Integrand size = 19, antiderivative size = 32 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\frac {6 (a+b x)^{7/6}}{7 (b c-a d) (c+d x)^{7/6}} \] Output:
6/7*(b*x+a)^(7/6)/(-a*d+b*c)/(d*x+c)^(7/6)
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\frac {6 (a+b x)^{7/6}}{7 (b c-a d) (c+d x)^{7/6}} \] Input:
Integrate[(a + b*x)^(1/6)/(c + d*x)^(13/6),x]
Output:
(6*(a + b*x)^(7/6))/(7*(b*c - a*d)*(c + d*x)^(7/6))
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {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.
\(\displaystyle \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {6 (a+b x)^{7/6}}{7 (c+d x)^{7/6} (b c-a d)}\) |
Input:
Int[(a + b*x)^(1/6)/(c + d*x)^(13/6),x]
Output:
(6*(a + b*x)^(7/6))/(7*(b*c - a*d)*(c + d*x)^(7/6))
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]
Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}}}{7 \left (x d +c \right )^{\frac {7}{6}} \left (a d -b c \right )}\) | \(27\) |
orering | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}}}{7 \left (x d +c \right )^{\frac {7}{6}} \left (a d -b c \right )}\) | \(27\) |
Input:
int((b*x+a)^(1/6)/(d*x+c)^(13/6),x,method=_RETURNVERBOSE)
Output:
-6/7*(b*x+a)^(7/6)/(d*x+c)^(7/6)/(a*d-b*c)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\frac {6 \, {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{7 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )}} \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(13/6),x, algorithm="fricas")
Output:
6/7*(b*x + a)^(7/6)*(d*x + c)^(5/6)/(b*c^3 - a*c^2*d + (b*c*d^2 - a*d^3)*x ^2 + 2*(b*c^2*d - a*c*d^2)*x)
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\int \frac {\sqrt [6]{a + b x}}{\left (c + d x\right )^{\frac {13}{6}}}\, dx \] Input:
integrate((b*x+a)**(1/6)/(d*x+c)**(13/6),x)
Output:
Integral((a + b*x)**(1/6)/(c + d*x)**(13/6), x)
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(13/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/6)/(d*x + c)^(13/6), x)
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(13/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/6)/(d*x + c)^(13/6), x)
Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.06 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=-\frac {\left (\frac {6\,a\,{\left (a+b\,x\right )}^{1/6}}{7\,a\,d^3-7\,b\,c\,d^2}+\frac {6\,b\,x\,{\left (a+b\,x\right )}^{1/6}}{7\,a\,d^3-7\,b\,c\,d^2}\right )\,{\left (c+d\,x\right )}^{5/6}}{x^2-\frac {7\,b\,c^3-7\,a\,c^2\,d}{7\,a\,d^3-7\,b\,c\,d^2}+\frac {14\,c\,d\,x\,\left (a\,d-b\,c\right )}{7\,a\,d^3-7\,b\,c\,d^2}} \] Input:
int((a + b*x)^(1/6)/(c + d*x)^(13/6),x)
Output:
-(((6*a*(a + b*x)^(1/6))/(7*a*d^3 - 7*b*c*d^2) + (6*b*x*(a + b*x)^(1/6))/( 7*a*d^3 - 7*b*c*d^2))*(c + d*x)^(5/6))/(x^2 - (7*b*c^3 - 7*a*c^2*d)/(7*a*d ^3 - 7*b*c*d^2) + (14*c*d*x*(a*d - b*c))/(7*a*d^3 - 7*b*c*d^2))
Time = 6.31 (sec) , antiderivative size = 494, normalized size of antiderivative = 15.44 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx=\frac {-\frac {2 \left (d x +c \right ) \left (b x +a \right )^{\frac {3}{2}} a b d}{3}+\frac {2 \left (d x +c \right ) \left (b x +a \right )^{\frac {3}{2}} b^{2} c}{3}-\frac {6 \sqrt {b x +a}\, a^{3} d^{2}}{7}+\frac {48 \sqrt {b x +a}\, a^{2} b c d}{7}+\frac {30 \sqrt {b x +a}\, a^{2} b \,d^{2} x}{7}-6 \sqrt {b x +a}\, a \,b^{2} c^{2}+\frac {12 \sqrt {b x +a}\, a \,b^{2} c d x}{7}+\frac {36 \sqrt {b x +a}\, a \,b^{2} d^{2} x^{2}}{7}-6 \sqrt {b x +a}\, b^{3} c^{2} x -\frac {36 \sqrt {b x +a}\, b^{3} c d \,x^{2}}{7}+8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) a \,b^{2} c +8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) a \,b^{2} d x +8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{3} c x +8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{3} d \,x^{2}-8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) a \,b^{2} c -8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) a \,b^{2} d x -8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{3} c x -8 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{3} d \,x^{2}}{\left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{3}} d \left (a^{2} b \,d^{3} x^{2}-2 a \,b^{2} c \,d^{2} x^{2}+b^{3} c^{2} d \,x^{2}+a^{3} d^{3} x -a^{2} b c \,d^{2} x -a \,b^{2} c^{2} d x +b^{3} c^{3} x +a^{3} c \,d^{2}-2 a^{2} b \,c^{2} d +a \,b^{2} c^{3}\right )} \] Input:
int((b*x+a)^(1/6)/(d*x+c)^(13/6),x)
Output:
(2*( - 7*(c + d*x)*(a + b*x)**(3/2)*a*b*d + 7*(c + d*x)*(a + b*x)**(3/2)*b **2*c - 9*sqrt(a + b*x)*a**3*d**2 + 72*sqrt(a + b*x)*a**2*b*c*d + 45*sqrt( a + b*x)*a**2*b*d**2*x - 63*sqrt(a + b*x)*a*b**2*c**2 + 18*sqrt(a + b*x)*a *b**2*c*d*x + 54*sqrt(a + b*x)*a*b**2*d**2*x**2 - 63*sqrt(a + b*x)*b**3*c* *2*x - 54*sqrt(a + b*x)*b**3*c*d*x**2 + 84*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**2*c + 84*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6)) *a*b**2*d*x + 84*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*c*x + 84*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*d*x**2 - 84*(c + d*x )*sqrt(a + b*x)*log((c + d*x)**(1/6))*a*b**2*c - 84*(c + d*x)*sqrt(a + b*x )*log((c + d*x)**(1/6))*a*b**2*d*x - 84*(c + d*x)*sqrt(a + b*x)*log((c + d *x)**(1/6))*b**3*c*x - 84*(c + d*x)*sqrt(a + b*x)*log((c + d*x)**(1/6))*b* *3*d*x**2))/(21*(c + d*x)**(1/6)*(a + b*x)**(1/3)*d*(a**3*c*d**2 + a**3*d* *3*x - 2*a**2*b*c**2*d - a**2*b*c*d**2*x + a**2*b*d**3*x**2 + a*b**2*c**3 - a*b**2*c**2*d*x - 2*a*b**2*c*d**2*x**2 + b**3*c**3*x + b**3*c**2*d*x**2) )