Integrand size = 19, antiderivative size = 136 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\frac {6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac {108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac {1296 b^2 (a+b x)^{11/6}}{11339 (b c-a d)^3 (c+d x)^{17/6}}+\frac {7776 b^3 (a+b x)^{11/6}}{124729 (b c-a d)^4 (c+d x)^{11/6}} \]
6/29*(b*x+a)^(11/6)/(-a*d+b*c)/(d*x+c)^(29/6)+108/667*b*(b*x+a)^(11/6)/(-a *d+b*c)^2/(d*x+c)^(23/6)+1296/11339*b^2*(b*x+a)^(11/6)/(-a*d+b*c)^3/(d*x+c )^(17/6)+7776/124729*b^3*(b*x+a)^(11/6)/(-a*d+b*c)^4/(d*x+c)^(11/6)
Time = 1.65 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\frac {6 (a+b x)^{11/6} \left (-4301 a^3 d^3+561 a^2 b d^2 (29 c+6 d x)-33 a b^2 d \left (667 c^2+348 c d x+72 d^2 x^2\right )+b^3 \left (11339 c^3+12006 c^2 d x+6264 c d^2 x^2+1296 d^3 x^3\right )\right )}{124729 (b c-a d)^4 (c+d x)^{29/6}} \]
(6*(a + b*x)^(11/6)*(-4301*a^3*d^3 + 561*a^2*b*d^2*(29*c + 6*d*x) - 33*a*b ^2*d*(667*c^2 + 348*c*d*x + 72*d^2*x^2) + b^3*(11339*c^3 + 12006*c^2*d*x + 6264*c*d^2*x^2 + 1296*d^3*x^3)))/(124729*(b*c - a*d)^4*(c + d*x)^(29/6))
Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}}dx}{29 (b c-a d)}+\frac {6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \left (\frac {12 b \int \frac {(a+b x)^{5/6}}{(c+d x)^{23/6}}dx}{23 (b c-a d)}+\frac {6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)}\right )}{29 (b c-a d)}+\frac {6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \left (\frac {12 b \left (\frac {6 b \int \frac {(a+b x)^{5/6}}{(c+d x)^{17/6}}dx}{17 (b c-a d)}+\frac {6 (a+b x)^{11/6}}{17 (c+d x)^{17/6} (b c-a d)}\right )}{23 (b c-a d)}+\frac {6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)}\right )}{29 (b c-a d)}+\frac {6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)}+\frac {18 b \left (\frac {6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)}+\frac {12 b \left (\frac {36 b (a+b x)^{11/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac {6 (a+b x)^{11/6}}{17 (c+d x)^{17/6} (b c-a d)}\right )}{23 (b c-a d)}\right )}{29 (b c-a d)}\) |
(6*(a + b*x)^(11/6))/(29*(b*c - a*d)*(c + d*x)^(29/6)) + (18*b*((6*(a + b* x)^(11/6))/(23*(b*c - a*d)*(c + d*x)^(23/6)) + (12*b*((6*(a + b*x)^(11/6)) /(17*(b*c - a*d)*(c + d*x)^(17/6)) + (36*b*(a + b*x)^(11/6))/(187*(b*c - a *d)^2*(c + d*x)^(11/6))))/(23*(b*c - a*d))))/(29*(b*c - a*d))
3.18.85.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.90 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
| method | result | size |
| gosper | \(-\frac {6 \left (b x +a \right )^{\frac {11}{6}} \left (-1296 d^{3} x^{3} b^{3}+2376 x^{2} a \,b^{2} d^{3}-6264 x^{2} b^{3} c \,d^{2}-3366 x \,a^{2} b \,d^{3}+11484 x a \,b^{2} c \,d^{2}-12006 x \,b^{3} c^{2} d +4301 a^{3} d^{3}-16269 a^{2} b c \,d^{2}+22011 a \,b^{2} c^{2} d -11339 b^{3} c^{3}\right )}{124729 \left (d x +c \right )^{\frac {29}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
-6/124729*(b*x+a)^(11/6)*(-1296*b^3*d^3*x^3+2376*a*b^2*d^3*x^2-6264*b^3*c* d^2*x^2-3366*a^2*b*d^3*x+11484*a*b^2*c*d^2*x-12006*b^3*c^2*d*x+4301*a^3*d^ 3-16269*a^2*b*c*d^2+22011*a*b^2*c^2*d-11339*b^3*c^3)/(d*x+c)^(29/6)/(a^4*d ^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.92 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\frac {6 \, {\left (1296 \, b^{4} d^{3} x^{4} + 11339 \, a b^{3} c^{3} - 22011 \, a^{2} b^{2} c^{2} d + 16269 \, a^{3} b c d^{2} - 4301 \, a^{4} d^{3} + 216 \, {\left (29 \, b^{4} c d^{2} - 5 \, a b^{3} d^{3}\right )} x^{3} + 18 \, {\left (667 \, b^{4} c^{2} d - 290 \, a b^{3} c d^{2} + 55 \, a^{2} b^{2} d^{3}\right )} x^{2} + {\left (11339 \, b^{4} c^{3} - 10005 \, a b^{3} c^{2} d + 4785 \, a^{2} b^{2} c d^{2} - 935 \, a^{3} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{124729 \, {\left (b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4} + {\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} x^{5} + 5 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} x^{4} + 10 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} x^{3} + 10 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} x^{2} + 5 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )} x\right )}} \]
6/124729*(1296*b^4*d^3*x^4 + 11339*a*b^3*c^3 - 22011*a^2*b^2*c^2*d + 16269 *a^3*b*c*d^2 - 4301*a^4*d^3 + 216*(29*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 18*(6 67*b^4*c^2*d - 290*a*b^3*c*d^2 + 55*a^2*b^2*d^3)*x^2 + (11339*b^4*c^3 - 10 005*a*b^3*c^2*d + 4785*a^2*b^2*c*d^2 - 935*a^3*b*d^3)*x)*(b*x + a)^(5/6)*( d*x + c)^(1/6)/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6* d^3 + a^4*c^5*d^4 + (b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4 *a^3*b*c*d^8 + a^4*d^9)*x^5 + 5*(b^4*c^5*d^4 - 4*a*b^3*c^4*d^5 + 6*a^2*b^2 *c^3*d^6 - 4*a^3*b*c^2*d^7 + a^4*c*d^8)*x^4 + 10*(b^4*c^6*d^3 - 4*a*b^3*c^ 5*d^4 + 6*a^2*b^2*c^4*d^5 - 4*a^3*b*c^3*d^6 + a^4*c^2*d^7)*x^3 + 10*(b^4*c ^7*d^2 - 4*a*b^3*c^6*d^3 + 6*a^2*b^2*c^5*d^4 - 4*a^3*b*c^4*d^5 + a^4*c^3*d ^6)*x^2 + 5*(b^4*c^8*d - 4*a*b^3*c^7*d^2 + 6*a^2*b^2*c^6*d^3 - 4*a^3*b*c^5 *d^4 + a^4*c^4*d^5)*x)
Timed out. \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {35}{6}}} \,d x } \]
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {35}{6}}} \,d x } \]
Time = 1.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.23 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {7776\,b^4\,x^4\,{\left (a+b\,x\right )}^{5/6}}{124729\,d^2\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{5/6}\,\left (25806\,a^4\,d^3-97614\,a^3\,b\,c\,d^2+132066\,a^2\,b^2\,c^2\,d-68034\,a\,b^3\,c^3\right )}{124729\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{5/6}\,\left (-5610\,a^3\,b\,d^3+28710\,a^2\,b^2\,c\,d^2-60030\,a\,b^3\,c^2\,d+68034\,b^4\,c^3\right )}{124729\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^2\,x^2\,{\left (a+b\,x\right )}^{5/6}\,\left (55\,a^2\,d^2-290\,a\,b\,c\,d+667\,b^2\,c^2\right )}{124729\,d^4\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^3\,x^3\,\left (5\,a\,d-29\,b\,c\right )\,{\left (a+b\,x\right )}^{5/6}}{124729\,d^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \]
((c + d*x)^(1/6)*((7776*b^4*x^4*(a + b*x)^(5/6))/(124729*d^2*(a*d - b*c)^4 ) - ((a + b*x)^(5/6)*(25806*a^4*d^3 - 68034*a*b^3*c^3 + 132066*a^2*b^2*c^2 *d - 97614*a^3*b*c*d^2))/(124729*d^5*(a*d - b*c)^4) + (x*(a + b*x)^(5/6)*( 68034*b^4*c^3 - 5610*a^3*b*d^3 + 28710*a^2*b^2*c*d^2 - 60030*a*b^3*c^2*d)) /(124729*d^5*(a*d - b*c)^4) + (108*b^2*x^2*(a + b*x)^(5/6)*(55*a^2*d^2 + 6 67*b^2*c^2 - 290*a*b*c*d))/(124729*d^4*(a*d - b*c)^4) - (1296*b^3*x^3*(5*a *d - 29*b*c)*(a + b*x)^(5/6))/(124729*d^3*(a*d - b*c)^4)))/(x^5 + c^5/d^5 + (5*c*x^4)/d + (5*c^4*x)/d^4 + (10*c^2*x^3)/d^2 + (10*c^3*x^2)/d^3)
\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx=\int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (d x +c \right )^{\frac {5}{6}} \left (d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}\right )}d x \]
int((a + b*x)**(5/6)/((c + d*x)**(5/6)*(c**5 + 5*c**4*d*x + 10*c**3*d**2*x **2 + 10*c**2*d**3*x**3 + 5*c*d**4*x**4 + d**5*x**5)),x)