Integrand size = 19, antiderivative size = 136 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {1296 b^2 (a+b x)^{7/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}+\frac {7776 b^3 (a+b x)^{7/6}}{43225 (b c-a d)^4 (c+d x)^{7/6}} \] Output:
6/25*(b*x+a)^(7/6)/(-a*d+b*c)/(d*x+c)^(25/6)+108/475*b*(b*x+a)^(7/6)/(-a*d +b*c)^2/(d*x+c)^(19/6)+1296/6175*b^2*(b*x+a)^(7/6)/(-a*d+b*c)^3/(d*x+c)^(1 3/6)+7776/43225*b^3*(b*x+a)^(7/6)/(-a*d+b*c)^4/(d*x+c)^(7/6)
Time = 1.00 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {6 (a+b x)^{7/6} \left (-1729 a^3 d^3+273 a^2 b d^2 (25 c+6 d x)-21 a b^2 d \left (475 c^2+300 c d x+72 d^2 x^2\right )+b^3 \left (6175 c^3+8550 c^2 d x+5400 c d^2 x^2+1296 d^3 x^3\right )\right )}{43225 (b c-a d)^4 (c+d x)^{25/6}} \] Input:
Integrate[(a + b*x)^(1/6)/(c + d*x)^(31/6),x]
Output:
(6*(a + b*x)^(7/6)*(-1729*a^3*d^3 + 273*a^2*b*d^2*(25*c + 6*d*x) - 21*a*b^ 2*d*(475*c^2 + 300*c*d*x + 72*d^2*x^2) + b^3*(6175*c^3 + 8550*c^2*d*x + 54 00*c*d^2*x^2 + 1296*d^3*x^3)))/(43225*(b*c - a*d)^4*(c + d*x)^(25/6))
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.
| \(\displaystyle \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}}dx}{25 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \left (\frac {12 b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}}dx}{19 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)}\right )}{25 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \left (\frac {12 b \left (\frac {6 b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}}dx}{13 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)}\right )}{19 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)}\right )}{25 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)}+\frac {18 b \left (\frac {6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)}+\frac {12 b \left (\frac {36 b (a+b x)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)}\right )}{19 (b c-a d)}\right )}{25 (b c-a d)}\) |
Input:
Int[(a + b*x)^(1/6)/(c + d*x)^(31/6),x]
Output:
(6*(a + b*x)^(7/6))/(25*(b*c - a*d)*(c + d*x)^(25/6)) + (18*b*((6*(a + b*x )^(7/6))/(19*(b*c - a*d)*(c + d*x)^(19/6)) + (12*b*((6*(a + b*x)^(7/6))/(1 3*(b*c - a*d)*(c + d*x)^(13/6)) + (36*b*(a + b*x)^(7/6))/(91*(b*c - a*d)^2 *(c + d*x)^(7/6))))/(19*(b*c - a*d))))/(25*(b*c - a*d))
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 = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
| method | result | size |
| gosper | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (-1296 d^{3} x^{3} b^{3}+1512 x^{2} a \,b^{2} d^{3}-5400 x^{2} b^{3} c \,d^{2}-1638 x \,a^{2} b \,d^{3}+6300 x a \,b^{2} c \,d^{2}-8550 x \,b^{3} c^{2} d +1729 a^{3} d^{3}-6825 a^{2} b c \,d^{2}+9975 a \,b^{2} c^{2} d -6175 b^{3} c^{3}\right )}{43225 \left (x d +c \right )^{\frac {25}{6}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\) | \(171\) |
| orering | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (-1296 d^{3} x^{3} b^{3}+1512 x^{2} a \,b^{2} d^{3}-5400 x^{2} b^{3} c \,d^{2}-1638 x \,a^{2} b \,d^{3}+6300 x a \,b^{2} c \,d^{2}-8550 x \,b^{3} c^{2} d +1729 a^{3} d^{3}-6825 a^{2} b c \,d^{2}+9975 a \,b^{2} c^{2} d -6175 b^{3} c^{3}\right )}{43225 \left (x d +c \right )^{\frac {25}{6}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\) | \(171\) |
Input:
int((b*x+a)^(1/6)/(d*x+c)^(31/6),x,method=_RETURNVERBOSE)
Output:
-6/43225*(b*x+a)^(7/6)*(-1296*b^3*d^3*x^3+1512*a*b^2*d^3*x^2-5400*b^3*c*d^ 2*x^2-1638*a^2*b*d^3*x+6300*a*b^2*c*d^2*x-8550*b^3*c^2*d*x+1729*a^3*d^3-68 25*a^2*b*c*d^2+9975*a*b^2*c^2*d-6175*b^3*c^3)/(d*x+c)^(25/6)/(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 = 0.12 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.92 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {6 \, {\left (1296 \, b^{4} d^{3} x^{4} + 6175 \, a b^{3} c^{3} - 9975 \, a^{2} b^{2} c^{2} d + 6825 \, a^{3} b c d^{2} - 1729 \, a^{4} d^{3} + 216 \, {\left (25 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 18 \, {\left (475 \, b^{4} c^{2} d - 50 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )} x^{2} + {\left (6175 \, b^{4} c^{3} - 1425 \, a b^{3} c^{2} d + 525 \, a^{2} b^{2} c d^{2} - 91 \, a^{3} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{43225 \, {\left (b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4} + {\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} x^{5} + 5 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} x^{4} + 10 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} x^{3} + 10 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} x^{2} + 5 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )} x\right )}} \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(31/6),x, algorithm="fricas")
