Integrand size = 22, antiderivative size = 103 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {a^2 (b c-a d)}{9 b^4 \left (a+b x^6\right )^{3/2}}+\frac {a (2 b c-3 a d)}{3 b^4 \sqrt {a+b x^6}}+\frac {(b c-3 a d) \sqrt {a+b x^6}}{3 b^4}+\frac {d \left (a+b x^6\right )^{3/2}}{9 b^4} \] Output:
-1/9*a^2*(-a*d+b*c)/b^4/(b*x^6+a)^(3/2)+1/3*a*(-3*a*d+2*b*c)/b^4/(b*x^6+a) ^(1/2)+1/3*(-3*a*d+b*c)*(b*x^6+a)^(1/2)/b^4+1/9*d*(b*x^6+a)^(3/2)/b^4
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-16 a^3 d+8 a^2 b \left (c-3 d x^6\right )-6 a b^2 x^6 \left (-2 c+d x^6\right )+b^3 x^{12} \left (3 c+d x^6\right )}{9 b^4 \left (a+b x^6\right )^{3/2}} \] Input:
Integrate[(x^17*(c + d*x^6))/(a + b*x^6)^(5/2),x]
Output:
(-16*a^3*d + 8*a^2*b*(c - 3*d*x^6) - 6*a*b^2*x^6*(-2*c + d*x^6) + b^3*x^12 *(3*c + d*x^6))/(9*b^4*(a + b*x^6)^(3/2))
Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 86, 2009}
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.
\(\displaystyle \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{6} \int \frac {x^{12} \left (d x^6+c\right )}{\left (b x^6+a\right )^{5/2}}dx^6\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{6} \int \left (-\frac {(a d-b c) a^2}{b^3 \left (b x^6+a\right )^{5/2}}+\frac {(3 a d-2 b c) a}{b^3 \left (b x^6+a\right )^{3/2}}+\frac {d \sqrt {b x^6+a}}{b^3}+\frac {b c-3 a d}{b^3 \sqrt {b x^6+a}}\right )dx^6\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \left (-\frac {2 a^2 (b c-a d)}{3 b^4 \left (a+b x^6\right )^{3/2}}+\frac {2 a (2 b c-3 a d)}{b^4 \sqrt {a+b x^6}}+\frac {2 \sqrt {a+b x^6} (b c-3 a d)}{b^4}+\frac {2 d \left (a+b x^6\right )^{3/2}}{3 b^4}\right )\) |
Input:
Int[(x^17*(c + d*x^6))/(a + b*x^6)^(5/2),x]
Output:
((-2*a^2*(b*c - a*d))/(3*b^4*(a + b*x^6)^(3/2)) + (2*a*(2*b*c - 3*a*d))/(b ^4*Sqrt[a + b*x^6]) + (2*(b*c - 3*a*d)*Sqrt[a + b*x^6])/b^4 + (2*d*(a + b* x^6)^(3/2))/(3*b^4))/6
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(\frac {\left (d \,x^{18}+3 c \,x^{12}\right ) b^{3}+12 \left (-\frac {d \,x^{6}}{2}+c \right ) x^{6} a \,b^{2}+8 a^{2} \left (-3 d \,x^{6}+c \right ) b -16 a^{3} d}{9 \left (b \,x^{6}+a \right )^{\frac {3}{2}} b^{4}}\) | \(69\) |
gosper | \(-\frac {-d \,x^{18} b^{3}+6 a \,b^{2} d \,x^{12}-3 b^{3} c \,x^{12}+24 a^{2} b d \,x^{6}-12 a \,b^{2} c \,x^{6}+16 a^{3} d -8 a^{2} b c}{9 \left (b \,x^{6}+a \right )^{\frac {3}{2}} b^{4}}\) | \(77\) |
trager | \(-\frac {-d \,x^{18} b^{3}+6 a \,b^{2} d \,x^{12}-3 b^{3} c \,x^{12}+24 a^{2} b d \,x^{6}-12 a \,b^{2} c \,x^{6}+16 a^{3} d -8 a^{2} b c}{9 \left (b \,x^{6}+a \right )^{\frac {3}{2}} b^{4}}\) | \(77\) |
orering | \(-\frac {-d \,x^{18} b^{3}+6 a \,b^{2} d \,x^{12}-3 b^{3} c \,x^{12}+24 a^{2} b d \,x^{6}-12 a \,b^{2} c \,x^{6}+16 a^{3} d -8 a^{2} b c}{9 \left (b \,x^{6}+a \right )^{\frac {3}{2}} b^{4}}\) | \(77\) |
