Integrand size = 19, antiderivative size = 101 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=\frac {6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac {72 b (a+b x)^{5/6}}{187 (b c-a d)^2 (c+d x)^{11/6}}+\frac {432 b^2 (a+b x)^{5/6}}{935 (b c-a d)^3 (c+d x)^{5/6}} \] Output:
6/17*(b*x+a)^(5/6)/(-a*d+b*c)/(d*x+c)^(17/6)+72/187*b*(b*x+a)^(5/6)/(-a*d+ b*c)^2/(d*x+c)^(11/6)+432/935*b^2*(b*x+a)^(5/6)/(-a*d+b*c)^3/(d*x+c)^(5/6)
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=\frac {6 (a+b x)^{5/6} \left (55 a^2 d^2-10 a b d (17 c+6 d x)+b^2 \left (187 c^2+204 c d x+72 d^2 x^2\right )\right )}{935 (b c-a d)^3 (c+d x)^{17/6}} \] Input:
Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]
Output:
(6*(a + b*x)^(5/6)*(55*a^2*d^2 - 10*a*b*d*(17*c + 6*d*x) + b^2*(187*c^2 + 204*c*d*x + 72*d^2*x^2)))/(935*(b*c - a*d)^3*(c + d*x)^(17/6))
Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {12 b \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}}dx}{17 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {12 b \left (\frac {6 b \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}}dx}{11 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)}\right )}{17 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {12 b \left (\frac {36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)}\right )}{17 (b c-a d)}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)}\) |
Input:
Int[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]
Output:
(6*(a + b*x)^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (12*b*((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))))/(17*(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 = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (72 d^{2} x^{2} b^{2}-60 x a b \,d^{2}+204 x \,b^{2} c d +55 a^{2} d^{2}-170 a b c d +187 b^{2} c^{2}\right )}{935 \left (x d +c \right )^{\frac {17}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
orering | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (72 d^{2} x^{2} b^{2}-60 x a b \,d^{2}+204 x \,b^{2} c d +55 a^{2} d^{2}-170 a b c d +187 b^{2} c^{2}\right )}{935 \left (x d +c \right )^{\frac {17}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
Input:
int(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x,method=_RETURNVERBOSE)
Output:
-6/935*(b*x+a)^(5/6)*(72*b^2*d^2*x^2-60*a*b*d^2*x+204*b^2*c*d*x+55*a^2*d^2 -170*a*b*c*d+187*b^2*c^2)/(d*x+c)^(17/6)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^ 2*d-b^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (83) = 166\).
Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=\frac {6 \, {\left (72 \, b^{2} d^{2} x^{2} + 187 \, b^{2} c^{2} - 170 \, a b c d + 55 \, a^{2} d^{2} + 12 \, {\left (17 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{935 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{3} + 3 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} + 3 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\right )} x\right )}} \] Input:
integrate(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x, algorithm="fricas")
Output:
6/935*(72*b^2*d^2*x^2 + 187*b^2*c^2 - 170*a*b*c*d + 55*a^2*d^2 + 12*(17*b^ 2*c*d - 5*a*b*d^2)*x)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(b^3*c^6 - 3*a*b^2*c ^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3* a^2*b*c*d^5 - a^3*d^6)*x^3 + 3*(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^ 2*d^4 - a^3*c*d^5)*x^2 + 3*(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4)*x)
Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x+a)**(1/6)/(d*x+c)**(23/6),x)
Output:
Timed out
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {23}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)
\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {23}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)
Time = 0.61 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.01 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {330\,a^3\,d^2-1020\,a^2\,b\,c\,d+1122\,a\,b^2\,c^2}{935\,d^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x\,\left (-30\,a^2\,b\,d^2+204\,a\,b^2\,c\,d+1122\,b^3\,c^2\right )}{935\,d^3\,{\left (a\,d-b\,c\right )}^3}+\frac {432\,b^3\,x^3}{935\,d\,{\left (a\,d-b\,c\right )}^3}+\frac {72\,b^2\,x^2\,\left (a\,d+17\,b\,c\right )}{935\,d^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^3\,{\left (a+b\,x\right )}^{1/6}+\frac {c^3\,{\left (a+b\,x\right )}^{1/6}}{d^3}+\frac {3\,c\,x^2\,{\left (a+b\,x\right )}^{1/6}}{d}+\frac {3\,c^2\,x\,{\left (a+b\,x\right )}^{1/6}}{d^2}} \] Input:
int(1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x)
Output:
-((c + d*x)^(1/6)*((330*a^3*d^2 + 1122*a*b^2*c^2 - 1020*a^2*b*c*d)/(935*d^ 3*(a*d - b*c)^3) + (x*(1122*b^3*c^2 - 30*a^2*b*d^2 + 204*a*b^2*c*d))/(935* d^3*(a*d - b*c)^3) + (432*b^3*x^3)/(935*d*(a*d - b*c)^3) + (72*b^2*x^2*(a* d + 17*b*c))/(935*d^2*(a*d - b*c)^3)))/(x^3*(a + b*x)^(1/6) + (c^3*(a + b* x)^(1/6))/d^3 + (3*c*x^2*(a + b*x)^(1/6))/d + (3*c^2*x*(a + b*x)^(1/6))/d^ 2)
Time = 1.46 (sec) , antiderivative size = 894, normalized size of antiderivative = 8.85 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx =\text {Too large to display} \] Input:
int(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x)
Output:
(2*(a + b*x)**(1/3)*( - 165*d*sqrt(a + b*x)*a**3*c*d**3 - 165*d*sqrt(a + b *x)*a**3*d**4*x + 765*d*sqrt(a + b*x)*a**2*b*c**2*d**2 + 1035*d*sqrt(a + b *x)*a**2*b*c*d**3*x + 270*d*sqrt(a + b*x)*a**2*b*d**4*x**2 - 1683*d*sqrt(a + b*x)*a*b**2*c**3*d - 3519*d*sqrt(a + b*x)*a*b**2*c**2*d**2*x - 2484*d*s qrt(a + b*x)*a*b**2*c*d**3*x**2 - 648*d*sqrt(a + b*x)*a*b**2*d**4*x**3 - 2 805*d*sqrt(a + b*x)*b**3*c**4 - 12903*d*sqrt(a + b*x)*b**3*c**3*d*x - 2111 4*d*sqrt(a + b*x)*b**3*c**2*d**2*x**2 - 14904*d*sqrt(a + b*x)*b**3*c*d**3* x**3 - 3888*d*sqrt(a + b*x)*b**3*d**4*x**4 - 187*(c + d*x)**2*sqrt(a + b*x )*a**2*b*d**3 + 748*(c + d*x)**2*sqrt(a + b*x)*a*b**2*c*d**2 + 374*(c + d* x)**2*sqrt(a + b*x)*a*b**2*d**3*x - 561*(c + d*x)**2*sqrt(a + b*x)*b**3*c* *2*d - 374*(c + d*x)**2*sqrt(a + b*x)*b**3*c*d**2*x - 4488*sqrt(a + b*x)*l og((a + b*x)**(1/6))*b**3*c**4*d - 17952*sqrt(a + b*x)*log((a + b*x)**(1/6 ))*b**3*c**3*d**2*x - 26928*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*c**2* d**3*x**2 - 17952*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*c*d**4*x**3 - 4 488*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*d**5*x**4 + 4488*sqrt(a + b*x )*log((c + d*x)**(1/6))*b**3*c**4*d + 17952*sqrt(a + b*x)*log((c + d*x)**( 1/6))*b**3*c**3*d**2*x + 26928*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**3*c* *2*d**3*x**2 + 17952*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**3*c*d**4*x**3 + 4488*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**3*d**5*x**4))/(935*(c + d*x) **(5/6)*d**2*(a**4*c**3*d**3 + 3*a**4*c**2*d**4*x + 3*a**4*c*d**5*x**2 ...