Integrand size = 44, antiderivative size = 37 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=\frac {820125 x^4-182574 x^{12}+5796 x^{20}-49 x^{28}}{4 \left (243-63 x^8+x^{16}\right )^2} \] Output:
1/4*(-49*x^28+5796*x^20-182574*x^12+820125*x^4)/(x^16-63*x^8+243)^2
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=\frac {x^4 \left (820125-182574 x^8+5796 x^{16}-49 x^{24}\right )}{4 \left (243-63 x^8+x^{16}\right )^2} \] Input:
Integrate[(199290375*x^3 + 21907179*x^11 - 10200897*x^19 + 464373*x^27 - 8 127*x^35 + 49*x^43)/(243 - 63*x^8 + x^16)^3,x]
Output:
(x^4*(820125 - 182574*x^8 + 5796*x^16 - 49*x^24))/(4*(243 - 63*x^8 + x^16) ^2)
Time = 0.78 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2311, 9, 2292, 2206, 27, 2204}
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.
\(\displaystyle \int \frac {49 x^{43}-8127 x^{35}+464373 x^{27}-10200897 x^{19}+21907179 x^{11}+199290375 x^3}{\left (x^{16}-63 x^8+243\right )^3} \, dx\) |
\(\Big \downarrow \) 2311 |
\(\displaystyle \int \frac {x \left (49 x^{42}-8127 x^{34}+464373 x^{26}-10200897 x^{18}+21907179 x^{10}+199290375 x^2\right )}{\left (x^{16}-63 x^8+243\right )^3}dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {x^3 \left (49 x^{40}-8127 x^{32}+464373 x^{24}-10200897 x^{16}+21907179 x^8+199290375\right )}{\left (x^{16}-63 x^8+243\right )^3}dx\) |
\(\Big \downarrow \) 2292 |
\(\displaystyle \frac {1}{4} \int \frac {49 x^{40}-8127 x^{32}+464373 x^{24}-10200897 x^{16}+21907179 x^8+199290375}{\left (x^{16}-63 x^8+243\right )^3}dx^4\) |
\(\Big \downarrow \) 2206 |
\(\displaystyle \frac {1}{4} \left (\frac {161838 x^4}{\left (x^{16}-63 x^8+243\right )^2}-\frac {\int -\frac {20391588 \left (7 x^{24}-720 x^{16}+19278 x^8+94041\right )}{\left (x^{16}-63 x^8+243\right )^2}dx^4}{2913084}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (7 \int \frac {7 x^{24}-720 x^{16}+19278 x^8+94041}{\left (x^{16}-63 x^8+243\right )^2}dx^4+\frac {161838 x^4}{\left (x^{16}-63 x^8+243\right )^2}\right )\) |
\(\Big \downarrow \) 2204 |
\(\displaystyle \frac {1}{4} \left (\frac {7 \left (387-7 x^8\right ) x^4}{x^{16}-63 x^8+243}+\frac {161838 x^4}{\left (x^{16}-63 x^8+243\right )^2}\right )\) |
Input:
Int[(199290375*x^3 + 21907179*x^11 - 10200897*x^19 + 464373*x^27 - 8127*x^ 35 + 49*x^43)/(243 - 63*x^8 + x^16)^3,x]
Output:
((161838*x^4)/(243 - 63*x^8 + x^16)^2 + (7*x^4*(387 - 7*x^8))/(243 - 63*x^ 8 + x^16))/4
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{d = Coeff[Px, x, 0], e = Coeff[Px, x, 2], f = Coeff[Px, x, 4], g = Coeff[Px, x, 6]}, Simp[x*(3*a*d + (a*e - b*d*(2*p + 3))*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(3*a^2)), x] /; EqQ[3*a^2*g - c*(4*p + 7)*(a*e - b*d*(2*p + 3)), 0] && EqQ[3*a^2*f - 3*a*c*d*(4*p + 5) - b*(2*p + 5)*(a*e - b*d*(2*p + 3)), 0]] / ; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && EqQ[Expon[Px, x], 6]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Sy mbol] :> With[{g = GCD[m + 1, n]}, Simp[1/g Subst[Int[x^((m + 1)/g - 1)*( Px /. x -> x^(1/g))*(a + b*x^(n/g) + c*x^(2*(n/g)))^p, x], x, x^g], x] /; N eQ[g, 1]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Px, x^n] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{ a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(-\frac {\left (49 x^{24}-5796 x^{16}+182574 x^{8}-820125\right ) x^{4}}{4 \left (x^{16}-63 x^{8}+243\right )^{2}}\) | \(35\) |
risch | \(\frac {\frac {820125}{4} x^{4}-\frac {91287}{2} x^{12}+1449 x^{20}-\frac {49}{4} x^{28}}{\left (x^{16}-63 x^{8}+243\right )^{2}}\) | \(35\) |
