Integrand size = 19, antiderivative size = 84 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\frac {6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {19}{6},\frac {11}{6},-\frac {d (a+b x)}{b c-a d}\right )}{5 (b c-a d)^3 \sqrt [6]{c+d x}} \]
6/5*b^2*(b*x+a)^(5/6)*(b*(d*x+c)/(-a*d+b*c))^(1/6)*hypergeom([5/6, 19/6],[ 11/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c)^3/(d*x+c)^(1/6)
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\frac {6 b (a+b x)^{5/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {19}{6},\frac {11}{6},\frac {d (a+b x)}{-b c+a d}\right )}{5 (b c-a d)^2 (c+d x)^{7/6}} \]
(6*b*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[5 /6, 19/6, 11/6, (d*(a + b*x))/(-(b*c) + a*d)])/(5*(b*c - a*d)^2*(c + d*x)^ (7/6))
Time = 0.18 (sec) , antiderivative size = 84, 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.
\(\displaystyle \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt [6]{a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{19/6}}dx}{\sqrt [6]{c+d x} (b c-a d)^3}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {19}{6},\frac {11}{6},-\frac {d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)^3}\) |
(6*b^2*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1 [5/6, 19/6, 11/6, -((d*(a + b*x))/(b*c - a*d))])/(5*(b*c - a*d)^3*(c + d*x )^(1/6))
3.19.17.3.1 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])
\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {19}{6}}}d x\]
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(b*d^4*x^5 + a*c^4 + (4*b*c*d^3 + a*d^4)*x^4 + 2*(3*b*c^2*d^2 + 2*a*c*d^3)*x^3 + 2*(2*b*c^3*d + 3*a*c^2*d^2 )*x^2 + (b*c^4 + 4*a*c^3*d)*x), x)
Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{1/6}\,{\left (c+d\,x\right )}^{19/6}} \,d x \]
Time = 22.03 (sec) , antiderivative size = 375, normalized size of antiderivative = 4.46 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx=\frac {2 \left (d x +c \right )^{\frac {5}{6}} \left (b x +a \right )^{\frac {5}{6}} \left (-12 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c^{2}-24 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c d x -12 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} d^{2} x^{2}+12 \,\mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c^{2}+24 \,\mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c d x +12 \,\mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} d^{2} x^{2}-3 a^{2} d^{2}+7 a b c d +a b \,d^{2} x -4 b^{2} c^{2}-b^{2} c d x \right )}{15 d \left (a^{2} b \,d^{5} x^{4}-2 a \,b^{2} c \,d^{4} x^{4}+b^{3} c^{2} d^{3} x^{4}+a^{3} d^{5} x^{3}+a^{2} b c \,d^{4} x^{3}-5 a \,b^{2} c^{2} d^{3} x^{3}+3 b^{3} c^{3} d^{2} x^{3}+3 a^{3} c \,d^{4} x^{2}-3 a^{2} b \,c^{2} d^{3} x^{2}-3 a \,b^{2} c^{3} d^{2} x^{2}+3 b^{3} c^{4} d \,x^{2}+3 a^{3} c^{2} d^{3} x -5 a^{2} b \,c^{3} d^{2} x +a \,b^{2} c^{4} d x +b^{3} c^{5} x +a^{3} c^{3} d^{2}-2 a^{2} b \,c^{4} d +a \,b^{2} c^{5}\right )} \]
(2*(c + d*x)**(5/6)*(a + b*x)**(5/6)*( - 12*log((a + b*x)**(1/6))*b**2*c** 2 - 24*log((a + b*x)**(1/6))*b**2*c*d*x - 12*log((a + b*x)**(1/6))*b**2*d* *2*x**2 + 12*log((c + d*x)**(1/6))*b**2*c**2 + 24*log((c + d*x)**(1/6))*b* *2*c*d*x + 12*log((c + d*x)**(1/6))*b**2*d**2*x**2 - 3*a**2*d**2 + 7*a*b*c *d + a*b*d**2*x - 4*b**2*c**2 - b**2*c*d*x))/(15*d*(a**3*c**3*d**2 + 3*a** 3*c**2*d**3*x + 3*a**3*c*d**4*x**2 + a**3*d**5*x**3 - 2*a**2*b*c**4*d - 5* a**2*b*c**3*d**2*x - 3*a**2*b*c**2*d**3*x**2 + a**2*b*c*d**4*x**3 + a**2*b *d**5*x**4 + a*b**2*c**5 + a*b**2*c**4*d*x - 3*a*b**2*c**3*d**2*x**2 - 5*a *b**2*c**2*d**3*x**3 - 2*a*b**2*c*d**4*x**4 + b**3*c**5*x + 3*b**3*c**4*d* x**2 + 3*b**3*c**3*d**2*x**3 + b**3*c**2*d**3*x**4))