Integrand size = 29, antiderivative size = 333 \[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {e x}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
-e*x/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)+c*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))*(d+(2*a*e-b *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)+c*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))*(d+(-2*a*e+b*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 = 0.64 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.39 \[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {\frac {4 e x}{\sqrt {d+e x^2}}-\text {RootSum}\left [a e^4+4 b d e^2 \text {$\#$1}^2-4 a e^3 \text {$\#$1}^2+16 c d^2 \text {$\#$1}^4-8 b d e \text {$\#$1}^4+6 a e^2 \text {$\#$1}^4+4 b d \text {$\#$1}^6-4 a e \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a e^4 \log (x)+a e^4 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )-4 c d^2 e \log (x) \text {$\#$1}^2+3 a e^3 \log (x) \text {$\#$1}^2+4 c d^2 e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 c d^2 \log (x) \text {$\#$1}^4-3 a e^2 \log (x) \text {$\#$1}^4-4 c d^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+a e \log (x) \text {$\#$1}^6-a e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{b d e^2 \text {$\#$1}-a e^3 \text {$\#$1}+8 c d^2 \text {$\#$1}^3-4 b d e \text {$\#$1}^3+3 a e^2 \text {$\#$1}^3+3 b d \text {$\#$1}^5-3 a e \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 c d^2-4 b d e+4 a e^2} \]
-(((4*e*x)/Sqrt[d + e*x^2] - 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^4*Log[x]) + a*e^4*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1 ] - 4*c*d^2*e*Log[x]*#1^2 + 3*a*e^3*Log[x]*#1^2 + 4*c*d^2*e*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 - 3*a*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x *#1]*#1^2 + 4*c*d^2*Log[x]*#1^4 - 3*a*e^2*Log[x]*#1^4 - 4*c*d^2*Log[-Sqrt[ d] + Sqrt[d + e*x^2] - x*#1]*#1^4 + 3*a*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 + a*e*Log[x]*#1^6 - a*e*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 = 0.73 (sec) , antiderivative size = 317, 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 = {1622, 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.
| \(\displaystyle \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1622 |
| \(\displaystyle \frac {\int \frac {c d x^2+a e}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\int \frac {c d x^2+a e}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}-\frac {e x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {\int \left (\frac {c d-\frac {c (2 a e-b d)}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}+\frac {c d+\frac {c (2 a e-b d)}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}\right )dx}{a e^2-b d e+c d^2}-\frac {e x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}{a e^2-b d e+c d^2}-\frac {e x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}\) |
-((e*x)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x^2])) + ((c*(d - (b*d - 2*a*e )/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(S qrt[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 + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sq rt[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)
3.4.96.3.1 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)*e*(f^2/(c*d^2 - b*d*e + a*e^2)) Int [(f*x)^(m - 2)*(d + e*x^2)^q, x], x] + Simp[f^2/(c*d^2 - b*d*e + a*e^2) I nt[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(Simp[a*e + c*d*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, 1] && LeQ[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 = 0.40 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(\frac {a \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (-b d e +2 c \,d^{2}+\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (a \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (b d e -2 c \,d^{2}+\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+2 \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, e x \right )}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (2 a \,e^{2}-2 b d e +2 c \,d^{2}\right )}\) | \(393\) |
| pseudoelliptic | \(\frac {a \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (-b d e +2 c \,d^{2}+\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (a \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (b d e -2 c \,d^{2}+\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+2 \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, e x \right )}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (2 a \,e^{2}-2 b d e +2 c \,d^{2}\right )}\) | \(393\) |
1/((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(a*((-2*a*e+b*d+(-4* d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*2^(1/2)*(e*x^2+d)^(1/2)*(-b*d*e+2*c*d^2 +(-4*d^2*(a*c-1/4*b^2))^(1/2)*e)*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)*(b*d*e-2*c*d^2+(-4*d^2* (a*c-1/4*b^2))^(1/2)*e)*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))+2*(-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)*e*x))/(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)/(-4*d^2*(a*c-1/4*b^2))^(1/2) /(2*a*e^2-2*b*d*e+2*c*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 14146 vs. \(2 (291) = 582\).
Time = 111.97 (sec) , antiderivative size = 14146, normalized size of antiderivative = 42.48 \[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {x^{2}}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {x^{2}}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {x^2}{{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]