Integrand size = 29, antiderivative size = 291 \[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{a x}-\frac {c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-(e*x^2+d)^(1/2)/a/x-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/(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/(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 = 0.73 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{a x}+\frac {\text {RootSum}\left [a e^4+4 b d e^2 \text {$\#$1}^2-4 a e^3 \text {$\#$1}^2+16 c d^2 \text {$\#$1}^4-8 b d e \text {$\#$1}^4+6 a e^2 \text {$\#$1}^4+4 b d \text {$\#$1}^6-4 a e \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {b d e^3 \log (x)-a e^4 \log (x)-b d e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )+a e^4 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )+4 c d^2 e \log (x) \text {$\#$1}^2-3 b d e^2 \log (x) \text {$\#$1}^2+3 a e^3 \log (x) \text {$\#$1}^2-4 c d^2 e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+3 b d e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-4 c d^2 \log (x) \text {$\#$1}^4+3 b d e \log (x) \text {$\#$1}^4-3 a e^2 \log (x) \text {$\#$1}^4+4 c d^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-3 b d e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-b d \log (x) \text {$\#$1}^6+a e \log (x) \text {$\#$1}^6+b d \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6-a e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{b d e^2 \text {$\#$1}-a e^3 \text {$\#$1}+8 c d^2 \text {$\#$1}^3-4 b d e \text {$\#$1}^3+3 a e^2 \text {$\#$1}^3+3 b d \text {$\#$1}^5-3 a e \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 a} \]
-(Sqrt[d + e*x^2]/(a*x)) + 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 & , (b*d*e^3*Log[x] - a*e^4*Log[x] - b*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1] + a*e^4*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1] + 4*c*d^2*e *Log[x]*#1^2 - 3*b*d*e^2*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*b*d*e^2*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*b*d*e*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*b*d*e*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 - b*d*Log[x]*#1^6 + a*e*Log[x]*#1^6 + b*d*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#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*a)
Time = 0.73 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1618, 242, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1618 |
| \(\displaystyle \frac {d \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{a}-\frac {\int \frac {c d x^2+b d-a e}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {\int \frac {c d x^2+b d-a e}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}-\frac {\sqrt {d+e x^2}}{a x}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle -\frac {\int \left (\frac {c d-\frac {c (b d-2 a e)}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}+\frac {c d+\frac {c (b d-2 a e)}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}\right )dx}{a}-\frac {\sqrt {d+e x^2}}{a x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}{a}-\frac {\sqrt {d+e x^2}}{a x}\) |
-(Sqrt[d + e*x^2]/(a*x)) - ((c*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTa n[(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 - (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 + 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]))/a
3.4.64.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4), x_Symbol] :> Simp[d/a Int[(f*x)^m*(d + e*x^2)^(q - 1), x], x] - Simp[1/(a*f^2) Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(Simp[b*d - 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] && GtQ[q, 0] && LtQ[m, 0]
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 = 0.77 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}}{a x}-\frac {\sqrt {2}\, \left (\frac {\left (a b d e +2 d^{2} a c -b^{2} d^{2}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (-a b d e -2 d^{2} a c +b^{2} d^{2}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 a \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\) | \(326\) |
| pseudoelliptic | \(\frac {-\frac {\sqrt {e \,x^{2}+d}}{x}-\frac {\left (a b d e +2 d^{2} a c -b^{2} d^{2}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}+\frac {\left (-a b d e -2 d^{2} a c +b^{2} d^{2}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}}{a}\) | \(342\) |
| default | \(\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+\frac {2 e \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{d}}{a}+\frac {\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {2}\, \left (\left (a e -b d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a c +b^{2}\right ) d^{2}-a b d e \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\left (\sqrt {2}\, \left (\left (a e -b d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (\left (2 a c -b^{2}\right ) d +a b e \right ) d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+2 \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{2 a \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(460\) |
-(e*x^2+d)^(1/2)/a/x-1/2*2^(1/2)/a/(-d^2*(4*a*c-b^2))^(1/2)*((a*b*d*e+2*d^ 2*a*c-b^2*d^2+(-d^2*(4*a*c-b^2))^(1/2)*a*e-(-d^2*(4*a*c-b^2))^(1/2)*b*d)/( (-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)*arctan(a/x*(e*x^2+d)^(1/2)* 2^(1/2)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2))-(-a*b*d*e-2*d^2*a *c+b^2*d^2+(-d^2*(4*a*c-b^2))^(1/2)*a*e-(-d^2*(4*a*c-b^2))^(1/2)*b*d)/((2* a*e-b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)*arctanh(a/x*(e*x^2+d)^(1/2)*2^( 1/2)/((2*a*e-b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2402 vs. \(2 (249) = 498\).
Time = 1.63 (sec) , antiderivative size = 2402, normalized size of antiderivative = 8.25 \[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
-1/4*(sqrt(1/2)*a*x*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + (a^3* b^2 - 4*a^4*c)*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a* b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a^2* b*c*d*e + (a^3*b^2*c - 4*a^4*c^2)*d*x^2*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2 *c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - 2*(a*b ^2*c - a^2*c^2)*d^2 + (4*a^2*b*c*e^2 + (b^3*c - a*b*c^2)*d^2 - (5*a*b^2*c - 4*a^2*c^2)*d*e)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((a^4*b^3 - 4*a^5*b*c) *x*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b* c)*d*e)/(a^6*b^2 - 4*a^7*c)) - ((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d - (a^2 *b^3 - 4*a^3*b*c)*e)*x)*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + ( a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2 *(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)))/x^2) - sqrt(1/2)*a*x*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a^2*b*c*d *e + (a^3*b^2*c - 4*a^4*c^2)*d*x^2*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - 2*(a*b^2*c - a^2*c^2)*d^2 + (4*a^2*b*c*e^2 + (b^3*c - a*b*c^2)*d^2 - (5*a*b^2*c - 4*a ^2*c^2)*d*e)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((a^4*b^3 - 4*a^5*b*c)*x*sq rt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)...
\[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {d + e x^{2}}}{x^{2} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {e\,x^2+d}}{x^2\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]