Integrand size = 19, antiderivative size = 101 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=\frac {6 (a+b x)^{7/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {72 b (a+b x)^{7/6}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {432 b^2 (a+b x)^{7/6}}{1729 (b c-a d)^3 (c+d x)^{7/6}} \] Output:
6/19*(b*x+a)^(7/6)/(-a*d+b*c)/(d*x+c)^(19/6)+72/247*b*(b*x+a)^(7/6)/(-a*d+ b*c)^2/(d*x+c)^(13/6)+432/1729*b^2*(b*x+a)^(7/6)/(-a*d+b*c)^3/(d*x+c)^(7/6 )
Time = 0.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=\frac {6 (a+b x)^{7/6} \left (91 a^2 d^2-14 a b d (19 c+6 d x)+b^2 \left (247 c^2+228 c d x+72 d^2 x^2\right )\right )}{1729 (b c-a d)^3 (c+d x)^{19/6}} \] Input:
Integrate[(a + b*x)^(1/6)/(c + d*x)^(25/6),x]
Output:
(6*(a + b*x)^(7/6)*(91*a^2*d^2 - 14*a*b*d*(19*c + 6*d*x) + b^2*(247*c^2 + 228*c*d*x + 72*d^2*x^2)))/(1729*(b*c - a*d)^3*(c + d*x)^(19/6))
Time = 0.19 (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.
\(\displaystyle \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {12 b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}}dx}{19 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {12 b \left (\frac {6 b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}}dx}{13 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)}\right )}{19 (b c-a d)}+\frac {6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)}+\frac {12 b \left (\frac {36 b (a+b x)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)}\right )}{19 (b c-a d)}\) |
Input:
Int[(a + b*x)^(1/6)/(c + d*x)^(25/6),x]
Output:
(6*(a + b*x)^(7/6))/(19*(b*c - a*d)*(c + d*x)^(19/6)) + (12*b*((6*(a + b*x )^(7/6))/(13*(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))
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 = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (72 d^{2} x^{2} b^{2}-84 x a b \,d^{2}+228 x \,b^{2} c d +91 a^{2} d^{2}-266 a b c d +247 b^{2} c^{2}\right )}{1729 \left (x d +c \right )^{\frac {19}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
orering | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (72 d^{2} x^{2} b^{2}-84 x a b \,d^{2}+228 x \,b^{2} c d +91 a^{2} d^{2}-266 a b c d +247 b^{2} c^{2}\right )}{1729 \left (x d +c \right )^{\frac {19}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
Input:
int((b*x+a)^(1/6)/(d*x+c)^(25/6),x,method=_RETURNVERBOSE)
Output:
-6/1729*(b*x+a)^(7/6)*(72*b^2*d^2*x^2-84*a*b*d^2*x+228*b^2*c*d*x+91*a^2*d^ 2-266*a*b*c*d+247*b^2*c^2)/(d*x+c)^(19/6)/(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. 338 vs. \(2 (83) = 166\).
Time = 0.13 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.35 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=\frac {6 \, {\left (72 \, b^{3} d^{2} x^{3} + 247 \, a b^{2} c^{2} - 266 \, a^{2} b c d + 91 \, a^{3} d^{2} + 12 \, {\left (19 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + {\left (247 \, b^{3} c^{2} - 38 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{1729 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3} + {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} x^{4} + 4 \, {\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} x^{3} + 6 \, {\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} x^{2} + 4 \, {\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )} x\right )}} \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(25/6),x, algorithm="fricas")
Output:
