Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {6 (a+b x)^{5/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {108 b (a+b x)^{5/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {1296 b^2 (a+b x)^{5/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}}+\frac {7776 b^3 (a+b x)^{5/6}}{21505 (b c-a d)^4 (c+d x)^{5/6}} \]
6/23*(b*x+a)^(5/6)/(-a*d+b*c)/(d*x+c)^(23/6)+108/391*b*(b*x+a)^(5/6)/(-a*d +b*c)^2/(d*x+c)^(17/6)+1296/4301*b^2*(b*x+a)^(5/6)/(-a*d+b*c)^3/(d*x+c)^(1 1/6)+7776/21505*b^3*(b*x+a)^(5/6)/(-a*d+b*c)^4/(d*x+c)^(5/6)
Time = 0.95 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {6 (a+b x)^{5/6} \left (-935 a^3 d^3+165 a^2 b d^2 (23 c+6 d x)-15 a b^2 d \left (391 c^2+276 c d x+72 d^2 x^2\right )+b^3 \left (4301 c^3+7038 c^2 d x+4968 c d^2 x^2+1296 d^3 x^3\right )\right )}{21505 (b c-a d)^4 (c+d x)^{23/6}} \]
(6*(a + b*x)^(5/6)*(-935*a^3*d^3 + 165*a^2*b*d^2*(23*c + 6*d*x) - 15*a*b^2 *d*(391*c^2 + 276*c*d*x + 72*d^2*x^2) + b^3*(4301*c^3 + 7038*c^2*d*x + 496 8*c*d^2*x^2 + 1296*d^3*x^3)))/(21505*(b*c - a*d)^4*(c + d*x)^(23/6))
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.
\(\displaystyle \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {18 b \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}}dx}{23 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {18 b \left (\frac {12 b \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}}dx}{17 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)}\right )}{23 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {18 b \left (\frac {12 b \left (\frac {6 b \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}}dx}{11 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)}\right )}{17 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)}\right )}{23 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {18 b \left (\frac {12 b \left (\frac {36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)}\right )}{17 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)}\right )}{23 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)}\) |
(6*(a + b*x)^(5/6))/(23*(b*c - a*d)*(c + d*x)^(23/6)) + (18*b*((6*(a + b*x )^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (12*b*((6*(a + b*x)^(5/6))/(1 1*(b*c - a*d)*(c + d*x)^(11/6)) + (36*b*(a + b*x)^(5/6))/(55*(b*c - a*d)^2 *(c + d*x)^(5/6))))/(17*(b*c - a*d))))/(23*(b*c - a*d))
3.19.11.3.1 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 = 0.92 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (-1296 d^{3} x^{3} b^{3}+1080 x^{2} a \,b^{2} d^{3}-4968 x^{2} b^{3} c \,d^{2}-990 x \,a^{2} b \,d^{3}+4140 x a \,b^{2} c \,d^{2}-7038 x \,b^{3} c^{2} d +935 a^{3} d^{3}-3795 a^{2} b c \,d^{2}+5865 a \,b^{2} c^{2} d -4301 b^{3} c^{3}\right )}{21505 \left (d x +c \right )^{\frac {23}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
