Integrand size = 19, antiderivative size = 58 \[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\frac {6 (a+b x)^{5/6} (c+d x)^{11/6} \operatorname {Hypergeometric2F1}\left (1,\frac {8}{3},\frac {11}{6},-\frac {d (a+b x)}{b c-a d}\right )}{5 (b c-a d)} \] Output:
6*(b*x+a)^(5/6)*(d*x+c)^(11/6)*hypergeom([1, 8/3],[11/6],-d*(b*x+a)/(-a*d+ b*c))/(-5*a*d+5*b*c)
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\frac {6 (a+b x)^{5/6} (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {5}{6},\frac {11}{6},\frac {d (a+b x)}{-b c+a d}\right )}{5 b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \] Input:
Integrate[(c + d*x)^(5/6)/(a + b*x)^(1/6),x]
Output:
(6*(a + b*x)^(5/6)*(c + d*x)^(5/6)*Hypergeometric2F1[-5/6, 5/6, 11/6, (d*( a + b*x))/(-(b*c) + a*d)])/(5*b*((b*(c + d*x))/(b*c - a*d))^(5/6))
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28, 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.
\(\displaystyle \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(c+d x)^{5/6} \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{5/6}}{\sqrt [6]{a+b x}}dx}{\left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {6 (a+b x)^{5/6} (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {5}{6},\frac {11}{6},-\frac {d (a+b x)}{b c-a d}\right )}{5 b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}}\) |
Input:
Int[(c + d*x)^(5/6)/(a + b*x)^(1/6),x]
Output:
(6*(a + b*x)^(5/6)*(c + d*x)^(5/6)*Hypergeometric2F1[-5/6, 5/6, 11/6, -((d *(a + b*x))/(b*c - a*d))])/(5*b*((b*(c + d*x))/(b*c - a*d))^(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])
\[\int \frac {\left (x d +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {1}{6}}}d x\]
Input:
int((d*x+c)^(5/6)/(b*x+a)^(1/6),x)
Output:
int((d*x+c)^(5/6)/(b*x+a)^(1/6),x)
\[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate((d*x+c)^(5/6)/(b*x+a)^(1/6),x, algorithm="fricas")
Output:
integral((d*x + c)^(5/6)/(b*x + a)^(1/6), x)
\[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{6}}}{\sqrt [6]{a + b x}}\, dx \] Input:
integrate((d*x+c)**(5/6)/(b*x+a)**(1/6),x)
Output:
Integral((c + d*x)**(5/6)/(a + b*x)**(1/6), x)
\[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate((d*x+c)^(5/6)/(b*x+a)^(1/6),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/6)/(b*x + a)^(1/6), x)
\[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate((d*x+c)^(5/6)/(b*x+a)^(1/6),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/6)/(b*x + a)^(1/6), x)
Timed out. \[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/6}}{{\left (a+b\,x\right )}^{1/6}} \,d x \] Input:
int((c + d*x)^(5/6)/(a + b*x)^(1/6),x)
Output:
int((c + d*x)^(5/6)/(a + b*x)^(1/6), x)
\[ \int \frac {(c+d x)^{5/6}}{\sqrt [6]{a+b x}} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {1}{6}}}d x \] Input:
int((d*x+c)^(5/6)/(b*x+a)^(1/6),x)
Output:
int((c + d*x)**(5/6)/(a + b*x)**(1/6),x)