Integrand size = 25, antiderivative size = 112 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=-\frac {d}{4 a x^4}-\frac {\left (b^2 d-2 a c d-a b e\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^2 \sqrt {b^2-4 a c}}-\frac {(b d-a e) \log (x)}{a^2}+\frac {(b d-a e) \log \left (a+b x^4+c x^8\right )}{8 a^2} \] Output:
-1/4*d/a/x^4-1/4*(-a*b*e-2*a*c*d+b^2*d)*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^( 1/2))/a^2/(-4*a*c+b^2)^(1/2)-(-a*e+b*d)*ln(x)/a^2+1/8*(-a*e+b*d)*ln(c*x^8+ b*x^4+a)/a^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=-\frac {d}{4 a x^4}+\frac {(-b d+a e) \log (x)}{a^2}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 d \log (x-\text {$\#$1})-a c d \log (x-\text {$\#$1})-a b e \log (x-\text {$\#$1})+b c d \log (x-\text {$\#$1}) \text {$\#$1}^4-a c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b+2 c \text {$\#$1}^4}\&\right ]}{4 a^2} \] Input:
Integrate[(d + e*x^4)/(x^5*(a + b*x^4 + c*x^8)),x]
Output:
-1/4*d/(a*x^4) + ((-(b*d) + a*e)*Log[x])/a^2 + RootSum[a + b*#1^4 + c*#1^8 & , (b^2*d*Log[x - #1] - a*c*d*Log[x - #1] - a*b*e*Log[x - #1] + b*c*d*Lo g[x - #1]*#1^4 - a*c*e*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a^2)
Time = 0.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1802, 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.
\(\displaystyle \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx\) |
\(\Big \downarrow \) 1802 |
\(\displaystyle \frac {1}{4} \int \frac {e x^4+d}{x^8 \left (c x^8+b x^4+a\right )}dx^4\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {1}{4} \int \left (\frac {d}{a x^8}+\frac {c (b d-a e) x^4+b^2 d-a c d-a b e}{a^2 \left (c x^8+b x^4+a\right )}+\frac {a e-b d}{a^2 x^4}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (-a b e-2 a c d+b^2 d\right )}{a^2 \sqrt {b^2-4 a c}}+\frac {(b d-a e) \log \left (a+b x^4+c x^8\right )}{2 a^2}-\frac {\log \left (x^4\right ) (b d-a e)}{a^2}-\frac {d}{a x^4}\right )\) |
Input:
Int[(d + e*x^4)/(x^5*(a + b*x^4 + c*x^8)),x]
Output:
(-(d/(a*x^4)) - ((b^2*d - 2*a*c*d - a*b*e)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b*d - a*e)*Log[x^4])/a^2 + ((b*d - a*e)*Log[a + b*x^4 + c*x^8])/(2*a^2))/4
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_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1 )/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {\frac {\left (a c e -c b d \right ) \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{4 c}+\frac {\left (a b e +a c d -d \,b^{2}-\frac {\left (a c e -c b d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a^{2}}-\frac {d}{4 a \,x^{4}}+\frac {\left (a e -b d \right ) \ln \left (x \right )}{a^{2}}\) | \(125\) |
risch | \(-\frac {d}{4 a \,x^{4}}+\frac {e \ln \left (x \right )}{a}-\frac {\ln \left (x \right ) b d}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{3} c -a^{2} b^{2}\right ) \textit {\_Z}^{2}+\left (4 a^{2} c e -a \,b^{2} e -4 a b c d +b^{3} d \right ) \textit {\_Z} +a c \,e^{2}-b c d e +c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (18 a^{3} c -5 a^{2} b^{2}\right ) \textit {\_R}^{2}+\left (9 a^{2} c e -8 a b c d \right ) \textit {\_R} +4 c^{2} d^{2}\right ) x^{4}-a^{3} b \,\textit {\_R}^{2}+\left (4 a^{2} b e +a^{2} c d -4 a \,b^{2} d \right ) \textit {\_R} -4 a c d e +4 b c \,d^{2}\right )\right )}{4}\) | \(197\) |
Input:
int((e*x^4+d)/x^5/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/2/a^2*(1/4*(a*c*e-b*c*d)/c*ln(c*x^8+b*x^4+a)+(a*b*e+a*c*d-d*b^2-1/2*(a* c*e-b*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2)))-1 /4*d/a/x^4+(a*e-b*d)/a^2*ln(x)
Time = 1.25 (sec) , antiderivative size = 385, normalized size of antiderivative = 3.44 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=\left [\frac {{\left (a b e - {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {b^{2} - 4 \, a c} x^{4} \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) + {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{4} \log \left (c x^{8} + b x^{4} + a\right ) - 8 \, {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{4} \log \left (x\right ) - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d}{8 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4}}, \frac {2 \, {\left (a b e - {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{4} \log \left (c x^{8} + b x^{4} + a\right ) - 8 \, {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{4} \log \left (x\right ) - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d}{8 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4}}\right ] \] Input:
