Integrand size = 22, antiderivative size = 87 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {1}{4 a c x^4}-\frac {(b c+a d) \log (x)}{a^2 c^2}+\frac {b^2 \log \left (a+b x^4\right )}{4 a^2 (b c-a d)}-\frac {d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)} \] Output:
-1/4/a/c/x^4-(a*d+b*c)*ln(x)/a^2/c^2+1/4*b^2*ln(b*x^4+a)/a^2/(-a*d+b*c)-1/ 4*d^2*ln(d*x^4+c)/c^2/(-a*d+b*c)
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {1}{4 a c x^4}+\frac {(-b c-a d) \log (x)}{a^2 c^2}-\frac {b^2 \log \left (a+b x^4\right )}{4 a^2 (-b c+a d)}-\frac {d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)} \] Input:
Integrate[1/(x^5*(a + b*x^4)*(c + d*x^4)),x]
Output:
-1/4*1/(a*c*x^4) + ((-(b*c) - a*d)*Log[x])/(a^2*c^2) - (b^2*Log[a + b*x^4] )/(4*a^2*(-(b*c) + a*d)) - (d^2*Log[c + d*x^4])/(4*c^2*(b*c - a*d))
Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 93, 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.
\(\displaystyle \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{4} \int \frac {1}{x^8 \left (b x^4+a\right ) \left (d x^4+c\right )}dx^4\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \frac {1}{4} \int \left (-\frac {b^3}{a^2 (a d-b c) \left (b x^4+a\right )}-\frac {d^3}{c^2 (b c-a d) \left (d x^4+c\right )}+\frac {-b c-a d}{a^2 c^2 x^4}+\frac {1}{a c x^8}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (\frac {b^2 \log \left (a+b x^4\right )}{a^2 (b c-a d)}-\frac {\log \left (x^4\right ) (a d+b c)}{a^2 c^2}-\frac {d^2 \log \left (c+d x^4\right )}{c^2 (b c-a d)}-\frac {1}{a c x^4}\right )\) |
Input:
Int[1/(x^5*(a + b*x^4)*(c + d*x^4)),x]
Output:
(-(1/(a*c*x^4)) - ((b*c + a*d)*Log[x^4])/(a^2*c^2) + (b^2*Log[a + b*x^4])/ (a^2*(b*c - a*d)) - (d^2*Log[c + d*x^4])/(c^2*(b*c - a*d)))/4
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
norman | \(-\frac {1}{4 a c \,x^{4}}-\frac {b^{2} \ln \left (b \,x^{4}+a \right )}{4 a^{2} \left (a d -c b \right )}+\frac {d^{2} \ln \left (d \,x^{4}+c \right )}{4 c^{2} \left (a d -c b \right )}-\frac {\left (a d +c b \right ) \ln \left (x \right )}{a^{2} c^{2}}\) | \(82\) |
default | \(-\frac {b^{2} \ln \left (b \,x^{4}+a \right )}{4 a^{2} \left (a d -c b \right )}-\frac {1}{4 a c \,x^{4}}+\frac {\left (-a d -c b \right ) \ln \left (x \right )}{a^{2} c^{2}}+\frac {d^{2} \ln \left (d \,x^{4}+c \right )}{4 c^{2} \left (a d -c b \right )}\) | \(83\) |
risch | \(-\frac {1}{4 a c \,x^{4}}-\frac {\ln \left (x \right ) d}{a \,c^{2}}-\frac {\ln \left (x \right ) b}{a^{2} c}+\frac {d^{2} \ln \left (-d \,x^{4}-c \right )}{4 c^{2} \left (a d -c b \right )}-\frac {b^{2} \ln \left (b \,x^{4}+a \right )}{4 a^{2} \left (a d -c b \right )}\) | \(90\) |
parallelrisch | \(-\frac {4 \ln \left (x \right ) x^{4} a^{2} d^{2}-4 \ln \left (x \right ) x^{4} b^{2} c^{2}+b^{2} \ln \left (b \,x^{4}+a \right ) c^{2} x^{4}-d^{2} \ln \left (d \,x^{4}+c \right ) a^{2} x^{4}+a^{2} c d -a b \,c^{2}}{4 a^{2} c^{2} x^{4} \left (a d -c b \right )}\) | \(99\) |
Input:
int(1/x^5/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
Output:
-1/4/a/c/x^4-1/4*b^2/a^2/(a*d-b*c)*ln(b*x^4+a)+1/4*d^2/c^2/(a*d-b*c)*ln(d* x^4+c)-(a*d+b*c)*ln(x)/a^2/c^2
Time = 4.48 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b^{2} c^{2} x^{4} \log \left (b x^{4} + a\right ) - a^{2} d^{2} x^{4} \log \left (d x^{4} + c\right ) - 4 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} \log \left (x\right ) - a b c^{2} + a^{2} c d}{4 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \] Input:
integrate(1/x^5/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
Output:
1/4*(b^2*c^2*x^4*log(b*x^4 + a) - a^2*d^2*x^4*log(d*x^4 + c) - 4*(b^2*c^2 - a^2*d^2)*x^4*log(x) - a*b*c^2 + a^2*c*d)/((a^2*b*c^3 - a^3*c^2*d)*x^4)
Timed out. \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**5/(b*x**4+a)/(d*x**4+c),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b^{2} \log \left (b x^{4} + a\right )}{4 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \log \left (d x^{4} + c\right )}{4 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{4}\right )}{4 \, a^{2} c^{2}} - \frac {1}{4 \, a c x^{4}} \] Input:
integrate(1/x^5/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
Output:
1/4*b^2*log(b*x^4 + a)/(a^2*b*c - a^3*d) - 1/4*d^2*log(d*x^4 + c)/(b*c^3 - a*c^2*d) - 1/4*(b*c + a*d)*log(x^4)/(a^2*c^2) - 1/4/(a*c*x^4)
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b^{3} \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (a^{2} b^{2} c - a^{3} b d\right )}} - \frac {d^{3} \log \left ({\left | d x^{4} + c \right |}\right )}{4 \, {\left (b c^{3} d - a c^{2} d^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{4}\right )}{4 \, a^{2} c^{2}} + \frac {b c x^{4} + a d x^{4} - a c}{4 \, a^{2} c^{2} x^{4}} \] Input:
integrate(1/x^5/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
Output:
1/4*b^3*log(abs(b*x^4 + a))/(a^2*b^2*c - a^3*b*d) - 1/4*d^3*log(abs(d*x^4 + c))/(b*c^3*d - a*c^2*d^2) - 1/4*(b*c + a*d)*log(x^4)/(a^2*c^2) + 1/4*(b* c*x^4 + a*d*x^4 - a*c)/(a^2*c^2*x^4)
Time = 5.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {b^2\,\ln \left (b\,x^4+a\right )}{4\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {d^2\,\ln \left (d\,x^4+c\right )}{4\,\left (b\,c^3-a\,c^2\,d\right )}-\frac {1}{4\,a\,c\,x^4}-\frac {\ln \left (x\right )\,\left (a\,d+b\,c\right )}{a^2\,c^2} \] Input:
int(1/(x^5*(a + b*x^4)*(c + d*x^4)),x)
Output:
- (b^2*log(a + b*x^4))/(4*(a^3*d - a^2*b*c)) - (d^2*log(c + d*x^4))/(4*(b* c^3 - a*c^2*d)) - 1/(4*a*c*x^4) - (log(x)*(a*d + b*c))/(a^2*c^2)
Time = 0.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.13 \[ \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-\mathrm {log}\left (-b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right ) b^{2} c^{2} x^{4}+\mathrm {log}\left (-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right ) a^{2} d^{2} x^{4}-\mathrm {log}\left (b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right ) b^{2} c^{2} x^{4}+\mathrm {log}\left (d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right ) a^{2} d^{2} x^{4}-4 \,\mathrm {log}\left (x \right ) a^{2} d^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{4}-a^{2} c d +a b \,c^{2}}{4 a^{2} c^{2} x^{4} \left (a d -b c \right )} \] Input:
int(1/x^5/(b*x^4+a)/(d*x^4+c),x)
Output:
( - log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*b**2*c**2 *x**4 + log( - d**(1/4)*c**(1/4)*sqrt(2)*x + sqrt(c) + sqrt(d)*x**2)*a**2* d**2*x**4 - log(b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*b**2 *c**2*x**4 + log(d**(1/4)*c**(1/4)*sqrt(2)*x + sqrt(c) + sqrt(d)*x**2)*a** 2*d**2*x**4 - 4*log(x)*a**2*d**2*x**4 + 4*log(x)*b**2*c**2*x**4 - a**2*c*d + a*b*c**2)/(4*a**2*c**2*x**4*(a*d - b*c))