Integrand size = 25, antiderivative size = 267 \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=-\frac {\sqrt {c} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )} \] Output:
-1/4*c^(1/2)*(e-(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x^ 2/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^ 2-b*d*e+c*d^2)-1/4*c^(1/2)*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1 /2)*c^(1/2)*x^2/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/(b+(-4*a*c+b^2)^(1/2 ))^(1/2)/(a*e^2-b*d*e+c*d^2)+1/2*e^(3/2)*arctan(e^(1/2)*x^2/d^(1/2))/d^(1/ 2)/(a*e^2-b*d*e+c*d^2)
Time = 0.39 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.06 \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\frac {1}{4} \left (\frac {\sqrt {2} \sqrt {c} \left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (-c d^2+e (b d-a e)\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \left (-c d^2+e (b d-a e)\right )}+\frac {2 e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\sqrt {d} \left (c d^2-b d e+a e^2\right )}\right ) \] Input:
Integrate[x/((d + e*x^4)*(a + b*x^4 + c*x^8)),x]
Output:
((Sqrt[2]*Sqrt[c]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqr t[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b ^2 - 4*a*c]]*(-(c*d^2) + e*(b*d - a*e))) + (Sqrt[2]*Sqrt[c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4* a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + e*(b*d - a*e))) + (2*e^(3/2)*ArcTan[(Sqrt[e]*x^2)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d *e + a*e^2)))/4
Time = 0.66 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1814, 1484, 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.
\(\displaystyle \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx\) |
\(\Big \downarrow \) 1814 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx^2\) |
\(\Big \downarrow \) 1484 |
\(\displaystyle \frac {1}{2} \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\sqrt {d} \left (a e^2-b d e+c d^2\right )}\right )\) |
Input:
Int[x/((d + e*x^4)*(a + b*x^4 + c*x^8)),x]
Output:
(-((Sqrt[c]*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]* x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c *d^2 - b*d*e + a*e^2))) - (Sqrt[c]*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*A rcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) + (e^(3/2)*ArcTan[(Sqrt[e]*x ^2)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2)))/2
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e _.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Sub st[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k)))^ p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.32 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {2 c \left (-\frac {\left (\sqrt {-4 a c +b^{2}}\, e +e b -2 c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (\sqrt {-4 a c +b^{2}}\, e -e b +2 c d \right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}+\frac {e^{2} \arctan \left (\frac {e \,x^{2}}{\sqrt {d e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {d e}}\) | \(220\) |
risch | \(\text {Expression too large to display}\) | \(2858\) |
Input:
int(x/(e*x^4+d)/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
Output:
-2/(a*e^2-b*d*e+c*d^2)*c*(-1/8*((-4*a*c+b^2)^(1/2)*e+e*b-2*c*d)/(-4*a*c+b^ 2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^2*2^(1/2)/( (-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*((-4*a*c+b^2)^(1/2)*e-e*b+2*c*d)/(-4 *a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^2*2^(1 /2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))+1/2*e^2/(a*e^2-b*d*e+c*d^2)/(d*e)^( 1/2)*arctan(e*x^2/(d*e)^(1/2))
Timed out. \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:
integrate(x/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:
integrate(x/(e*x**4+d)/(c*x**8+b*x**4+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x/(e*x^4+d)/(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(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 7671 vs. \(2 (217) = 434\).
Time = 2.32 (sec) , antiderivative size = 7671, normalized size of antiderivative = 28.73 \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:
integrate(x/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="giac")
Output:
1/2*e^2*arctan(e*x^2/sqrt(d*e))/((c*d^2 - b*d*e + a*e^2)*sqrt(d*e)) + 1/16 *(2*(2*b^3*c^5 - 8*a*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*b*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b *c^5 - 2*(b^2 - 4*a*c)*b*c^5)*d^5 - 5*(2*b^4*c^4 - 8*a*b^2*c^5 - sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 4*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 2*sqrt(2)*sqrt(b^ 2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^4 - 2*(b^2 - 4*a*c)*b^2*c^4)*d^ 4*e + 4*(2*b^5*c^3 - 6*a*b^3*c^4 - 8*a^2*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b ^2 - 4*a*c)*c)*b^3*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*b*c^4 - 2*(b^2 - 4*a*c)*b^3*c^3 - 2*(b^2 - 4*a*c)*a*b*c^4)*d^3* e^2 - (2*b^6*c^2 + 4*a*b^4*c^3 - 48*a^2*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr...