-6/21505*(b*x+a)^(5/6)*(-1296*b^3*d^3*x^3+1080*a*b^2*d^3*x^2-4968*b^3*c*d^ 2*x^2-990*a^2*b*d^3*x+4140*a*b^2*c*d^2*x-7038*b^3*c^2*d*x+935*a^3*d^3-3795 *a^2*b*c*d^2+5865*a*b^2*c^2*d-4301*b^3*c^3)/(d*x+c)^(23/6)/(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. 420 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.09 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {6 \, {\left (1296 \, b^{3} d^{3} x^{3} + 4301 \, b^{3} c^{3} - 5865 \, a b^{2} c^{2} d + 3795 \, a^{2} b c d^{2} - 935 \, a^{3} d^{3} + 216 \, {\left (23 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (391 \, b^{3} c^{2} d - 230 \, a b^{2} c d^{2} + 55 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{21505 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4} + {\left (b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{4} + 4 \, {\left (b^{4} c^{5} d^{3} - 4 \, a b^{3} c^{4} d^{4} + 6 \, a^{2} b^{2} c^{3} d^{5} - 4 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )} x^{3} + 6 \, {\left (b^{4} c^{6} d^{2} - 4 \, a b^{3} c^{5} d^{3} + 6 \, a^{2} b^{2} c^{4} d^{4} - 4 \, a^{3} b c^{3} d^{5} + a^{4} c^{2} d^{6}\right )} x^{2} + 4 \, {\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )} x\right )}} \]
6/21505*(1296*b^3*d^3*x^3 + 4301*b^3*c^3 - 5865*a*b^2*c^2*d + 3795*a^2*b*c *d^2 - 935*a^3*d^3 + 216*(23*b^3*c*d^2 - 5*a*b^2*d^3)*x^2 + 18*(391*b^3*c^ 2*d - 230*a*b^2*c*d^2 + 55*a^2*b*d^3)*x)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/( b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^ 4 + (b^4*c^4*d^4 - 4*a*b^3*c^3*d^5 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a ^4*d^8)*x^4 + 4*(b^4*c^5*d^3 - 4*a*b^3*c^4*d^4 + 6*a^2*b^2*c^3*d^5 - 4*a^3 *b*c^2*d^6 + a^4*c*d^7)*x^3 + 6*(b^4*c^6*d^2 - 4*a*b^3*c^5*d^3 + 6*a^2*b^2 *c^4*d^4 - 4*a^3*b*c^3*d^5 + a^4*c^2*d^6)*x^2 + 4*(b^4*c^7*d - 4*a*b^3*c^6 *d^2 + 6*a^2*b^2*c^5*d^3 - 4*a^3*b*c^4*d^4 + a^4*c^3*d^5)*x)
Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {29}{6}}} \,d x } \]
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {29}{6}}} \,d x } \]
Time = 1.38 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.15 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {7776\,b^4\,x^4}{21505\,d\,{\left (a\,d-b\,c\right )}^4}-\frac {5610\,a^4\,d^3-22770\,a^3\,b\,c\,d^2+35190\,a^2\,b^2\,c^2\,d-25806\,a\,b^3\,c^3}{21505\,d^4\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (330\,a^3\,b\,d^3-2070\,a^2\,b^2\,c\,d^2+7038\,a\,b^3\,c^2\,d+25806\,b^4\,c^3\right )}{21505\,d^4\,{\left (a\,d-b\,c\right )}^4}+\frac {1296\,b^3\,x^3\,\left (a\,d+23\,b\,c\right )}{21505\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^2\,x^2\,\left (-5\,a^2\,d^2+46\,a\,b\,c\,d+391\,b^2\,c^2\right )}{21505\,d^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^4\,{\left (a+b\,x\right )}^{1/6}+\frac {c^4\,{\left (a+b\,x\right )}^{1/6}}{d^4}+\frac {6\,c^2\,x^2\,{\left (a+b\,x\right )}^{1/6}}{d^2}+\frac {4\,c\,x^3\,{\left (a+b\,x\right )}^{1/6}}{d}+\frac {4\,c^3\,x\,{\left (a+b\,x\right )}^{1/6}}{d^3}} \]
((c + d*x)^(1/6)*((7776*b^4*x^4)/(21505*d*(a*d - b*c)^4) - (5610*a^4*d^3 - 25806*a*b^3*c^3 + 35190*a^2*b^2*c^2*d - 22770*a^3*b*c*d^2)/(21505*d^4*(a* d - b*c)^4) + (x*(25806*b^4*c^3 + 330*a^3*b*d^3 - 2070*a^2*b^2*c*d^2 + 703 8*a*b^3*c^2*d))/(21505*d^4*(a*d - b*c)^4) + (1296*b^3*x^3*(a*d + 23*b*c))/ (21505*d^2*(a*d - b*c)^4) + (108*b^2*x^2*(391*b^2*c^2 - 5*a^2*d^2 + 46*a*b *c*d))/(21505*d^3*(a*d - b*c)^4)))/(x^4*(a + b*x)^(1/6) + (c^4*(a + b*x)^( 1/6))/d^4 + (6*c^2*x^2*(a + b*x)^(1/6))/d^2 + (4*c*x^3*(a + b*x)^(1/6))/d + (4*c^3*x*(a + b*x)^(1/6))/d^3)