Integrand size = 29, antiderivative size = 373 \[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{3 a x^3}+\frac {2 e \sqrt {d+e x^2}}{3 a d x}+\frac {(b d-a e) \sqrt {d+e x^2}}{a^2 d x}+\frac {c \left (b d-a e+\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {c \left (b d-a e-\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-1/3*(e*x^2+d)^(1/2)/a/x^3+2/3*e*(e*x^2+d)^(1/2)/a/d/x+(-a*e+b*d)*(e*x^2+d )^(1/2)/a^2/d/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))*(b*d-a*e+(-a*b*e-2*a*c*d+b^2*d)/(-4* a*c+b^2)^(1/2))/a^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))*(b*d-a*e+(a*b*e+2*a*c*d-b^2*d)/(-4*a*c +b^2)^(1/2))/a^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)
Leaf count is larger than twice the leaf count of optimal. \(7477\) vs. \(2(373)=746\).
Time = 16.38 (sec) , antiderivative size = 7477, normalized size of antiderivative = 20.05 \[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
Time = 1.59 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1618, 245, 242, 2246, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1618 |
| \(\displaystyle \frac {d \int \frac {1}{x^4 \sqrt {e x^2+d}}dx}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {d \left (-\frac {2 e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {d \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 2246 |
| \(\displaystyle \frac {d \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{a}-\frac {\int \left (\frac {b d-a e}{a x^2 \sqrt {e x^2+d}}+\frac {-d b^2+a e b-c (b d-a e) x^2+a c d}{a \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}\right )dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{a}-\frac {-\frac {c \left (\frac {-a b e-2 a c d+b^2 d}{\sqrt {b^2-4 a c}}-a e+b d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {c \left (-\frac {-a b e-2 a c d+b^2 d}{\sqrt {b^2-4 a c}}-a e+b d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x^2} (b d-a e)}{a d x}}{a}\) |
(d*(-1/3*Sqrt[d + e*x^2]/(d*x^3) + (2*e*Sqrt[d + e*x^2])/(3*d^2*x)))/a - ( -(((b*d - a*e)*Sqrt[d + e*x^2])/(a*d*x)) - (c*(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])])/(a*Sqrt[b - Sqrt[b^ 2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (c*(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])])/(a*Sqrt [b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a
3.4.65.3.1 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[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 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_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x ] && PolyQ[Px, x] && IntegerQ[p]
Time = 0.91 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}\, \left (a e \,x^{2}-3 b d \,x^{2}+d a \right )}{3 a^{2} x^{3} d}-\frac {\sqrt {2}\, \left (\left (\frac {\left (\left (b e +c d \right ) a -b^{2} d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (a^{2} c e -\frac {b \left (b e +3 c d \right ) a}{2}+\frac {b^{3} d}{2}\right )\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right ) \left (\frac {\left (\left (-b e -c d \right ) a +b^{2} d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (a^{2} c e -\frac {b \left (b e +3 c d \right ) a}{2}+\frac {b^{3} d}{2}\right )\right )\right )}{a^{2} \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(398\) |
| pseudoelliptic | \(-\frac {\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\frac {\left (d \left (a c -b^{2}\right )+a b e \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (\frac {\left (-3 a b c +b^{3}\right ) d}{2}+a e \left (a c -\frac {b^{2}}{2}\right )\right )\right ) \sqrt {2}\, d \,x^{3} \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (\sqrt {2}\, d \,x^{3} \left (\frac {\left (\left (-a c +b^{2}\right ) d -a b e \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (\frac {\left (-3 a b c +b^{3}\right ) d}{2}+a e \left (a c -\frac {b^{2}}{2}\right )\right )\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\frac {\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\left (-3 b \,x^{2}+a \right ) d +a e \,x^{2}\right )}{3}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a^{2} x^{3} d}\) | \(450\) |
| default | \(-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 a d \,x^{3}}-\frac {b \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+\frac {2 e \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{d}\right )}{a^{2}}-\frac {\sqrt {2}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\frac {\left (d \left (a c -b^{2}\right )+a b e \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (\frac {\left (-3 a b c +b^{3}\right ) d}{2}+a e \left (a c -\frac {b^{2}}{2}\right )\right )\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (\sqrt {2}\, \left (\frac {\left (\left (-a c +b^{2}\right ) d -a b e \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (\frac {\left (-3 a b c +b^{3}\right ) d}{2}+a e \left (a c -\frac {b^{2}}{2}\right )\right )\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-b \sqrt {e}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{a^{2} \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(517\) |
-1/3*(e*x^2+d)^(1/2)*(a*e*x^2-3*b*d*x^2+a*d)/a^2/x^3/d-1/a^2/((2*a*e-b*d+( -4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-4*d^2*(a*c-1/ 4*b^2))^(1/2))*a)^(1/2)*((1/2*((b*e+c*d)*a-b^2*d)*(-4*d^2*(a*c-1/4*b^2))^( 1/2)+d*(a^2*c*e-1/2*b*(b*e+3*c*d)*a+1/2*b^3*d))*((-2*a*e+b*d+(-4*d^2*(a*c- 1/4*b^2))^(1/2))*a)^(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)*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))*(1/2*((-b*e-c*d)*a+b^2*d)*(-4*d^2*(a*c-1/4* b^2))^(1/2)+d*(a^2*c*e-1/2*b*(b*e+3*c*d)*a+1/2*b^3*d)))/(-4*d^2*(a*c-1/4*b ^2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 4095 vs. \(2 (323) = 646\).
Time = 6.51 (sec) , antiderivative size = 4095, normalized size of antiderivative = 10.98 \[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
1/12*(3*sqrt(1/2)*a^2*d*x^3*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a* b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e - (a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^ 6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5 *c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2 *c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c))*log(((a^5*b^2*c^2 - 4*a^6*c^3)*d*x^2*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d* e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)) + 2*(a*b^4*c^2 - 3*a^2*b^2*c^3 + a^3*c^4)*d^2 - 2*(a^2*b^3*c^2 - 2*a^3*b*c^3 )*d*e - ((b^5*c^2 - 3*a*b^3*c^3 + a^2*b*c^4)*d^2 - (5*a*b^4*c^2 - 14*a^2*b ^2*c^3 + 4*a^3*c^4)*d*e + 4*(a^2*b^3*c^2 - 2*a^3*b*c^3)*e^2)*x^2 + 2*sqrt( 1/2)*sqrt(e*x^2 + d)*((a^6*b^4 - 6*a^7*b^2*c + 8*a^8*c^2)*x*sqrt(((b^8 - 6 *a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^ 2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^ 4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)) + ((a*b^7 - 7*a^2*b^5*c + 13*a^3*b^ 3*c^2 - 4*a^4*b*c^3)*d - (a^2*b^6 - 6*a^3*b^4*c + 8*a^4*b^2*c^2)*e)*x)*sqr t(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)* e - (a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^ 2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^ 3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^1...
\[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {d + e x^{2}}}{x^{4} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {e\,x^2+d}}{x^4\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]