Time = 82.70 (sec) , antiderivative size = 104763, normalized size of antiderivative = 392.37 \[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:
int(x/((d + e*x^4)*(a + b*x^4 + c*x^8)),x)
Output:
(atan((((((-d*e^3)^(1/2)*(((-d*e^3)^(1/2)*(x^2*(320*b^5*c^7*e^12 + 320*c^1 2*d^5*e^7 - 2240*a*b^3*c^8*e^12 + 3840*a^2*b*c^9*e^12 + 448*a*c^11*d^3*e^9 - 4224*a^2*c^10*d*e^11 - 640*b*c^11*d^4*e^8 - 640*b^4*c^8*d*e^11 + 320*b^ 2*c^10*d^3*e^9 + 320*b^3*c^9*d^2*e^10 - 1728*a*b*c^10*d^2*e^10 + 3648*a*b^ 2*c^9*d*e^11) - ((-d*e^3)^(1/2)*(3072*a*c^12*d^6*e^7 - ((-d*e^3)^(1/2)*(x^ 2*(65536*a^5*c^9*e^14 + 2048*b^10*c^4*e^14 + 2048*c^14*d^10*e^4 - 28672*a* b^8*c^5*e^14 + 6144*a*c^13*d^8*e^6 - 8192*b*c^13*d^9*e^5 - 8192*b^9*c^5*d* e^13 + 149504*a^2*b^6*c^6*e^14 - 339968*a^3*b^4*c^7*e^14 + 262144*a^4*b^2* c^8*e^14 - 104448*a^2*c^12*d^6*e^8 - 120832*a^3*c^11*d^4*e^10 + 446464*a^4 *c^10*d^2*e^12 + 12288*b^2*c^12*d^8*e^6 - 30720*b^4*c^10*d^6*e^8 + 49152*b ^5*c^9*d^5*e^9 - 30720*b^6*c^8*d^4*e^10 + 12288*b^8*c^6*d^2*e^12 - 106496* a^2*b^2*c^10*d^4*e^10 - 331776*a^2*b^3*c^9*d^3*e^11 + 618496*a^2*b^4*c^8*d ^2*e^12 - 979968*a^3*b^2*c^9*d^2*e^12 - 51200*a*b*c^12*d^7*e^7 + 104448*a* b^7*c^6*d*e^13 - 733184*a^4*b*c^9*d*e^13 + 177152*a*b^2*c^11*d^6*e^8 - 271 360*a*b^3*c^10*d^5*e^9 + 158720*a*b^4*c^9*d^4*e^10 + 54272*a*b^5*c^8*d^3*e ^11 - 149504*a*b^6*c^7*d^2*e^12 + 262144*a^2*b*c^11*d^5*e^9 - 487424*a^2*b ^5*c^7*d*e^13 + 456704*a^3*b*c^10*d^3*e^11 + 986112*a^3*b^3*c^8*d*e^13) + ((-d*e^3)^(1/2)*(((-d*e^3)^(1/2)*(x^2*(9437184*a^7*b*c^8*e^16 - 11010048*a ^7*c^9*d*e^15 + 49152*a^3*b^9*c^4*e^16 - 737280*a^4*b^7*c^5*e^16 + 4128768 *a^5*b^5*c^6*e^16 - 10223616*a^6*b^3*c^7*e^16 - 655360*a^2*c^14*d^11*e^...
\[ \int \frac {x}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\int \frac {x}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )}d x \] Input:
int(x/(e*x^4+d)/(c*x^8+b*x^4+a),x)
Output:
int(x/(e*x^4+d)/(c*x^8+b*x^4+a),x)