risch | \(-\frac {\left (-x^{6} b d +8 a d -3 c b \right ) \sqrt {b \,x^{6}+a}}{9 b^{4}}-\frac {\sqrt {b \,x^{6}+a}\, \left (9 a b d \,x^{6}-6 b^{2} c \,x^{6}+8 a^{2} d -5 a b c \right ) a}{9 b^{4} \left (x^{12} b^{2}+2 a \,x^{6} b +a^{2}\right )}\) | \(96\) |
Input:
int(x^17*(d*x^6+c)/(b*x^6+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/9*((d*x^18+3*c*x^12)*b^3+12*(-1/2*d*x^6+c)*x^6*a*b^2+8*a^2*(-3*d*x^6+c)* b-16*a^3*d)/(b*x^6+a)^(3/2)/b^4
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {{\left (b^{3} d x^{18} + 3 \, {\left (b^{3} c - 2 \, a b^{2} d\right )} x^{12} + 12 \, {\left (a b^{2} c - 2 \, a^{2} b d\right )} x^{6} + 8 \, a^{2} b c - 16 \, a^{3} d\right )} \sqrt {b x^{6} + a}}{9 \, {\left (b^{6} x^{12} + 2 \, a b^{5} x^{6} + a^{2} b^{4}\right )}} \] Input:
integrate(x^17*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
Output:
1/9*(b^3*d*x^18 + 3*(b^3*c - 2*a*b^2*d)*x^12 + 12*(a*b^2*c - 2*a^2*b*d)*x^ 6 + 8*a^2*b*c - 16*a^3*d)*sqrt(b*x^6 + a)/(b^6*x^12 + 2*a*b^5*x^6 + a^2*b^ 4)
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (90) = 180\).
Time = 2.27 (sec) , antiderivative size = 337, normalized size of antiderivative = 3.27 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\begin {cases} - \frac {16 a^{3} d}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} + \frac {8 a^{2} b c}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} - \frac {24 a^{2} b d x^{6}}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} + \frac {12 a b^{2} c x^{6}}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} - \frac {6 a b^{2} d x^{12}}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} + \frac {3 b^{3} c x^{12}}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} + \frac {b^{3} d x^{18}}{9 a b^{4} \sqrt {a + b x^{6}} + 9 b^{5} x^{6} \sqrt {a + b x^{6}}} & \text {for}\: b \neq 0 \\\frac {\frac {c x^{18}}{18} + \frac {d x^{24}}{24}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**17*(d*x**6+c)/(b*x**6+a)**(5/2),x)
Output:
Piecewise((-16*a**3*d/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6*sqrt(a + b* x**6)) + 8*a**2*b*c/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6*sqrt(a + b*x* *6)) - 24*a**2*b*d*x**6/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6*sqrt(a + b*x**6)) + 12*a*b**2*c*x**6/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6*sqrt( a + b*x**6)) - 6*a*b**2*d*x**12/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6*s qrt(a + b*x**6)) + 3*b**3*c*x**12/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6 *sqrt(a + b*x**6)) + b**3*d*x**18/(9*a*b**4*sqrt(a + b*x**6) + 9*b**5*x**6 *sqrt(a + b*x**6)), Ne(b, 0)), ((c*x**18/18 + d*x**24/24)/a**(5/2), True))