6/1729*(72*b^3*d^2*x^3 + 247*a*b^2*c^2 - 266*a^2*b*c*d + 91*a^3*d^2 + 12*( 19*b^3*c*d - a*b^2*d^2)*x^2 + (247*b^3*c^2 - 38*a*b^2*c*d + 7*a^2*b*d^2)*x )*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d ^2 - a^3*c^4*d^3 + (b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^ 7)*x^4 + 4*(b^3*c^4*d^3 - 3*a*b^2*c^3*d^4 + 3*a^2*b*c^2*d^5 - a^3*c*d^6)*x ^3 + 6*(b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)*x^2 + 4*(b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4)*x)
Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**(1/6)/(d*x+c)**(25/6),x)
Output:
Timed out
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {25}{6}}} \,d x } \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(25/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/6)/(d*x + c)^(25/6), x)
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {25}{6}}} \,d x } \] Input:
integrate((b*x+a)^(1/6)/(d*x+c)^(25/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/6)/(d*x + c)^(25/6), x)
Time = 0.51 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {{\left (a+b\,x\right )}^{1/6}\,\left (546\,a^3\,d^2-1596\,a^2\,b\,c\,d+1482\,a\,b^2\,c^2\right )}{1729\,d^4\,{\left (a\,d-b\,c\right )}^3}+\frac {432\,b^3\,x^3\,{\left (a+b\,x\right )}^{1/6}}{1729\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {x\,{\left (a+b\,x\right )}^{1/6}\,\left (42\,a^2\,b\,d^2-228\,a\,b^2\,c\,d+1482\,b^3\,c^2\right )}{1729\,d^4\,{\left (a\,d-b\,c\right )}^3}-\frac {72\,b^2\,x^2\,\left (a\,d-19\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{1729\,d^3\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \] Input:
int((a + b*x)^(1/6)/(c + d*x)^(25/6),x)
Output:
-((c + d*x)^(5/6)*(((a + b*x)^(1/6)*(546*a^3*d^2 + 1482*a*b^2*c^2 - 1596*a ^2*b*c*d))/(1729*d^4*(a*d - b*c)^3) + (432*b^3*x^3*(a + b*x)^(1/6))/(1729* d^2*(a*d - b*c)^3) + (x*(a + b*x)^(1/6)*(1482*b^3*c^2 + 42*a^2*b*d^2 - 228 *a*b^2*c*d))/(1729*d^4*(a*d - b*c)^3) - (72*b^2*x^2*(a*d - 19*b*c)*(a + b* x)^(1/6))/(1729*d^3*(a*d - b*c)^3)))/(x^4 + c^4/d^4 + (4*c*x^3)/d + (4*c^3 *x)/d^3 + (6*c^2*x^2)/d^2)
Time = 6.98 (sec) , antiderivative size = 1430, normalized size of antiderivative = 14.16 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^(1/6)/(d*x+c)^(25/6),x)
Output:
(2*( - 1235*(c + d*x)*(a + b*x)**(3/2)*a**3*b*d**3 + 5681*(c + d*x)*(a + b *x)**(3/2)*a**2*b**2*c*d**2 + 1976*(c + d*x)*(a + b*x)**(3/2)*a**2*b**2*d* *3*x - 11609*(c + d*x)*(a + b*x)**(3/2)*a*b**3*c**2*d - 11856*(c + d*x)*(a + b*x)**(3/2)*a*b**3*c*d**2*x - 3952*(c + d*x)*(a + b*x)**(3/2)*a*b**3*d* *3*x**2 + 7163*(c + d*x)*(a + b*x)**(3/2)*b**4*c**3 + 9880*(c + d*x)*(a + b*x)**(3/2)*b**4*c**2*d*x + 3952*(c + d*x)*(a + b*x)**(3/2)*b**4*c*d**2*x* *2 - 1365*sqrt(a + b*x)*a**5*d**4 + 7350*sqrt(a + b*x)*a**4*b*c*d**3 + 525 *sqrt(a + b*x)*a**4*b*d**4*x - 17100*sqrt(a + b*x)*a**3*b**2*c**2*d**2 - 4 800*sqrt(a + b*x)*a**3*b**2*c*d**3*x - 1350*sqrt(a + b*x)*a**3*b**2*d**4*x **2 + 37050*sqrt(a + b*x)*a**2*b**3*c**3*d + 59850*sqrt(a + b*x)*a**2*b**3 *c**2*d**2*x + 52650*sqrt(a + b*x)*a**2*b**3*c*d**3*x**2 + 16200*sqrt(a + b*x)*a**2*b**3*d**4*x**3 - 25935*sqrt(a + b*x)*a*b**4*c**4 - 29640*sqrt(a + b*x)*a*b**4*c**3*d*x + 15390*sqrt(a + b*x)*a*b**4*c**2*d**2*x**2 + 45360 *sqrt(a + b*x)*a*b**4*c*d**3*x**3 + 19440*sqrt(a + b*x)*a*b**4*d**4*x**4 - 25935*sqrt(a + b*x)*b**5*c**4*x - 66690*sqrt(a + b*x)*b**5*c**3*d*x**2 - 61560*sqrt(a + b*x)*b**5*c**2*d**2*x**3 - 19440*sqrt(a + b*x)*b**5*c*d**3* x**4 + 47424*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**4*c**3 + 1 42272*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**4*c**2*d*x + 1422 72*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**4*c*d**2*x**2 + 4742 4*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*a*b**4*d**3*x**3 + 4742...