integrate((e*x^4+d)/x^5/(c*x^8+b*x^4+a),x, algorithm="fricas")
Output:
[1/8*((a*b*e - (b^2 - 2*a*c)*d)*sqrt(b^2 - 4*a*c)*x^4*log((2*c^2*x^8 + 2*b *c*x^4 + b^2 - 2*a*c + (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a )) + ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^4*log(c*x^8 + b*x^4 + a) - 8*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^4*log(x) - 2*(a*b^2 - 4*a^ 2*c)*d)/((a^2*b^2 - 4*a^3*c)*x^4), 1/8*(2*(a*b*e - (b^2 - 2*a*c)*d)*sqrt(- b^2 + 4*a*c)*x^4*arctan(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^4*log(c*x^8 + b*x^4 + a) - 8* ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^4*log(x) - 2*(a*b^2 - 4*a^2*c) *d)/((a^2*b^2 - 4*a^3*c)*x^4)]
Timed out. \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((e*x**4+d)/x**5/(c*x**8+b*x**4+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x^4+d)/x^5/(c*x^8+b*x^4+a),x, algorithm="maxima")
Output:
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 = 1.01 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.12 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=\frac {{\left (b d - a e\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, a^{2}} - \frac {{\left (b d - a e\right )} \log \left (x^{4}\right )}{4 \, a^{2}} + \frac {{\left (b^{2} d - 2 \, a c d - a b e\right )} \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} + \frac {b d x^{4} - a e x^{4} - a d}{4 \, a^{2} x^{4}} \] Input:
integrate((e*x^4+d)/x^5/(c*x^8+b*x^4+a),x, algorithm="giac")
Output:
1/8*(b*d - a*e)*log(c*x^8 + b*x^4 + a)/a^2 - 1/4*(b*d - a*e)*log(x^4)/a^2 + 1/4*(b^2*d - 2*a*c*d - a*b*e)*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/( sqrt(-b^2 + 4*a*c)*a^2) + 1/4*(b*d*x^4 - a*e*x^4 - a*d)/(a^2*x^4)
Time = 43.40 (sec) , antiderivative size = 14100, normalized size of antiderivative = 125.89 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:
int((d + e*x^4)/(x^5*(a + b*x^4 + c*x^8)),x)
Output:
(log(x)*(a*e - b*d))/a^2 - d/(4*a*x^4) - (log((((((((a*e - b*d + a^2*(-(a* b*e - b^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2))*((((256*b^3*c^4*(a*b* e - b^2*d + a*c*d))/a + (64*b^2*c^5*x^4*(7*b^2*d + 9*a*b*e - 54*a*c*d))/a + (32*b^3*c^4*(a*b + 5*b^2*x^4 - 18*a*c*x^4)*(a*e - b*d + a^2*(-(a*b*e - b ^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2)))/a^2)*(a*e - b*d + a^2*(-(a* b*e - b^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2)))/(8*a^2) + (32*b^2*c^ 5*d*(8*a*b*e - 8*b^2*d + 3*a*c*d))/a^2 - (16*b*c^6*d*x^4*(13*b^2*d - 27*a* b*e + 54*a*c*d))/a^2))/(8*a^2) - (4*c^7*d^2*x^4*(31*b^2*d - 27*a*b*e + 18* a*c*d))/a^3 + (16*b*c^6*d^2*(6*a*b*e - 6*b^2*d + a*c*d))/a^3)*(a*e - b*d + a^2*(-(a*b*e - b^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2)))/(8*a^2) + (c^7*d^3*(16*a*b*e - 16*b^2*d + a*c*d))/a^4 + (c^8*d^3*x^4*(9*a*e - 20*b*d ))/a^4)*(a*e - b*d + a^2*(-(a*b*e - b^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)) )^(1/2)))/(8*a^2) + (c^8*d^4*(a*e - b*d))/a^5 - (c^9*d^5*x^4)/a^5)*((((c^7 *d^3*(16*a*b*e - 16*b^2*d + a*c*d))/a^4 - ((((b*d - a*e + a^2*(-(a*b*e - b ^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2))*((((256*b^3*c^4*(a*b*e - b^2 *d + a*c*d))/a + (64*b^2*c^5*x^4*(7*b^2*d + 9*a*b*e - 54*a*c*d))/a - (32*b ^3*c^4*(a*b + 5*b^2*x^4 - 18*a*c*x^4)*(b*d - a*e + a^2*(-(a*b*e - b^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2)))/a^2)*(b*d - a*e + a^2*(-(a*b*e - b ^2*d + 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2)))/(8*a^2) - (32*b^2*c^5*d*(8* a*b*e - 8*b^2*d + 3*a*c*d))/a^2 + (16*b*c^6*d*x^4*(13*b^2*d - 27*a*b*e ...
\[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )} \, dx=\int \frac {e \,x^{4}+d}{x^{5} \left (c \,x^{8}+b \,x^{4}+a \right )}d x \] Input:
int((e*x^4+d)/x^5/(c*x^8+b*x^4+a),x)
Output:
int((e*x^4+d)/x^5/(c*x^8+b*x^4+a),x)