default | \(\frac {-49 x^{28}+5796 x^{20}-182574 x^{12}+820125 x^{4}}{4 \left (x^{16}-63 x^{8}+243\right )^{2}}\) | \(36\) |
parallelrisch | \(\frac {-49 x^{28}+5796 x^{20}-182574 x^{12}+820125 x^{4}}{4 \left (x^{16}-63 x^{8}+243\right )^{2}}\) | \(36\) |
orering | \(-\frac {x \left (49 x^{24}-5796 x^{16}+182574 x^{8}-820125\right ) \left (49 x^{43}-8127 x^{35}+464373 x^{27}-10200897 x^{19}+21907179 x^{11}+199290375 x^{3}\right )}{4 \left (x^{16}-63 x^{8}+243\right )^{2} \left (49 x^{40}-8127 x^{32}+464373 x^{24}-10200897 x^{16}+21907179 x^{8}+199290375\right )}\) | \(93\) |
Input:
int((49*x^43-8127*x^35+464373*x^27-10200897*x^19+21907179*x^11+199290375*x ^3)/(x^16-63*x^8+243)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*(49*x^24-5796*x^16+182574*x^8-820125)*x^4/(x^16-63*x^8+243)^2
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=-\frac {49 \, x^{28} - 5796 \, x^{20} + 182574 \, x^{12} - 820125 \, x^{4}}{4 \, {\left (x^{32} - 126 \, x^{24} + 4455 \, x^{16} - 30618 \, x^{8} + 59049\right )}} \] Input:
integrate((49*x^43-8127*x^35+464373*x^27-10200897*x^19+21907179*x^11+19929 0375*x^3)/(x^16-63*x^8+243)^3,x, algorithm="fricas")
Output:
-1/4*(49*x^28 - 5796*x^20 + 182574*x^12 - 820125*x^4)/(x^32 - 126*x^24 + 4 455*x^16 - 30618*x^8 + 59049)
Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=\frac {- 49 x^{28} + 5796 x^{20} - 182574 x^{12} + 820125 x^{4}}{4 x^{32} - 504 x^{24} + 17820 x^{16} - 122472 x^{8} + 236196} \] Input:
integrate((49*x**43-8127*x**35+464373*x**27-10200897*x**19+21907179*x**11+ 199290375*x**3)/(x**16-63*x**8+243)**3,x)
Output:
(-49*x**28 + 5796*x**20 - 182574*x**12 + 820125*x**4)/(4*x**32 - 504*x**24 + 17820*x**16 - 122472*x**8 + 236196)
Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=-\frac {49 \, x^{28} - 5796 \, x^{20} + 182574 \, x^{12} - 820125 \, x^{4}}{4 \, {\left (x^{32} - 126 \, x^{24} + 4455 \, x^{16} - 30618 \, x^{8} + 59049\right )}} \] Input:
integrate((49*x^43-8127*x^35+464373*x^27-10200897*x^19+21907179*x^11+19929 0375*x^3)/(x^16-63*x^8+243)^3,x, algorithm="maxima")
Output:
-1/4*(49*x^28 - 5796*x^20 + 182574*x^12 - 820125*x^4)/(x^32 - 126*x^24 + 4 455*x^16 - 30618*x^8 + 59049)
Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=-\frac {49 \, x^{28} - 5796 \, x^{20} + 182574 \, x^{12} - 820125 \, x^{4}}{4 \, {\left (x^{16} - 63 \, x^{8} + 243\right )}^{2}} \] Input:
integrate((49*x^43-8127*x^35+464373*x^27-10200897*x^19+21907179*x^11+19929 0375*x^3)/(x^16-63*x^8+243)^3,x, algorithm="giac")
Output:
-1/4*(49*x^28 - 5796*x^20 + 182574*x^12 - 820125*x^4)/(x^16 - 63*x^8 + 243 )^2
Time = 9.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=-\frac {x^4\,\left (49\,x^{24}-5796\,x^{16}+182574\,x^8-820125\right )}{4\,{\left (x^{16}-63\,x^8+243\right )}^2} \] Input:
int((199290375*x^3 + 21907179*x^11 - 10200897*x^19 + 464373*x^27 - 8127*x^ 35 + 49*x^43)/(x^16 - 63*x^8 + 243)^3,x)
Output:
-(x^4*(182574*x^8 - 5796*x^16 + 49*x^24 - 820125))/(4*(x^16 - 63*x^8 + 243 )^2)
Time = 2.90 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {199290375 x^3+21907179 x^{11}-10200897 x^{19}+464373 x^{27}-8127 x^{35}+49 x^{43}}{\left (243-63 x^8+x^{16}\right )^3} \, dx=\frac {x^{4} \left (-49 x^{24}+5796 x^{16}-182574 x^{8}+820125\right )}{4 x^{32}-504 x^{24}+17820 x^{16}-122472 x^{8}+236196} \] Input:
int((49*x^43-8127*x^35+464373*x^27-10200897*x^19+21907179*x^11+199290375*x ^3)/(x^16-63*x^8+243)^3,x)
Output:
(x**4*( - 49*x**24 + 5796*x**16 - 182574*x**8 + 820125))/(4*(x**32 - 126*x **24 + 4455*x**16 - 30618*x**8 + 59049))