Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {6 (a+b x)^{5/6}}{11 (b c-a d) (c+d x)^{11/6}}+\frac {36 b (a+b x)^{5/6}}{55 (b c-a d)^2 (c+d x)^{5/6}} \] Output:
6/11*(b*x+a)^(5/6)/(-a*d+b*c)/(d*x+c)^(11/6)+36/55*b*(b*x+a)^(5/6)/(-a*d+b *c)^2/(d*x+c)^(5/6)
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {6 (a+b x)^{5/6} (11 b c-5 a d+6 b d x)}{55 (b c-a d)^2 (c+d x)^{11/6}} \] Input:
Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(17/6)),x]
Output:
(6*(a + b*x)^(5/6)*(11*b*c - 5*a*d + 6*b*d*x))/(55*(b*c - a*d)^2*(c + d*x) ^(11/6))
Time = 0.16 (sec) , antiderivative size = 66, 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 = {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.
\(\displaystyle \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {6 b \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}}dx}{11 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)}\) |
Input:
Int[1/((a + b*x)^(1/6)*(c + d*x)^(17/6)),x]
Output:
(6*(a + b*x)^(5/6))/(11*(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))
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 = 54, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (-6 b d x +5 a d -11 b c \right )}{55 \left (x d +c \right )^{\frac {11}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
orering | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (-6 b d x +5 a d -11 b c \right )}{55 \left (x d +c \right )^{\frac {11}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
Input:
int(1/(b*x+a)^(1/6)/(d*x+c)^(17/6),x,method=_RETURNVERBOSE)
Output:
-6/55*(b*x+a)^(5/6)*(-6*b*d*x+5*a*d-11*b*c)/(d*x+c)^(11/6)/(a^2*d^2-2*a*b* c*d+b^2*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {6 \, {\left (6 \, b d x + 11 \, b c - 5 \, a d\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{55 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \] Input:
integrate(1/(b*x+a)^(1/6)/(d*x+c)^(17/6),x, algorithm="fricas")
Output:
6/55*(6*b*d*x + 11*b*c - 5*a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2 *(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x)
Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x+a)**(1/6)/(d*x+c)**(17/6),x)
Output:
Timed out
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/6)/(d*x+c)^(17/6),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)), x)
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/6)/(d*x+c)^(17/6),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)), x)
Time = 0.51 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {x\,\left (66\,c\,b^2+6\,a\,d\,b\right )}{55\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {30\,a^2\,d-66\,a\,b\,c}{55\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {36\,b^2\,x^2}{55\,d\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,{\left (a+b\,x\right )}^{1/6}+\frac {c^2\,{\left (a+b\,x\right )}^{1/6}}{d^2}+\frac {2\,c\,x\,{\left (a+b\,x\right )}^{1/6}}{d}} \] Input:
int(1/((a + b*x)^(1/6)*(c + d*x)^(17/6)),x)
Output:
((c + d*x)^(1/6)*((x*(66*b^2*c + 6*a*b*d))/(55*d^2*(a*d - b*c)^2) - (30*a^ 2*d - 66*a*b*c)/(55*d^2*(a*d - b*c)^2) + (36*b^2*x^2)/(55*d*(a*d - b*c)^2) ))/(x^2*(a + b*x)^(1/6) + (c^2*(a + b*x)^(1/6))/d^2 + (2*c*x*(a + b*x)^(1/ 6))/d)
Time = 1.04 (sec) , antiderivative size = 541, normalized size of antiderivative = 8.20 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {2 \left (b x +a \right )^{\frac {1}{3}} \left (-45 d^{3} \sqrt {b x +a}\, a^{2} c -45 d^{4} \sqrt {b x +a}\, a^{2} x +198 d^{2} \sqrt {b x +a}\, a b \,c^{2}+306 d^{3} \sqrt {b x +a}\, a b c x +108 d^{4} \sqrt {b x +a}\, a b \,x^{2}+495 d \sqrt {b x +a}\, b^{2} c^{3}+1683 d^{2} \sqrt {b x +a}\, b^{2} c^{2} x +1836 d^{3} \sqrt {b x +a}\, b^{2} c \,x^{2}+648 d^{4} \sqrt {b x +a}\, b^{2} x^{3}-55 \left (d x +c \right )^{2} \sqrt {b x +a}\, a b \,d^{2}+55 \left (d x +c \right )^{2} \sqrt {b x +a}\, b^{2} c d +660 \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c^{3} d +1980 \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c^{2} d^{2} x +1980 \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} c \,d^{3} x^{2}+660 \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{2} d^{4} x^{3}-660 \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c^{3} d -1980 \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c^{2} d^{2} x -1980 \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} c \,d^{3} x^{2}-660 \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{2} d^{4} x^{3}\right )}{165 \left (d x +c \right )^{\frac {5}{6}} d^{2} \left (a^{2} b \,d^{4} x^{3}-2 a \,b^{2} c \,d^{3} x^{3}+b^{3} c^{2} d^{2} x^{3}+a^{3} d^{4} x^{2}-3 a \,b^{2} c^{2} d^{2} x^{2}+2 b^{3} c^{3} d \,x^{2}+2 a^{3} c \,d^{3} x -3 a^{2} b \,c^{2} d^{2} x +b^{3} c^{4} x +a^{3} c^{2} d^{2}-2 a^{2} b \,c^{3} d +a \,b^{2} c^{4}\right )} \] Input:
int(1/(b*x+a)^(1/6)/(d*x+c)^(17/6),x)
Output:
(2*(a + b*x)**(1/3)*( - 45*d*sqrt(a + b*x)*a**2*c*d**2 - 45*d*sqrt(a + b*x )*a**2*d**3*x + 198*d*sqrt(a + b*x)*a*b*c**2*d + 306*d*sqrt(a + b*x)*a*b*c *d**2*x + 108*d*sqrt(a + b*x)*a*b*d**3*x**2 + 495*d*sqrt(a + b*x)*b**2*c** 3 + 1683*d*sqrt(a + b*x)*b**2*c**2*d*x + 1836*d*sqrt(a + b*x)*b**2*c*d**2* x**2 + 648*d*sqrt(a + b*x)*b**2*d**3*x**3 - 55*(c + d*x)**2*sqrt(a + b*x)* a*b*d**2 + 55*(c + d*x)**2*sqrt(a + b*x)*b**2*c*d + 660*sqrt(a + b*x)*log( (a + b*x)**(1/6))*b**2*c**3*d + 1980*sqrt(a + b*x)*log((a + b*x)**(1/6))*b **2*c**2*d**2*x + 1980*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**2*c*d**3*x** 2 + 660*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**2*d**4*x**3 - 660*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**2*c**3*d - 1980*sqrt(a + b*x)*log((c + d*x)* *(1/6))*b**2*c**2*d**2*x - 1980*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**2*c *d**3*x**2 - 660*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**2*d**4*x**3))/(165 *(c + d*x)**(5/6)*d**2*(a**3*c**2*d**2 + 2*a**3*c*d**3*x + a**3*d**4*x**2 - 2*a**2*b*c**3*d - 3*a**2*b*c**2*d**2*x + a**2*b*d**4*x**3 + a*b**2*c**4 - 3*a*b**2*c**2*d**2*x**2 - 2*a*b**2*c*d**3*x**3 + b**3*c**4*x + 2*b**3*c* *3*d*x**2 + b**3*c**2*d**2*x**3))