Integrand size = 19, antiderivative size = 82 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\frac {6 b (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {17}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 (b c-a d)^2 (c+d x)^{5/6}} \]
6/7*b*(b*x+a)^(7/6)*(b*(d*x+c)/(-a*d+b*c))^(5/6)*hypergeom([7/6, 17/6],[13 /6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c)^2/(d*x+c)^(5/6)
Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\frac {6 b (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {17}{6},\frac {13}{6},\frac {d (a+b x)}{-b c+a d}\right )}{7 (b c-a d)^2 (c+d x)^{5/6}} \]
(6*b*(a + b*x)^(7/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[7 /6, 17/6, 13/6, (d*(a + b*x))/(-(b*c) + a*d)])/(7*(b*c - a*d)^2*(c + d*x)^ (5/6))
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {b^2 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \int \frac {\sqrt [6]{a+b x}}{\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{17/6}}dx}{(c+d x)^{5/6} (b c-a d)^2}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {6 b (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {17}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 (c+d x)^{5/6} (b c-a d)^2}\) |
(6*b*(a + b*x)^(7/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[7 /6, 17/6, 13/6, -((d*(a + b*x))/(b*c - a*d))])/(7*(b*c - a*d)^2*(c + d*x)^ (5/6))
3.18.71.3.1 Defintions of rubi rules used
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])
\[\int \frac {\left (b x +a \right )^{\frac {1}{6}}}{\left (d x +c \right )^{\frac {17}{6}}}d x\]
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/6}}{{\left (c+d\,x\right )}^{17/6}} \,d x \]
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{17/6}} \, dx=\text {too large to display} \]
( - 2*(c + d*x)**(3/2)*(a + b*x)**(3/2)*a*b*d + 2*(c + d*x)**(3/2)*(a + b* x)**(3/2)*b**2*c + 3*(c + d*x)**(1/3)*(a + b*x)**(1/3)*int(((c + d*x)**(1/ 6)*(a + b*x)**(1/6))/(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**4*c* *2*d**3 + 6*(c + d*x)**(1/3)*(a + b*x)**(1/3)*int(((c + d*x)**(1/6)*(a + b *x)**(1/6))/(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**4*c*d**4*x + 3*(c + d*x)**(1/3)*(a + b*x)**(1/3)*int(((c + d*x)**(1/6)*(a + b*x)**(1/6) )/(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**4*d**5*x**2 - 9*(c + d* x)**(1/3)*(a + b*x)**(1/3)*int(((c + d*x)**(1/6)*(a + b*x)**(1/6))/(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*b*c**3*d**2 - 15*(c + d*x)**(1 /3)*(a + b*x)**(1/3)*int(((c + d*x)**(1/6)*(a + b*x)**(1/6))/(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*b*c**2*d**3*x - 3*(c + d*x)**(1/3)*( a + b*x)**(1/3)*int(((c + d*x)**(1/6)*(a + b*x)**(1/6))/(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*b*c*d**4*x**2 + 3*(c + d*x)**(1/3)*(a + b *x)**(1/3)*int(((c + d*x)**(1/6)*(a + b*x)**(1/6))/(a*c**3 + 3*a*c**2*d...