypergeom([1, 4/3],[19/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d +b*c)/(d*x+c)^(5/6)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\frac {6 (a+b x)^{13/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{11/6} \operatorname {Hypergeometric2F1}\left (\frac {11}{6},\frac {13}{6},\frac {19}{6},\frac {d (a+b x)}{-b c+a d}\right )}{13 b (c+d x)^{11/6}} \] Input:
Integrate[(a + b*x)^(7/6)/(c + d*x)^(11/6),x]
Output:
(6*(a + b*x)^(13/6)*((b*(c + d*x))/(b*c - a*d))^(11/6)*Hypergeometric2F1[1 1/6, 13/6, 19/6, (d*(a + b*x))/(-(b*c) + a*d)])/(13*b*(c + d*x)^(11/6))
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.40, 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.
| \(\displaystyle \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \int \frac {(a+b x)^{7/6}}{\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{11/6}}dx}{(c+d x)^{5/6} (b c-a d)}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {6 (a+b x)^{13/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {11}{6},\frac {13}{6},\frac {19}{6},-\frac {d (a+b x)}{b c-a d}\right )}{13 (c+d x)^{5/6} (b c-a d)}\) |
Input:
Int[(a + b*x)^(7/6)/(c + d*x)^(11/6),x]
Output:
(6*(a + b*x)^(13/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[11 /6, 13/6, 19/6, -((d*(a + b*x))/(b*c - a*d))])/(13*(b*c - a*d)*(c + d*x)^( 5/6))
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])
\[\int \frac {\left (b x +a \right )^{\frac {7}{6}}}{\left (x d +c \right )^{\frac {11}{6}}}d x\]
Input:
int((b*x+a)^(7/6)/(d*x+c)^(11/6),x)
Output:
int((b*x+a)^(7/6)/(d*x+c)^(11/6),x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(11/6),x, algorithm="fricas")
Output:
integral((b*x + a)^(7/6)*(d*x + c)^(1/6)/(d^2*x^2 + 2*c*d*x + c^2), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{6}}}{\left (c + d x\right )^{\frac {11}{6}}}\, dx \] Input:
integrate((b*x+a)**(7/6)/(d*x+c)**(11/6),x)
Output:
Integral((a + b*x)**(7/6)/(c + d*x)**(11/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(11/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(7/6)/(d*x + c)^(11/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(11/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(7/6)/(d*x + c)^(11/6), x)
Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/6}}{{\left (c+d\,x\right )}^{11/6}} \,d x \] Input:
int((a + b*x)^(7/6)/(c + d*x)^(11/6),x)
Output:
int((a + b*x)^(7/6)/(c + d*x)^(11/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx=\text {too large to display} \] Input:
int((b*x+a)^(7/6)/(d*x+c)^(11/6),x)
Output:
( - 6*(c + d*x)**(7/6)*(a + b*x)**(7/6)*log((a + b*x)**(1/6))*a*b + 6*(c + d*x)**(7/6)*(a + b*x)**(7/6)*log((c + d*x)**(1/6))*a*b + int(((a + b*x)** (1/6)*x)/((c + d*x)**(5/6)*c + (c + d*x)**(5/6)*d*x),x)*a**2*b*c*d**2 + in t(((a + b*x)**(1/6)*x)/((c + d*x)**(5/6)*c + (c + d*x)**(5/6)*d*x),x)*a**2 *b*d**3*x - int(((a + b*x)**(1/6)*x)/((c + d*x)**(5/6)*c + (c + d*x)**(5/6 )*d*x),x)*a*b**2*c**2*d + int(((a + b*x)**(1/6)*x)/((c + d*x)**(5/6)*c + ( c + d*x)**(5/6)*d*x),x)*a*b**2*d**3*x**2 - int(((a + b*x)**(1/6)*x)/((c + d*x)**(5/6)*c + (c + d*x)**(5/6)*d*x),x)*b**3*c**2*d*x - int(((a + b*x)**( 1/6)*x)/((c + d*x)**(5/6)*c + (c + d*x)**(5/6)*d*x),x)*b**3*c*d**2*x**2 + 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)*a**4*c*d** 2 + 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)*sqr t(a + b*x)*b*c*d*x**2 + sqrt(c + d*x)*sqrt(a + b*x)*b*d**2*x**3),x)*a**4*d **3*x - 2*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 +...