Integrand size = 13, antiderivative size = 69 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=-\frac {1}{4 a^3 x^4}-\frac {c}{8 a^2 \left (a+c x^4\right )^2}-\frac {c}{2 a^3 \left (a+c x^4\right )}-\frac {3 c \log (x)}{a^4}+\frac {3 c \log \left (a+c x^4\right )}{4 a^4} \] Output:
-1/4/a^3/x^4-1/8*c/a^2/(c*x^4+a)^2-1/2*c/a^3/(c*x^4+a)-3*c*ln(x)/a^4+3/4*c *ln(c*x^4+a)/a^4
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=-\frac {\frac {a \left (2 a^2+9 a c x^4+6 c^2 x^8\right )}{x^4 \left (a+c x^4\right )^2}+24 c \log (x)-6 c \log \left (a+c x^4\right )}{8 a^4} \] Input:
Integrate[1/(x^5*(a + c*x^4)^3),x]
Output:
-1/8*((a*(2*a^2 + 9*a*c*x^4 + 6*c^2*x^8))/(x^4*(a + c*x^4)^2) + 24*c*Log[x ] - 6*c*Log[a + c*x^4])/a^4
Time = 0.35 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {798, 54, 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.
\(\displaystyle \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {1}{x^8 \left (c x^4+a\right )^3}dx^4\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{4} \int \left (\frac {3 c^2}{a^4 \left (c x^4+a\right )}+\frac {2 c^2}{a^3 \left (c x^4+a\right )^2}+\frac {c^2}{a^2 \left (c x^4+a\right )^3}-\frac {3 c}{a^4 x^4}+\frac {1}{a^3 x^8}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {3 c \log \left (x^4\right )}{a^4}+\frac {3 c \log \left (a+c x^4\right )}{a^4}-\frac {2 c}{a^3 \left (a+c x^4\right )}-\frac {1}{a^3 x^4}-\frac {c}{2 a^2 \left (a+c x^4\right )^2}\right )\) |
Input:
Int[1/(x^5*(a + c*x^4)^3),x]
Output:
(-(1/(a^3*x^4)) - c/(2*a^2*(a + c*x^4)^2) - (2*c)/(a^3*(a + c*x^4)) - (3*c *Log[x^4])/a^4 + (3*c*Log[a + c*x^4])/a^4)/4
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.46 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {-\frac {1}{4 a}+\frac {3 c^{2} x^{8}}{2 a^{3}}+\frac {9 c^{3} x^{12}}{8 a^{4}}}{x^{4} \left (c \,x^{4}+a \right )^{2}}-\frac {3 c \ln \left (x \right )}{a^{4}}+\frac {3 c \ln \left (c \,x^{4}+a \right )}{4 a^{4}}\) | \(65\) |
risch | \(\frac {-\frac {3 c^{2} x^{8}}{4 a^{3}}-\frac {9 c \,x^{4}}{8 a^{2}}-\frac {1}{4 a}}{x^{4} \left (c \,x^{4}+a \right )^{2}}-\frac {3 c \ln \left (x \right )}{a^{4}}+\frac {3 c \ln \left (-c \,x^{4}-a \right )}{4 a^{4}}\) | \(66\) |
default | \(-\frac {1}{4 a^{3} x^{4}}-\frac {3 c \ln \left (x \right )}{a^{4}}+\frac {c^{2} \left (\frac {3 \ln \left (c \,x^{4}+a \right )}{2 c}-\frac {a}{c \left (c \,x^{4}+a \right )}-\frac {a^{2}}{4 c \left (c \,x^{4}+a \right )^{2}}\right )}{2 a^{4}}\) | \(72\) |
parallelrisch | \(-\frac {24 \ln \left (x \right ) x^{12} c^{3}-6 \ln \left (c \,x^{4}+a \right ) x^{12} c^{3}-9 c^{3} x^{12}+48 \ln \left (x \right ) x^{8} a \,c^{2}-12 \ln \left (c \,x^{4}+a \right ) x^{8} a \,c^{2}-12 a \,c^{2} x^{8}+24 \ln \left (x \right ) x^{4} a^{2} c -6 \ln \left (c \,x^{4}+a \right ) x^{4} a^{2} c +2 a^{3}}{8 a^{4} x^{4} \left (c \,x^{4}+a \right )^{2}}\) | \(123\) |
Input:
int(1/x^5/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(-1/4/a+3/2*c^2/a^3*x^8+9/8*c^3/a^4*x^12)/x^4/(c*x^4+a)^2-3*c*ln(x)/a^4+3/ 4*c*ln(c*x^4+a)/a^4
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=-\frac {6 \, a c^{2} x^{8} + 9 \, a^{2} c x^{4} + 2 \, a^{3} - 6 \, {\left (c^{3} x^{12} + 2 \, a c^{2} x^{8} + a^{2} c x^{4}\right )} \log \left (c x^{4} + a\right ) + 24 \, {\left (c^{3} x^{12} + 2 \, a c^{2} x^{8} + a^{2} c x^{4}\right )} \log \left (x\right )}{8 \, {\left (a^{4} c^{2} x^{12} + 2 \, a^{5} c x^{8} + a^{6} x^{4}\right )}} \] Input:
integrate(1/x^5/(c*x^4+a)^3,x, algorithm="fricas")
Output:
-1/8*(6*a*c^2*x^8 + 9*a^2*c*x^4 + 2*a^3 - 6*(c^3*x^12 + 2*a*c^2*x^8 + a^2* c*x^4)*log(c*x^4 + a) + 24*(c^3*x^12 + 2*a*c^2*x^8 + a^2*c*x^4)*log(x))/(a ^4*c^2*x^12 + 2*a^5*c*x^8 + a^6*x^4)
Time = 0.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=\frac {- 2 a^{2} - 9 a c x^{4} - 6 c^{2} x^{8}}{8 a^{5} x^{4} + 16 a^{4} c x^{8} + 8 a^{3} c^{2} x^{12}} - \frac {3 c \log {\left (x \right )}}{a^{4}} + \frac {3 c \log {\left (\frac {a}{c} + x^{4} \right )}}{4 a^{4}} \] Input:
integrate(1/x**5/(c*x**4+a)**3,x)
Output:
(-2*a**2 - 9*a*c*x**4 - 6*c**2*x**8)/(8*a**5*x**4 + 16*a**4*c*x**8 + 8*a** 3*c**2*x**12) - 3*c*log(x)/a**4 + 3*c*log(a/c + x**4)/(4*a**4)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=-\frac {6 \, c^{2} x^{8} + 9 \, a c x^{4} + 2 \, a^{2}}{8 \, {\left (a^{3} c^{2} x^{12} + 2 \, a^{4} c x^{8} + a^{5} x^{4}\right )}} + \frac {3 \, c \log \left (c x^{4} + a\right )}{4 \, a^{4}} - \frac {3 \, c \log \left (x^{4}\right )}{4 \, a^{4}} \] Input:
integrate(1/x^5/(c*x^4+a)^3,x, algorithm="maxima")
Output:
-1/8*(6*c^2*x^8 + 9*a*c*x^4 + 2*a^2)/(a^3*c^2*x^12 + 2*a^4*c*x^8 + a^5*x^4 ) + 3/4*c*log(c*x^4 + a)/a^4 - 3/4*c*log(x^4)/a^4
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=-\frac {3 \, c \log \left (x^{4}\right )}{4 \, a^{4}} + \frac {3 \, c \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{4}} - \frac {9 \, c^{3} x^{8} + 22 \, a c^{2} x^{4} + 14 \, a^{2} c}{8 \, {\left (c x^{4} + a\right )}^{2} a^{4}} + \frac {3 \, c x^{4} - a}{4 \, a^{4} x^{4}} \] Input:
integrate(1/x^5/(c*x^4+a)^3,x, algorithm="giac")
Output:
-3/4*c*log(x^4)/a^4 + 3/4*c*log(abs(c*x^4 + a))/a^4 - 1/8*(9*c^3*x^8 + 22* a*c^2*x^4 + 14*a^2*c)/((c*x^4 + a)^2*a^4) + 1/4*(3*c*x^4 - a)/(a^4*x^4)
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=\frac {3\,c\,\ln \left (c\,x^4+a\right )}{4\,a^4}-\frac {\frac {1}{4\,a}+\frac {9\,c\,x^4}{8\,a^2}+\frac {3\,c^2\,x^8}{4\,a^3}}{a^2\,x^4+2\,a\,c\,x^8+c^2\,x^{12}}-\frac {3\,c\,\ln \left (x\right )}{a^4} \] Input:
int(1/(x^5*(a + c*x^4)^3),x)
Output:
(3*c*log(a + c*x^4))/(4*a^4) - (1/(4*a) + (9*c*x^4)/(8*a^2) + (3*c^2*x^8)/ (4*a^3))/(a^2*x^4 + c^2*x^12 + 2*a*c*x^8) - (3*c*log(x))/a^4
Time = 0.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 3.74 \[ \int \frac {1}{x^5 \left (a+c x^4\right )^3} \, dx=\frac {6 \,\mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a^{2} c \,x^{4}+12 \,\mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a \,c^{2} x^{8}+6 \,\mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) c^{3} x^{12}+6 \,\mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a^{2} c \,x^{4}+12 \,\mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a \,c^{2} x^{8}+6 \,\mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) c^{3} x^{12}-24 \,\mathrm {log}\left (x \right ) a^{2} c \,x^{4}-48 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{8}-24 \,\mathrm {log}\left (x \right ) c^{3} x^{12}-2 a^{3}-6 a^{2} c \,x^{4}+3 c^{3} x^{12}}{8 a^{4} x^{4} \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )} \] Input:
int(1/x^5/(c*x^4+a)^3,x)
Output:
(6*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a**2*c*x** 4 + 12*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*c**2 *x**8 + 6*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*c** 3*x**12 + 6*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a**2 *c*x**4 + 12*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*c **2*x**8 + 6*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*c** 3*x**12 - 24*log(x)*a**2*c*x**4 - 48*log(x)*a*c**2*x**8 - 24*log(x)*c**3*x **12 - 2*a**3 - 6*a**2*c*x**4 + 3*c**3*x**12)/(8*a**4*x**4*(a**2 + 2*a*c*x **4 + c**2*x**8))