Integrand size = 13, antiderivative size = 71 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=-\frac {x^2}{8 c \left (a+c x^4\right )^2}+\frac {x^2}{16 a c \left (a+c x^4\right )}+\frac {\arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{3/2} c^{3/2}} \] Output:
-1/8*x^2/c/(c*x^4+a)^2+1/16*x^2/a/c/(c*x^4+a)+1/16*arctan(c^(1/2)*x^2/a^(1 /2))/a^(3/2)/c^(3/2)
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {\sqrt {a} \sqrt {c} x^2 \left (-a+c x^4\right )}{\left (a+c x^4\right )^2}+\arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{3/2} c^{3/2}} \] Input:
Integrate[x^5/(a + c*x^4)^3,x]
Output:
((Sqrt[a]*Sqrt[c]*x^2*(-a + c*x^4))/(a + c*x^4)^2 + ArcTan[(Sqrt[c]*x^2)/S qrt[a]])/(16*a^(3/2)*c^(3/2))
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {807, 252, 215, 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.
\(\displaystyle \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (c x^4+a\right )^3}dx^2\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{\left (c x^4+a\right )^2}dx^2}{4 c}-\frac {x^2}{4 c \left (a+c x^4\right )^2}\right )\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {1}{c x^4+a}dx^2}{2 a}+\frac {x^2}{2 a \left (a+c x^4\right )}}{4 c}-\frac {x^2}{4 c \left (a+c x^4\right )^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {\arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {x^2}{2 a \left (a+c x^4\right )}}{4 c}-\frac {x^2}{4 c \left (a+c x^4\right )^2}\right )\) |
Input:
Int[x^5/(a + c*x^4)^3,x]
Output:
(-1/4*x^2/(c*(a + c*x^4)^2) + (x^2/(2*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2 )/Sqrt[a]]/(2*a^(3/2)*Sqrt[c]))/(4*c))/2
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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 = 54, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {\frac {x^{6}}{8 a}-\frac {x^{2}}{8 c}}{2 \left (c \,x^{4}+a \right )^{2}}+\frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 a c \sqrt {a c}}\) | \(54\) |
risch | \(\frac {\frac {x^{6}}{16 a}-\frac {x^{2}}{16 c}}{\left (c \,x^{4}+a \right )^{2}}-\frac {\ln \left (x^{2} \sqrt {-a c}-a \right )}{32 \sqrt {-a c}\, c a}+\frac {\ln \left (x^{2} \sqrt {-a c}+a \right )}{32 \sqrt {-a c}\, c a}\) | \(85\) |
Input:
int(x^5/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*(1/8/a*x^6-1/8*x^2/c)/(c*x^4+a)^2+1/16/a/c/(a*c)^(1/2)*arctan(c*x^2/(a *c)^(1/2))
Time = 0.08 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.80 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=\left [\frac {2 \, a c^{2} x^{6} - 2 \, a^{2} c x^{2} - {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{32 \, {\left (a^{2} c^{4} x^{8} + 2 \, a^{3} c^{3} x^{4} + a^{4} c^{2}\right )}}, \frac {a c^{2} x^{6} - a^{2} c x^{2} - {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{16 \, {\left (a^{2} c^{4} x^{8} + 2 \, a^{3} c^{3} x^{4} + a^{4} c^{2}\right )}}\right ] \] Input:
integrate(x^5/(c*x^4+a)^3,x, algorithm="fricas")
Output:
[1/32*(2*a*c^2*x^6 - 2*a^2*c*x^2 - (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^2*c^4*x^8 + 2*a^3*c^3* x^4 + a^4*c^2), 1/16*(a*c^2*x^6 - a^2*c*x^2 - (c^2*x^8 + 2*a*c*x^4 + a^2)* sqrt(a*c)*arctan(sqrt(a*c)/(c*x^2)))/(a^2*c^4*x^8 + 2*a^3*c^3*x^4 + a^4*c^ 2)]
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.63 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x^{2} \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x^{2} \right )}}{32} + \frac {- a x^{2} + c x^{6}}{16 a^{3} c + 32 a^{2} c^{2} x^{4} + 16 a c^{3} x^{8}} \] Input:
integrate(x**5/(c*x**4+a)**3,x)
Output:
-sqrt(-1/(a**3*c**3))*log(-a**2*c*sqrt(-1/(a**3*c**3)) + x**2)/32 + sqrt(- 1/(a**3*c**3))*log(a**2*c*sqrt(-1/(a**3*c**3)) + x**2)/32 + (-a*x**2 + c*x **6)/(16*a**3*c + 32*a**2*c**2*x**4 + 16*a*c**3*x**8)
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=\frac {c x^{6} - a x^{2}}{16 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a c} \] Input:
integrate(x^5/(c*x^4+a)^3,x, algorithm="maxima")
Output:
1/16*(c*x^6 - a*x^2)/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c) + 1/16*arctan(c*x ^2/sqrt(a*c))/(sqrt(a*c)*a*c)
Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=\frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a c} + \frac {c x^{6} - a x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} a c} \] Input:
integrate(x^5/(c*x^4+a)^3,x, algorithm="giac")
Output:
1/16*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a*c) + 1/16*(c*x^6 - a*x^2)/((c*x^ 4 + a)^2*a*c)
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {x^6}{16\,a}-\frac {x^2}{16\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,a^{3/2}\,c^{3/2}} \] Input:
int(x^5/(a + c*x^4)^3,x)
Output:
(x^6/(16*a) - x^2/(16*c))/(a^2 + c^2*x^8 + 2*a*c*x^4) + atan((c^(1/2)*x^2) /a^(1/2))/(16*a^(3/2)*c^(3/2))
Time = 0.23 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.83 \[ \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx=\frac {-\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2} x^{8}-\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) a c \,x^{4}-\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2} x^{8}-a^{2} c \,x^{2}+a \,c^{2} x^{6}}{16 a^{2} c^{2} \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )} \] Input:
int(x^5/(c*x^4+a)^3,x)
Output:
( - 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 - 2*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqr t(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x**4 - 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 - 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 - 2*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 - sq rt(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 - a**2*c*x**2 + a*c**2*x**6)/(16*a**2*c**2*(a**2 + 2*a*c*x**4 + c**2*x**8))