Integrand size = 29, antiderivative size = 360 \[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {d x}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {\left (b d-a e-\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\left (b d-a e+\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
d*x/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)-arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^ (1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b*d-a*e+(a*b* e+2*a*c*d-b^2*d)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(2*c*d-e*(b-(-4*a *c+b^2)^(1/2)))^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-arctan(x*(2*c*d-e*(b+(- 4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b* d-a*e+(-a*b*e-2*a*c*d+b^2*d)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(b+(- 4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.25 \[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {\frac {4 d x}{\sqrt {d+e x^2}}-d \text {RootSum}\left [a e^4+4 b d e^2 \text {$\#$1}^2-4 a e^3 \text {$\#$1}^2+16 c d^2 \text {$\#$1}^4-8 b d e \text {$\#$1}^4+6 a e^2 \text {$\#$1}^4+4 b d \text {$\#$1}^6-4 a e \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a e^3 \log (x)+a e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )-4 b d e \log (x) \text {$\#$1}^2+7 a e^2 \log (x) \text {$\#$1}^2+4 b d e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-7 a e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b d \log (x) \text {$\#$1}^4-7 a e \log (x) \text {$\#$1}^4-4 b d \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+7 a e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+a \log (x) \text {$\#$1}^6-a \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{b d e^2 \text {$\#$1}-a e^3 \text {$\#$1}+8 c d^2 \text {$\#$1}^3-4 b d e \text {$\#$1}^3+3 a e^2 \text {$\#$1}^3+3 b d \text {$\#$1}^5-3 a e \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 c d^2-4 b d e+4 a e^2} \]
((4*d*x)/Sqrt[d + e*x^2] - d*RootSum[a*e^4 + 4*b*d*e^2*#1^2 - 4*a*e^3*#1^2 + 16*c*d^2*#1^4 - 8*b*d*e*#1^4 + 6*a*e^2*#1^4 + 4*b*d*#1^6 - 4*a*e*#1^6 + a*#1^8 & , (-(a*e^3*Log[x]) + a*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1 ] - 4*b*d*e*Log[x]*#1^2 + 7*a*e^2*Log[x]*#1^2 + 4*b*d*e*Log[-Sqrt[d] + Sqr t[d + e*x^2] - x*#1]*#1^2 - 7*a*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1] *#1^2 + 4*b*d*Log[x]*#1^4 - 7*a*e*Log[x]*#1^4 - 4*b*d*Log[-Sqrt[d] + Sqrt[ d + e*x^2] - x*#1]*#1^4 + 7*a*e*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^ 4 + a*Log[x]*#1^6 - a*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^6)/(b*d*e^ 2*#1 - a*e^3*#1 + 8*c*d^2*#1^3 - 4*b*d*e*#1^3 + 3*a*e^2*#1^3 + 3*b*d*#1^5 - 3*a*e*#1^5 + a*#1^7) & ])/(4*c*d^2 - 4*b*d*e + 4*a*e^2)
Time = 0.98 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1620, 208, 2256, 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.
| \(\displaystyle \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1620 |
| \(\displaystyle \frac {d^2 \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx}{a e^2-b d e+c d^2}-\frac {\int \frac {(b d-a e) x^2+a d}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {d x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {(b d-a e) x^2+a d}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {d x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \left (\frac {b d-a e-\frac {-d b^2+a e b+2 a c d}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}+\frac {b d-a e+\frac {-d b^2+a e b+2 a c d}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}\right )dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {\left (-\frac {-a b e-2 a c d+b^2 d}{\sqrt {b^2-4 a c}}-a e+b d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (\frac {-a b e-2 a c d+b^2 d}{\sqrt {b^2-4 a c}}-a e+b d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}{a e^2-b d e+c d^2}\) |
(d*x)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x^2]) - (((b*d - a*e - (b^2*d - 2*a*c*d - a*b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4 *a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqr t[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((b*d - a*e + ( b^2*d - 2*a*c*d - a*b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt [b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt [b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(c*d^2 - b*d*e + a*e^2)
3.4.95.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[d^2*(f^4/(c*d^2 - b*d*e + a*e^2)) Int[(f *x)^(m - 4)*(d + e*x^2)^q, x], x] - Simp[f^4/(c*d^2 - b*d*e + a*e^2) Int[ (f*x)^(m - 4)*(d + e*x^2)^(q + 1)*(Simp[a*d + (b*d - a*e)*x^2, x]/(a + b*x^ 2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 3]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 )^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Time = 0.54 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(\frac {d \left (\sqrt {2}\, a \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \left (a e -\frac {b d}{2}-\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (a \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (a e -\frac {b d}{2}+\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+x \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\right )\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right )}\) | \(382\) |
| default | \(\frac {x}{c d \sqrt {e \,x^{2}+d}}+\frac {\sqrt {2}\, a c \,d^{2} \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \left (a e -\frac {b d}{2}-\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (a c \,d^{2} \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (a e -\frac {b d}{2}+\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, e x \left (a e -b d \right )\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{c \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(423\) |
1/((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)/(e*x^2+d)^(1/2)/((2* a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*d/(-4*d^2*(a*c-1/4*b^2))^(1 /2)*(2^(1/2)*a*((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(e*x^2+ d)^(1/2)*(a*e-1/2*b*d-1/2*(-4*d^2*(a*c-1/4*b^2))^(1/2))*arctanh(a/x*(e*x^2 +d)^(1/2)*2^(1/2)/((2*a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2))+((2* a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(a*2^(1/2)*(e*x^2+d)^(1/2)* (a*e-1/2*b*d+1/2*(-4*d^2*(a*c-1/4*b^2))^(1/2))*arctan(a/x*(e*x^2+d)^(1/2)* 2^(1/2)/((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2))+x*(-4*d^2*(a* c-1/4*b^2))^(1/2)*((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)))/(a *e^2-b*d*e+c*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 14462 vs. \(2 (318) = 636\).
Time = 156.12 (sec) , antiderivative size = 14462, normalized size of antiderivative = 40.17 \[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {x^{4}}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {x^{4}}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {x^4}{{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]