Integrand size = 27, antiderivative size = 268 \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{4 a d x^4}+\frac {b d+a e}{2 a^2 d^2 x^2}+\frac {\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^2 d^2+a b d e-a \left (c d^2-a e^2\right )\right ) \log (x)}{a^3 d^3}-\frac {e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (c d^2-b d e+a e^2\right )} \]
-1/4/a/d/x^4+1/2*(a*e+b*d)/a^2/d^2/x^2+(b^2*d^2+a*b*d*e-a*(-a*e^2+c*d^2))* ln(x)/a^3/d^3-1/2*e^4*ln(e*x^2+d)/d^3/(a*e^2-b*d*e+c*d^2)-1/4*(2*a*b*c*e-a *c^2*d-b^3*e+b^2*c*d)*ln(c*x^4+b*x^2+a)/a^3/(a*e^2-b*d*e+c*d^2)+1/2*(-2*a^ 2*c^2*e+4*a*b^2*c*e-3*a*b*c^2*d-b^4*e+b^3*c*d)*arctanh((2*c*x^2+b)/(-4*a*c +b^2)^(1/2))/a^3/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)
Time = 0.28 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\frac {1}{4} \left (-\frac {1}{a d x^4}+\frac {2 (b d+a e)}{a^2 d^2 x^2}+\frac {4 \left (b^2 d^2+a b d e+a \left (-c d^2+a e^2\right )\right ) \log (x)}{a^3 d^3}-\frac {\left (b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )-b^2 c \left (\sqrt {b^2-4 a c} d+4 a e\right )+a b c \left (3 c d-2 \sqrt {b^2-4 a c} e\right )+b^3 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{a^3 \sqrt {b^2-4 a c} \left (-c d^2+e (b d-a e)\right )}-\frac {\left (-b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 c \left (-\sqrt {b^2-4 a c} d+4 a e\right )+b^3 \left (c d+\sqrt {b^2-4 a c} e\right )-a b c \left (3 c d+2 \sqrt {b^2-4 a c} e\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{a^3 \sqrt {b^2-4 a c} \left (-c d^2+e (b d-a e)\right )}-\frac {2 e^4 \log \left (d+e x^2\right )}{c d^5+d^3 e (-b d+a e)}\right ) \]
(-(1/(a*d*x^4)) + (2*(b*d + a*e))/(a^2*d^2*x^2) + (4*(b^2*d^2 + a*b*d*e + a*(-(c*d^2) + a*e^2))*Log[x])/(a^3*d^3) - ((b^4*e + a*c^2*(Sqrt[b^2 - 4*a* c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*Sqr t[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^3*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) - ( (-(b^4*e) + a*c^2*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a* c]*d) + 4*a*e) + b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b ^2 - 4*a*c]*e))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^3*Sqrt[b^2 - 4*a* c]*(-(c*d^2) + e*(b*d - a*e))) - (2*e^4*Log[d + e*x^2])/(c*d^5 + d^3*e*(-( b*d) + a*e)))/4
Time = 0.65 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1578, 1200, 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.
