Integrand size = 19, antiderivative size = 101 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {72 b \sqrt [6]{a+b x}}{91 (b c-a d)^2 (c+d x)^{7/6}}+\frac {432 b^2 \sqrt [6]{a+b x}}{91 (b c-a d)^3 \sqrt [6]{c+d x}} \] Output:
6/13*(b*x+a)^(1/6)/(-a*d+b*c)/(d*x+c)^(13/6)+72/91*b*(b*x+a)^(1/6)/(-a*d+b *c)^2/(d*x+c)^(7/6)+432/91*b^2*(b*x+a)^(1/6)/(-a*d+b*c)^3/(d*x+c)^(1/6)
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (7 a^2 d^2-2 a b d (13 c+6 d x)+b^2 \left (91 c^2+156 c d x+72 d^2 x^2\right )\right )}{91 (b c-a d)^3 (c+d x)^{13/6}} \] Input:
Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x]
Output:
(6*(a + b*x)^(1/6)*(7*a^2*d^2 - 2*a*b*d*(13*c + 6*d*x) + b^2*(91*c^2 + 156 *c*d*x + 72*d^2*x^2)))/(91*(b*c - a*d)^3*(c + d*x)^(13/6))
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.
\(\displaystyle \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {12 b \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}}dx}{13 (b c-a d)}+\frac {6 \sqrt [6]{a+b x}}{13 (c+d x)^{13/6} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {12 b \left (\frac {6 b \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}}dx}{7 (b c-a d)}+\frac {6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)}\right )}{13 (b c-a d)}+\frac {6 \sqrt [6]{a+b x}}{13 (c+d x)^{13/6} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {12 b \left (\frac {36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)}\right )}{13 (b c-a d)}+\frac {6 \sqrt [6]{a+b x}}{13 (c+d x)^{13/6} (b c-a d)}\) |
Input:
Int[1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x]
Output:
(6*(a + b*x)^(1/6))/(13*(b*c - a*d)*(c + d*x)^(13/6)) + (12*b*((6*(a + b*x )^(1/6))/(7*(b*c - a*d)*(c + d*x)^(7/6)) + (36*b*(a + b*x)^(1/6))/(7*(b*c - a*d)^2*(c + d*x)^(1/6))))/(13*(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 = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (72 d^{2} x^{2} b^{2}-12 x a b \,d^{2}+156 x \,b^{2} c d +7 a^{2} d^{2}-26 a b c d +91 b^{2} c^{2}\right )}{91 \left (x d +c \right )^{\frac {13}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
orering | \(-\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (72 d^{2} x^{2} b^{2}-12 x a b \,d^{2}+156 x \,b^{2} c d +7 a^{2} d^{2}-26 a b c d +91 b^{2} c^{2}\right )}{91 \left (x d +c \right )^{\frac {13}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
Input:
int(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x,method=_RETURNVERBOSE)
Output:
-6/91*(b*x+a)^(1/6)*(72*b^2*d^2*x^2-12*a*b*d^2*x+156*b^2*c*d*x+7*a^2*d^2-2 6*a*b*c*d+91*b^2*c^2)/(d*x+c)^(13/6)/(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 = 0.10 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {6 \, {\left (72 \, b^{2} d^{2} x^{2} + 91 \, b^{2} c^{2} - 26 \, a b c d + 7 \, a^{2} d^{2} + 12 \, {\left (13 \, b^{2} c d - a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{91 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{3} + 3 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} + 3 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\right )} x\right )}} \] Input:
integrate(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="fricas")
Output:
6/91*(72*b^2*d^2*x^2 + 91*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2 + 12*(13*b^2*c* d - a*b*d^2)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(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. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x+a)**(5/6)/(d*x+c)**(19/6),x)
Output:
Timed out
\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(5/6)*(d*x + c)^(19/6)), x)
\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(5/6)*(d*x + c)^(19/6)), x)
Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{19/6}} \,d x \] Input:
int(1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x)
Output:
int(1/((a + b*x)^(5/6)*(c + d*x)^(19/6)), x)
Time = 0.97 (sec) , antiderivative size = 638, normalized size of antiderivative = 6.32 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx =\text {Too large to display} \] Input:
int(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x)
Output:
(2*(c + d*x)**(5/6)*( - (a + b*x)**(3/2)*a**2*d**2 + 4*(a + b*x)**(3/2)*a* b*c*d + 2*(a + b*x)**(3/2)*a*b*d**2*x - 3*(a + b*x)**(3/2)*b**2*c**2 - 2*( a + b*x)**(3/2)*b**2*c*d*x - 24*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**2 *c**2 - 48*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**2*c*d*x - 24*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**2*d**2*x**2 - 24*sqrt(a + b*x)*log((a + b* x)**(1/6))*b**3*c**2*x - 48*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*c*d*x **2 - 24*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*d**2*x**3 + 24*sqrt(a + b*x)*log((c + d*x)**(1/6))*a*b**2*c**2 + 48*sqrt(a + b*x)*log((c + d*x)**( 1/6))*a*b**2*c*d*x + 24*sqrt(a + b*x)*log((c + d*x)**(1/6))*a*b**2*d**2*x* *2 + 24*sqrt(a + b*x)*log((c + d*x)**(1/6))*b**3*c**2*x + 48*sqrt(a + b*x) *log((c + d*x)**(1/6))*b**3*c*d*x**2 + 24*sqrt(a + b*x)*log((c + d*x)**(1/ 6))*b**3*d**2*x**3))/(5*(a + b*x)**(1/3)*(a**4*c**3*d**3 + 3*a**4*c**2*d** 4*x + 3*a**4*c*d**5*x**2 + a**4*d**6*x**3 - 3*a**3*b*c**4*d**2 - 8*a**3*b* c**3*d**3*x - 6*a**3*b*c**2*d**4*x**2 + a**3*b*d**6*x**4 + 3*a**2*b**2*c** 5*d + 6*a**2*b**2*c**4*d**2*x - 6*a**2*b**2*c**2*d**4*x**3 - 3*a**2*b**2*c *d**5*x**4 - a*b**3*c**6 + 6*a*b**3*c**4*d**2*x**2 + 8*a*b**3*c**3*d**3*x* *3 + 3*a*b**3*c**2*d**4*x**4 - b**4*c**6*x - 3*b**4*c**5*d*x**2 - 3*b**4*c **4*d**2*x**3 - b**4*c**3*d**3*x**4))