Integrand size = 13, antiderivative size = 68 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=-\frac {x^6}{8 c \left (a+c x^4\right )^2}-\frac {3 x^2}{16 c^2 \left (a+c x^4\right )}+\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 \sqrt {a} c^{5/2}} \] Output:
-1/8*x^6/c/(c*x^4+a)^2-3/16*x^2/c^2/(c*x^4+a)+3/16*arctan(c^(1/2)*x^2/a^(1 /2))/a^(1/2)/c^(5/2)
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=\frac {1}{16} \left (\frac {-3 a x^2-5 c x^6}{c^2 \left (a+c x^4\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}\right ) \] Input:
Integrate[x^9/(a + c*x^4)^3,x]
Output:
((-3*a*x^2 - 5*c*x^6)/(c^2*(a + c*x^4)^2) + (3*ArcTan[(Sqrt[c]*x^2)/Sqrt[a ]])/(Sqrt[a]*c^(5/2)))/16
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {807, 252, 252, 218}
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.
\(\displaystyle \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (c x^4+a\right )^3}dx^2\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \int \frac {x^4}{\left (c x^4+a\right )^2}dx^2}{4 c}-\frac {x^6}{4 c \left (a+c x^4\right )^2}\right )\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {\int \frac {1}{c x^4+a}dx^2}{2 c}-\frac {x^2}{2 c \left (a+c x^4\right )}\right )}{4 c}-\frac {x^6}{4 c \left (a+c x^4\right )^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}-\frac {x^2}{2 c \left (a+c x^4\right )}\right )}{4 c}-\frac {x^6}{4 c \left (a+c x^4\right )^2}\right )\) |
Input:
Int[x^9/(a + c*x^4)^3,x]
Output:
(-1/4*x^6/(c*(a + c*x^4)^2) + (3*(-1/2*x^2/(c*(a + c*x^4)) + ArcTan[(Sqrt[ c]*x^2)/Sqrt[a]]/(2*Sqrt[a]*c^(3/2))))/(4*c))/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.48 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {-\frac {5 x^{6}}{8 c}-\frac {3 a \,x^{2}}{8 c^{2}}}{2 \left (c \,x^{4}+a \right )^{2}}+\frac {3 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 c^{2} \sqrt {a c}}\) | \(52\) |
risch | \(\frac {-\frac {5 x^{6}}{16 c}-\frac {3 a \,x^{2}}{16 c^{2}}}{\left (c \,x^{4}+a \right )^{2}}-\frac {3 \ln \left (x^{2} \sqrt {-a c}-a \right )}{32 \sqrt {-a c}\, c^{2}}+\frac {3 \ln \left (x^{2} \sqrt {-a c}+a \right )}{32 \sqrt {-a c}\, c^{2}}\) | \(80\) |
Input:
int(x^9/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*(-5/8*x^6/c-3/8*a*x^2/c^2)/(c*x^4+a)^2+3/16/c^2/(a*c)^(1/2)*arctan(c*x ^2/(a*c)^(1/2))
Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.88 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=\left [-\frac {10 \, a c^{2} x^{6} + 6 \, a^{2} c x^{2} + 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{32 \, {\left (a c^{5} x^{8} + 2 \, a^{2} c^{4} x^{4} + a^{3} c^{3}\right )}}, -\frac {5 \, a c^{2} x^{6} + 3 \, a^{2} c x^{2} + 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{16 \, {\left (a c^{5} x^{8} + 2 \, a^{2} c^{4} x^{4} + a^{3} c^{3}\right )}}\right ] \] Input:
integrate(x^9/(c*x^4+a)^3,x, algorithm="fricas")
Output:
[-1/32*(10*a*c^2*x^6 + 6*a^2*c*x^2 + 3*(c^2*x^8 + 2*a*c*x^4 + a^2)*sqrt(-a *c)*log((c*x^4 - 2*sqrt(-a*c)*x^2 - a)/(c*x^4 + a)))/(a*c^5*x^8 + 2*a^2*c^ 4*x^4 + a^3*c^3), -1/16*(5*a*c^2*x^6 + 3*a^2*c*x^2 + 3*(c^2*x^8 + 2*a*c*x^ 4 + a^2)*sqrt(a*c)*arctan(sqrt(a*c)/(c*x^2)))/(a*c^5*x^8 + 2*a^2*c^4*x^4 + a^3*c^3)]
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.71 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=- \frac {3 \sqrt {- \frac {1}{a c^{5}}} \log {\left (- a c^{2} \sqrt {- \frac {1}{a c^{5}}} + x^{2} \right )}}{32} + \frac {3 \sqrt {- \frac {1}{a c^{5}}} \log {\left (a c^{2} \sqrt {- \frac {1}{a c^{5}}} + x^{2} \right )}}{32} + \frac {- 3 a x^{2} - 5 c x^{6}}{16 a^{2} c^{2} + 32 a c^{3} x^{4} + 16 c^{4} x^{8}} \] Input:
integrate(x**9/(c*x**4+a)**3,x)
Output:
-3*sqrt(-1/(a*c**5))*log(-a*c**2*sqrt(-1/(a*c**5)) + x**2)/32 + 3*sqrt(-1/ (a*c**5))*log(a*c**2*sqrt(-1/(a*c**5)) + x**2)/32 + (-3*a*x**2 - 5*c*x**6) /(16*a**2*c**2 + 32*a*c**3*x**4 + 16*c**4*x**8)
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=-\frac {5 \, c x^{6} + 3 \, a x^{2}}{16 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} + \frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} c^{2}} \] Input:
integrate(x^9/(c*x^4+a)^3,x, algorithm="maxima")
Output:
-1/16*(5*c*x^6 + 3*a*x^2)/(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2) + 3/16*arctan( c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2)
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} c^{2}} - \frac {5 \, c x^{6} + 3 \, a x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} c^{2}} \] Input:
integrate(x^9/(c*x^4+a)^3,x, algorithm="giac")
Output:
3/16*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2) - 1/16*(5*c*x^6 + 3*a*x^2)/(( c*x^4 + a)^2*c^2)
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,\sqrt {a}\,c^{5/2}}-\frac {\frac {5\,x^6}{16\,c}+\frac {3\,a\,x^2}{16\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8} \] Input:
int(x^9/(a + c*x^4)^3,x)
Output:
(3*atan((c^(1/2)*x^2)/a^(1/2)))/(16*a^(1/2)*c^(5/2)) - ((5*x^6)/(16*c) + ( 3*a*x^2)/(16*c^2))/(a^2 + c^2*x^8 + 2*a*c*x^4)
Time = 0.19 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.01 \[ \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx=\frac {-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2} x^{8}-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2} x^{8}-3 a^{2} c \,x^{2}-5 a \,c^{2} x^{6}}{16 a \,c^{3} \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )} \] Input:
int(x^9/(c*x^4+a)^3,x)
Output:
( - 3*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**( 1/4)*a**(1/4)*sqrt(2)))*a**2 - 6*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x**4 - 3*sqrt(c)*sq rt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sq rt(2)))*c**2*x**8 - 3*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2* sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2 - 6*sqrt(c)*sqrt(a)*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 - 3*sqrt(c)*sqrt(a)*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 - 3*a**2*c*x**2 - 5*a*c**2*x**6)/(16*a*c **3*(a**2 + 2*a*c*x**4 + c**2*x**8))