\(\displaystyle \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (e x^2+d\right ) \left (c x^4+b x^2+a\right )}dx^2\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {e^5}{d^3 \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )}+\frac {e b^4-c d b^3-3 a c e b^2+2 a c^2 d b-c \left (-e b^3+c d b^2+2 a c e b-a c^2 d\right ) x^2+a^2 c^2 e}{a^3 \left (c d^2-b e d+a e^2\right ) \left (c x^4+b x^2+a\right )}+\frac {b^2 d^2+a b e d-a \left (c d^2-a e^2\right )}{a^3 d^3 x^2}+\frac {-b d-a e}{a^2 d^2 x^4}+\frac {1}{a d x^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right ) \left (a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 d^3}-\frac {\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x^2+c x^4\right )}{2 a^3 \left (a e^2-b d e+c d^2\right )}+\frac {a e+b d}{a^2 d^2 x^2}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right )}{a^3 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {e^4 \log \left (d+e x^2\right )}{d^3 \left (a e^2-b d e+c d^2\right )}-\frac {1}{2 a d x^4}\right )\) |
(-1/2*1/(a*d*x^4) + (b*d + a*e)/(a^2*d^2*x^2) + ((b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c] ])/(a^3*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + ((b^2*d^2 + a*b*d*e - a*(c*d^2 - a*e^2))*Log[x^2])/(a^3*d^3) - (e^4*Log[d + e*x^2])/(d^3*(c*d^2 - b*d*e + a*e^2)) - ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*Log[a + b*x^ 2 + c*x^4])/(2*a^3*(c*d^2 - b*d*e + a*e^2)))/2
3.4.2.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.54 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {1}{4 a d \,x^{4}}-\frac {-a e -b d}{2 a^{2} d^{2} x^{2}}+\frac {\left (e^{2} a^{2}+a b d e -d^{2} a c +b^{2} d^{2}\right ) \ln \left (x \right )}{d^{3} a^{3}}+\frac {\frac {\left (-2 a b \,c^{2} e +a \,c^{3} d +b^{3} c e -b^{2} c^{2} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{2} c^{2} e -3 a \,b^{2} c e +2 a b \,c^{2} d +b^{4} e -b^{3} c d -\frac {\left (-2 a b \,c^{2} e +a \,c^{3} d +b^{3} c e -b^{2} c^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) a^{3}}-\frac {e^{4} \ln \left (e \,x^{2}+d \right )}{2 d^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )}\) | \(284\) |
risch | \(\text {Expression too large to display}\) | \(1325\) |
-1/4/a/d/x^4-1/2*(-a*e-b*d)/a^2/d^2/x^2+(a^2*e^2+a*b*d*e-a*c*d^2+b^2*d^2)/ d^3/a^3*ln(x)+1/2/(a*e^2-b*d*e+c*d^2)/a^3*(1/2*(-2*a*b*c^2*e+a*c^3*d+b^3*c *e-b^2*c^2*d)/c*ln(c*x^4+b*x^2+a)+2*(a^2*c^2*e-3*a*b^2*c*e+2*a*b*c^2*d+b^4 *e-b^3*c*d-1/2*(-2*a*b*c^2*e+a*c^3*d+b^3*c*e-b^2*c^2*d)*b/c)/(4*a*c-b^2)^( 1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2)))-1/2*e^4*ln(e*x^2+d)/d^3/(a*e^2 -b*d*e+c*d^2)