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.13 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {1}{9} \, d {\left (\frac {{\left (b x^{6} + a\right )}^{\frac {3}{2}}}{b^{4}} - \frac {9 \, \sqrt {b x^{6} + a} a}{b^{4}} - \frac {9 \, a^{2}}{\sqrt {b x^{6} + a} b^{4}} + \frac {a^{3}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{4}}\right )} + \frac {1}{9} \, c {\left (\frac {3 \, \sqrt {b x^{6} + a}}{b^{3}} + \frac {6 \, a}{\sqrt {b x^{6} + a} b^{3}} - \frac {a^{2}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{3}}\right )} \] Input:
integrate(x^17*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
Output:
1/9*d*((b*x^6 + a)^(3/2)/b^4 - 9*sqrt(b*x^6 + a)*a/b^4 - 9*a^2/(sqrt(b*x^6 + a)*b^4) + a^3/((b*x^6 + a)^(3/2)*b^4)) + 1/9*c*(3*sqrt(b*x^6 + a)/b^3 + 6*a/(sqrt(b*x^6 + a)*b^3) - a^2/((b*x^6 + a)^(3/2)*b^3))
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {6 \, {\left (b x^{6} + a\right )} a b c - a^{2} b c - 9 \, {\left (b x^{6} + a\right )} a^{2} d + a^{3} d}{9 \, {\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{4}} + \frac {3 \, \sqrt {b x^{6} + a} b^{9} c + {\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{8} d - 9 \, \sqrt {b x^{6} + a} a b^{8} d}{9 \, b^{12}} \] Input:
integrate(x^17*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
Output:
1/9*(6*(b*x^6 + a)*a*b*c - a^2*b*c - 9*(b*x^6 + a)*a^2*d + a^3*d)/((b*x^6 + a)^(3/2)*b^4) + 1/9*(3*sqrt(b*x^6 + a)*b^9*c + (b*x^6 + a)^(3/2)*b^8*d - 9*sqrt(b*x^6 + a)*a*b^8*d)/b^12
Time = 3.71 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {d\,{\left (b\,x^6+a\right )}^3+a^3\,d-9\,a\,d\,{\left (b\,x^6+a\right )}^2+3\,b\,c\,{\left (b\,x^6+a\right )}^2-9\,a^2\,d\,\left (b\,x^6+a\right )-a^2\,b\,c+6\,a\,b\,c\,\left (b\,x^6+a\right )}{9\,b^4\,{\left (b\,x^6+a\right )}^{3/2}} \] Input:
int((x^17*(c + d*x^6))/(a + b*x^6)^(5/2),x)
Output:
(d*(a + b*x^6)^3 + a^3*d - 9*a*d*(a + b*x^6)^2 + 3*b*c*(a + b*x^6)^2 - 9*a ^2*d*(a + b*x^6) - a^2*b*c + 6*a*b*c*(a + b*x^6))/(9*b^4*(a + b*x^6)^(3/2) )
Time = 0.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.91 \[ \int \frac {x^{17} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {\sqrt {b \,x^{6}+a}\, \left (b^{3} d \,x^{18}-6 a \,b^{2} d \,x^{12}+3 b^{3} c \,x^{12}-24 a^{2} b d \,x^{6}+12 a \,b^{2} c \,x^{6}-16 a^{3} d +8 a^{2} b c \right )}{9 b^{4} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:
int(x^17*(d*x^6+c)/(b*x^6+a)^(5/2),x)
Output:
(sqrt(a + b*x**6)*( - 16*a**3*d + 8*a**2*b*c - 24*a**2*b*d*x**6 + 12*a*b** 2*c*x**6 - 6*a*b**2*d*x**12 + 3*b**3*c*x**12 + b**3*d*x**18))/(9*b**4*(a** 2 + 2*a*b*x**6 + b**2*x**12))