Output:
6/43225*(1296*b^4*d^3*x^4 + 6175*a*b^3*c^3 - 9975*a^2*b^2*c^2*d + 6825*a^3 *b*c*d^2 - 1729*a^4*d^3 + 216*(25*b^4*c*d^2 - a*b^3*d^3)*x^3 + 18*(475*b^4 *c^2*d - 50*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + (6175*b^4*c^3 - 1425*a*b^3* c^2*d + 525*a^2*b^2*c*d^2 - 91*a^3*b*d^3)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/ 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. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**(1/6)/(d*x+c)**(31/6),x)
Output:
Timed out
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}} \,d x } \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(31/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/6)/(d*x + c)^(31/6), x)
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}} \,d x } \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(31/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/6)/(d*x + c)^(31/6), x)
Time = 0.62 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {7776\,b^4\,x^4\,{\left (a+b\,x\right )}^{1/6}}{43225\,d^2\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{1/6}\,\left (10374\,a^4\,d^3-40950\,a^3\,b\,c\,d^2+59850\,a^2\,b^2\,c^2\,d-37050\,a\,b^3\,c^3\right )}{43225\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{1/6}\,\left (-546\,a^3\,b\,d^3+3150\,a^2\,b^2\,c\,d^2-8550\,a\,b^3\,c^2\,d+37050\,b^4\,c^3\right )}{43225\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^2\,x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (7\,a^2\,d^2-50\,a\,b\,c\,d+475\,b^2\,c^2\right )}{43225\,d^4\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^3\,x^3\,\left (a\,d-25\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{43225\,d^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \] Input:
int((a + b*x)^(1/6)/(c + d*x)^(31/6),x)
Output:
((c + d*x)^(5/6)*((7776*b^4*x^4*(a + b*x)^(1/6))/(43225*d^2*(a*d - b*c)^4) - ((a + b*x)^(1/6)*(10374*a^4*d^3 - 37050*a*b^3*c^3 + 59850*a^2*b^2*c^2*d - 40950*a^3*b*c*d^2))/(43225*d^5*(a*d - b*c)^4) + (x*(a + b*x)^(1/6)*(370 50*b^4*c^3 - 546*a^3*b*d^3 + 3150*a^2*b^2*c*d^2 - 8550*a*b^3*c^2*d))/(4322 5*d^5*(a*d - b*c)^4) + (108*b^2*x^2*(a + b*x)^(1/6)*(7*a^2*d^2 + 475*b^2*c ^2 - 50*a*b*c*d))/(43225*d^4*(a*d - b*c)^4) - (1296*b^3*x^3*(a*d - 25*b*c) *(a + b*x)^(1/6))/(43225*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)
Time = 8.02 (sec) , antiderivative size = 2011, normalized size of antiderivative = 14.79 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^(1/6)/(d*x+c)^(31/6),x)
Output:
(2*( - 43225*(c + d*x)*(a + b*x)**(3/2)*a**4*b*d**4 + 234650*(c + d*x)*(a + b*x)**(3/2)*a**3*b**2*c*d**3 + 61750*(c + d*x)*(a + b*x)**(3/2)*a**3*b** 2*d**4*x - 543400*(c + d*x)*(a + b*x)**(3/2)*a**2*b**3*c**2*d**2 - 382850* (c + d*x)*(a + b*x)**(3/2)*a**2*b**3*c*d**3*x - 98800*(c + d*x)*(a + b*x)* *(3/2)*a**2*b**3*d**4*x**2 + 753350*(c + d*x)*(a + b*x)**(3/2)*a*b**4*c**3 *d + 1173250*(c + d*x)*(a + b*x)**(3/2)*a*b**4*c**2*d**2*x + 790400*(c + d *x)*(a + b*x)**(3/2)*a*b**4*c*d**3*x**2 + 197600*(c + d*x)*(a + b*x)**(3/2 )*a*b**4*d**4*x**3 - 401375*(c + d*x)*(a + b*x)**(3/2)*b**5*c**4 - 852150* (c + d*x)*(a + b*x)**(3/2)*b**5*c**3*d*x - 691600*(c + d*x)*(a + b*x)**(3/ 2)*b**5*c**2*d**2*x**2 - 197600*(c + d*x)*(a + b*x)**(3/2)*b**5*c*d**3*x** 3 - 46683*sqrt(a + b*x)*a**6*d**5 + 292383*sqrt(a + b*x)*a**5*b*c*d**4 + 1 2285*sqrt(a + b*x)*a**5*b*d**5*x - 784350*sqrt(a + b*x)*a**4*b**2*c**2*d** 3 - 106785*sqrt(a + b*x)*a**4*b**2*c*d**4*x - 22680*sqrt(a + b*x)*a**4*b** 2*d**5*x**2 + 1205550*sqrt(a + b*x)*a**3*b**3*c**3*d**2 + 479250*sqrt(a + b*x)*a**3*b**3*c**2*d**3*x + 265680*sqrt(a + b*x)*a**3*b**3*c*d**4*x**2 + 58320*sqrt(a + b*x)*a**3*b**3*d**5*x**3 - 1833975*sqrt(a + b*x)*a**2*b**4* c**4*d - 3719250*sqrt(a + b*x)*a**2*b**4*c**3*d**2*x - 4860000*sqrt(a + b* x)*a**2*b**4*c**2*d**3*x**2 - 2974320*sqrt(a + b*x)*a**2*b**4*c*d**4*x**3 - 699840*sqrt(a + b*x)*a**2*b**4*d**5*x**4 + 1167075*sqrt(a + b*x)*a*b**5* c**5 + 2167425*sqrt(a + b*x)*a*b**5*c**4*d*x + 615600*sqrt(a + b*x)*a*b...