Timed out. \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.60 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=-\frac {e^{5} \log \left ({\left | e x^{2} + d \right |}\right )}{2 \, {\left (c d^{5} e - b d^{4} e^{2} + a d^{3} e^{3}\right )}} - \frac {{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )}} - \frac {{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (b^{2} d^{2} - a c d^{2} + a b d e + a^{2} e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} d^{3}} - \frac {3 \, b^{2} d^{2} x^{4} - 3 \, a c d^{2} x^{4} + 3 \, a b d e x^{4} + 3 \, a^{2} e^{2} x^{4} - 2 \, a b d^{2} x^{2} - 2 \, a^{2} d e x^{2} + a^{2} d^{2}}{4 \, a^{3} d^{3} x^{4}} \]
-1/2*e^5*log(abs(e*x^2 + d))/(c*d^5*e - b*d^4*e^2 + a*d^3*e^3) - 1/4*(b^2* c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*log(c*x^4 + b*x^2 + a)/(a^3*c*d^2 - a^3 *b*d*e + a^4*e^2) - 1/2*(b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a ^2*c^2*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(b^2*d^2 - a*c*d^2 + a*b*d*e + a^2*e ^2)*log(x^2)/(a^3*d^3) - 1/4*(3*b^2*d^2*x^4 - 3*a*c*d^2*x^4 + 3*a*b*d*e*x^ 4 + 3*a^2*e^2*x^4 - 2*a*b*d^2*x^2 - 2*a^2*d*e*x^2 + a^2*d^2)/(a^3*d^3*x^4)
Time = 142.00 (sec) , antiderivative size = 10300, normalized size of antiderivative = 38.43 \[ \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
(log((c^8*e^8*(a^2*e^2 + b^2*d^2 - a*c*d^2 + a*b*d*e))/(a^6*d^6) - (c^9*e^ 9*x^2)/(a^5*d^5) - (((c^5*e^5*(4*a^3*b^3*e^6 + 4*b^3*c^3*d^6 + 4*b^6*d^3*e ^3 + 8*a*b^5*d^2*e^4 + 8*a^2*b^4*d*e^5 + 4*a^2*c^4*d^5*e + 16*a^4*c^2*d*e^ 5 - 19*a^3*c^3*d^3*e^3 - 4*a*b*c^4*d^6 - 12*a^4*b*c*e^6 + 36*a^2*b^2*c^2*d ^3*e^3 - 24*a*b^4*c*d^3*e^3 - 32*a^3*b^2*c*d*e^5 - 36*a^2*b^3*c*d^2*e^4 + 28*a^3*b*c^2*d^2*e^4))/(a^6*d^6) - (((4*a^4*b^6*c^2*e^12 - 24*a^5*b^4*c^3* e^12 + 36*a^6*b^2*c^4*e^12 - 4*a^3*c^9*d^8*e^4 + 64*a^4*c^8*d^6*e^6 - 144* a^5*c^7*d^4*e^8 + 96*a^6*c^6*d^2*e^10 + 4*b^4*c^8*d^10*e^2 + 8*b^7*c^5*d^7 *e^5 + 4*b^10*c^2*d^4*e^8 + 64*a^2*b^3*c^7*d^7*e^5 - 8*a^2*b^4*c^6*d^6*e^6 - 8*a^2*b^5*c^5*d^5*e^7 + 172*a^2*b^6*c^4*d^4*e^8 - 112*a^2*b^7*c^3*d^3*e ^9 + 16*a^2*b^8*c^2*d^2*e^10 - 72*a^3*b^2*c^7*d^6*e^6 + 56*a^3*b^3*c^6*d^5 *e^7 - 312*a^3*b^4*c^5*d^4*e^8 + 348*a^3*b^5*c^4*d^3*e^9 - 132*a^3*b^6*c^3 *d^2*e^10 + 324*a^4*b^2*c^6*d^4*e^8 - 428*a^4*b^3*c^5*d^3*e^9 + 344*a^4*b^ 4*c^4*d^2*e^10 - 300*a^5*b^2*c^5*d^2*e^10 - 96*a^6*b*c^5*d*e^11 - 4*a*b^2* c^9*d^10*e^2 - 4*a*b^3*c^8*d^9*e^3 - 48*a*b^5*c^6*d^7*e^5 + 8*a*b^6*c^5*d^ 6*e^6 - 44*a*b^8*c^3*d^4*e^8 + 12*a*b^9*c^2*d^3*e^9 + 8*a^2*b*c^9*d^9*e^3 - 24*a^3*b*c^8*d^7*e^5 + 12*a^3*b^7*c^2*d*e^11 - 88*a^4*b*c^7*d^5*e^7 - 88 *a^4*b^5*c^3*d*e^11 + 228*a^5*b*c^6*d^3*e^9 + 188*a^5*b^3*c^4*d*e^11)/(a^6 *d^6) + (x^2*(32*a^6*c^6*d*e^11 - 24*a^6*b*c^5*e^12 + 4*a^3*b^7*c^2*e^12 - 28*a^4*b^5*c^3*e^12 + 56*a^5*b^3*c^4*e^12 + 2*a^3*c^9*d^7*e^5 + 104*a^...