Integrand size = 13, antiderivative size = 170 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=-\frac {x^5}{8 c \left (a+c x^4\right )^2}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}-\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}} \] Output:
-1/8*x^5/c/(c*x^4+a)^2-5/32*x/c^2/(c*x^4+a)+5/128*arctan(-1+2^(1/2)*c^(1/4 )*x/a^(1/4))*2^(1/2)/a^(3/4)/c^(9/4)+5/128*arctan(1+2^(1/2)*c^(1/4)*x/a^(1 /4))*2^(1/2)/a^(3/4)/c^(9/4)+5/128*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1 /2)+c^(1/2)*x^2))*2^(1/2)/a^(3/4)/c^(9/4)
Time = 0.06 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {32 a \sqrt [4]{c} x}{\left (a+c x^4\right )^2}-\frac {72 \sqrt [4]{c} x}{a+c x^4}-\frac {10 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {10 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {5 \sqrt {2} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {5 \sqrt {2} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}}{256 c^{9/4}} \] Input:
Integrate[x^8/(a + c*x^4)^3,x]
Output:
((32*a*c^(1/4)*x)/(a + c*x^4)^2 - (72*c^(1/4)*x)/(a + c*x^4) - (10*Sqrt[2] *ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) + (10*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) - (5*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]* a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) + (5*Sqrt[2]*Log[Sqrt[a] + Sqrt[ 2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4))/(256*c^(9/4))
Time = 0.73 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.48, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {817, 817, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {5 \int \frac {x^4}{\left (c x^4+a\right )^2}dx}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {1}{c x^4+a}dx}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 \left (\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}}{4 c}-\frac {x}{4 c \left (a+c x^4\right )}\right )}{8 c}-\frac {x^5}{8 c \left (a+c x^4\right )^2}\) |
Input:
Int[x^8/(a + c*x^4)^3,x]
Output:
-1/8*x^5/(c*(a + c*x^4)^2) + (5*(-1/4*x/(c*(a + c*x^4)) + ((-(ArcTan[1 - ( Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[ 2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[ Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(Sqrt[2]*a^(1/4)*c^(1/4 )) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(2*Sqrt[2]*a^( 1/4)*c^(1/4)))/(2*Sqrt[a]))/(4*c)))/(8*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.50 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.32
| method | result | size |
| risch | \(\frac {-\frac {9 x^{5}}{32 c}-\frac {5 a x}{32 c^{2}}}{\left (c \,x^{4}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 c^{3}}\) | \(54\) |
| default | \(\frac {-\frac {9 x^{5}}{32 c}-\frac {5 a x}{32 c^{2}}}{\left (c \,x^{4}+a \right )^{2}}+\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} a}\) | \(132\) |
Input:
int(x^8/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(-9/32/c*x^5-5/32/c^2*a*x)/(c*x^4+a)^2+5/128/c^3*sum(1/_R^3*ln(x-_R),_R=Ro otOf(_Z^4*c+a))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.54 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=-\frac {36 \, c x^{5} - 5 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (a c^{2} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} + x\right ) + 5 \, {\left (-i \, c^{4} x^{8} - 2 i \, a c^{3} x^{4} - i \, a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (i \, a c^{2} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} + x\right ) + 5 \, {\left (i \, c^{4} x^{8} + 2 i \, a c^{3} x^{4} + i \, a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (-i \, a c^{2} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} + x\right ) + 5 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (-a c^{2} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} + x\right ) + 20 \, a x}{128 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \] Input:
integrate(x^8/(c*x^4+a)^3,x, algorithm="fricas")
Output:
-1/128*(36*c*x^5 - 5*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c^9))^(1/4 )*log(a*c^2*(-1/(a^3*c^9))^(1/4) + x) + 5*(-I*c^4*x^8 - 2*I*a*c^3*x^4 - I* a^2*c^2)*(-1/(a^3*c^9))^(1/4)*log(I*a*c^2*(-1/(a^3*c^9))^(1/4) + x) + 5*(I *c^4*x^8 + 2*I*a*c^3*x^4 + I*a^2*c^2)*(-1/(a^3*c^9))^(1/4)*log(-I*a*c^2*(- 1/(a^3*c^9))^(1/4) + x) + 5*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c^9 ))^(1/4)*log(-a*c^2*(-1/(a^3*c^9))^(1/4) + x) + 20*a*x)/(c^4*x^8 + 2*a*c^3 *x^4 + a^2*c^2)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.40 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=\frac {- 5 a x - 9 c x^{5}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} a^{3} c^{9} + 625, \left ( t \mapsto t \log {\left (\frac {128 t a c^{2}}{5} + x \right )} \right )\right )} \] Input:
integrate(x**8/(c*x**4+a)**3,x)
Output:
(-5*a*x - 9*c*x**5)/(32*a**2*c**2 + 64*a*c**3*x**4 + 32*c**4*x**8) + RootS um(268435456*_t**4*a**3*c**9 + 625, Lambda(_t, _t*log(128*_t*a*c**2/5 + x) ))
Time = 0.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.25 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=-\frac {9 \, c x^{5} + 5 \, a x}{32 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} + \frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{256 \, c^{2}} \] Input:
integrate(x^8/(c*x^4+a)^3,x, algorithm="maxima")
Output:
-1/32*(9*c*x^5 + 5*a*x)/(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2) + 5/256*(2*sqrt( 2)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a) *sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))) + 2*sqrt(2)*arctan(1/2*sqrt(2)* (2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sq rt(sqrt(a)*sqrt(c))) + sqrt(2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(1/4)) - sqrt(2)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)* c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(1/4)))/c^2
Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=\frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a c^{3}} + \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a c^{3}} + \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a c^{3}} - \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a c^{3}} - \frac {9 \, c x^{5} + 5 \, a x}{32 \, {\left (c x^{4} + a\right )}^{2} c^{2}} \] Input:
integrate(x^8/(c*x^4+a)^3,x, algorithm="giac")
Output:
5/128*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4)) /(a/c)^(1/4))/(a*c^3) + 5/128*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2* x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 5/256*sqrt(2)*(a*c^3)^(1/4 )*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 5/256*sqrt(2)*(a* c^3)^(1/4)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 1/32*(9* c*x^5 + 5*a*x)/((c*x^4 + a)^2*c^2)
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.48 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx=-\frac {\frac {9\,x^5}{32\,c}+\frac {5\,a\,x}{32\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {5\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,c^{9/4}}-\frac {5\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,c^{9/4}} \] Input:
int(x^8/(a + c*x^4)^3,x)
Output:
- ((9*x^5)/(32*c) + (5*a*x)/(32*c^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) - (5*ata n((c^(1/4)*x)/(-a)^(1/4)))/(64*(-a)^(3/4)*c^(9/4)) - (5*atanh((c^(1/4)*x)/ (-a)^(1/4)))/(64*(-a)^(3/4)*c^(9/4))
Time = 0.20 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.79 \[ \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^8/(c*x^4+a)^3,x)
Output:
( - 10*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2 - 20*c**(3/4)*a**(1/4)*sqrt(2)*ata n((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a *c*x**4 - 10*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2 *sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*x**8 + 10*c**(3/4)*a**(1/4)* sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)* sqrt(2)))*a**2 + 20*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt (2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x**4 + 10*c**(3/4)*a** (1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a** (1/4)*sqrt(2)))*c**2*x**8 - 5*c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4)*a* *(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a**2 - 10*c**(3/4)*a**(1/4)*sqr t(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*c*x**4 - 5*c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a ) + sqrt(c)*x**2)*c**2*x**8 + 5*c**(3/4)*a**(1/4)*sqrt(2)*log(c**(1/4)*a** (1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a**2 + 10*c**(3/4)*a**(1/4)*sqrt (2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*c*x**4 + 5 *c**(3/4)*a**(1/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqr t(c)*x**2)*c**2*x**8 - 40*a**2*c*x - 72*a*c**2*x**5)/(256*a*c**3*(a**2 + 2 *a*c*x**4 + c**2*x**8))