Integrand size = 29, antiderivative size = 390 \[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {x \sqrt {d+e x^2}}{2 c}-\frac {\left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {(c d-2 b e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}} \]
1/2*(-2*b*e+c*d)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c^2/e^(1/2)+1/2*x*(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))*(b*c*d-b^2*e+a*c*e+(-3*a*b*c*e+2*a*c^2*d +b^3*e-b^2*c*d)/(-4*a*c+b^2)^(1/2))/c^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*c*d-b^2*e+a*c*e+( 3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))/c^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 = 1.01 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.02 \[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {2 c x \sqrt {d+e x^2}+\frac {4 (c d-2 b e) \text {arctanh}\left (\frac {\sqrt {e} x}{-\sqrt {d}+\sqrt {d+e x^2}}\right )}{\sqrt {e}}+\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 c d e^3 \log (x)-a b e^4 \log (x)-a c d e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )+a b e^4 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )+4 b c d^2 e \log (x) \text {$\#$1}^2-4 b^2 d e^2 \log (x) \text {$\#$1}^2+a c d e^2 \log (x) \text {$\#$1}^2+3 a b e^3 \log (x) \text {$\#$1}^2-4 b c d^2 e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b^2 d e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a c d e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a b e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-4 b c d^2 \log (x) \text {$\#$1}^4+4 b^2 d e \log (x) \text {$\#$1}^4-a c d e \log (x) \text {$\#$1}^4-3 a b e^2 \log (x) \text {$\#$1}^4+4 b c d^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-4 b^2 d e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+a c d e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a b e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-a c d \log (x) \text {$\#$1}^6+a b e \log (x) \text {$\#$1}^6+a c d \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6-a b 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^2} \]
(2*c*x*Sqrt[d + e*x^2] + (4*(c*d - 2*b*e)*ArcTanh[(Sqrt[e]*x)/(-Sqrt[d] + Sqrt[d + e*x^2])])/Sqrt[e] + 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*c*d*e^3*Log[x] - a*b*e^4*Log[x] - a*c*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1] + a*b*e^4*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1] + 4*b*c*d^2*e*Log[x]*#1^2 - 4*b^2*d*e^2*Log[x]*#1^2 + a*c*d*e^2*Log[x]*#1^2 + 3*a*b*e^3*Log[x]*#1^2 - 4*b*c*d^2*e*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x* #1]*#1^2 + 4*b^2*d*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 - a*c*d *e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 - 3*a*b*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 - 4*b*c*d^2*Log[x]*#1^4 + 4*b^2*d*e*Log[x]* #1^4 - a*c*d*e*Log[x]*#1^4 - 3*a*b*e^2*Log[x]*#1^4 + 4*b*c*d^2*Log[-Sqrt[d ] + Sqrt[d + e*x^2] - x*#1]*#1^4 - 4*b^2*d*e*Log[-Sqrt[d] + Sqrt[d + e*x^2 ] - x*#1]*#1^4 + a*c*d*e*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 + 3*a *b*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 - a*c*d*Log[x]*#1^6 + a *b*e*Log[x]*#1^6 + a*c*d*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^6 - a*b *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^2)
Time = 1.84 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1614, 299, 224, 219, 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.
| \(\displaystyle \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1614 |
| \(\displaystyle \frac {\int \frac {c e x^2+c d-b e}{\sqrt {e x^2+d}}dx}{c^2}-\frac {\int \frac {\left (-e b^2+c d b+a c e\right ) x^2+a (c d-b e)}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {1}{2} (c d-2 b e) \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} c x \sqrt {d+e x^2}}{c^2}-\frac {\int \frac {\left (-e b^2+c d b+a c e\right ) x^2+a (c d-b e)}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{2} (c d-2 b e) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} c x \sqrt {d+e x^2}}{c^2}-\frac {\int \frac {\left (-e b^2+c d b+a c e\right ) x^2+a (c d-b e)}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (c d-2 b e)}{2 \sqrt {e}}+\frac {1}{2} c x \sqrt {d+e x^2}}{c^2}-\frac {\int \frac {\left (-e b^2+c d b+a c e\right ) x^2+a (c d-b e)}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (c d-2 b e)}{2 \sqrt {e}}+\frac {1}{2} c x \sqrt {d+e x^2}}{c^2}-\frac {\int \left (\frac {-e b^2+c d b+a c e-\frac {e b^3-c d b^2-3 a c e b+2 a c^2 d}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}+\frac {-e b^2+c d b+a c e+\frac {e b^3-c d b^2-3 a c e b+2 a c^2 d}{\sqrt {b^2-4 a c}}}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}\right )dx}{c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (c d-2 b e)}{2 \sqrt {e}}+\frac {1}{2} c x \sqrt {d+e x^2}}{c^2}-\frac {\frac {\left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}{c^2}\) |
-((((b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqr t[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]) + ((b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*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^2) + ((c*x*Sqrt[d + e*x^2])/2 + ((c*d - 2*b*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[ d + e*x^2]])/(2*Sqrt[e]))/c^2
3.4.61.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[f^4/c^2 Int[(f*x)^(m - 4)*(c*d - b*e + c *e*x^2)*(d + e*x^2)^(q - 1), x], x] - Simp[f^4/c^2 Int[(f*x)^(m - 4)*(d + e*x^2)^(q - 1)*(Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*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] && GtQ[q, 0] && 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 = 2.62 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {x \sqrt {e \,x^{2}+d}}{2 c}-\frac {\frac {\left (2 b e -c d \right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{c \sqrt {e}}+\frac {a \sqrt {2}\, \left (\frac {\left (2 a c d e -b^{2} d e +b c \,d^{2}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b e -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (-2 a c d e +b^{2} d e -b c \,d^{2}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b e -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{c \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}}{2 c}\) | \(360\) |
| default | \(\frac {\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}}{c}-\frac {a \sqrt {2}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\frac {\left (-b e +c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (a c e -\frac {b \left (b e -c d \right )}{2}\right )\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (\left (\frac {\left (b e -c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (a c e -\frac {b \left (b e -c d \right )}{2}\right )\right ) a \sqrt {2}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+b \sqrt {e}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{c^{2} \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(438\) |
| pseudoelliptic | \(-\frac {a \sqrt {2}\, \left (\frac {\left (c d \sqrt {e}-e^{\frac {3}{2}} b \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (\left (a c -\frac {b^{2}}{2}\right ) e^{\frac {3}{2}}+\frac {b \sqrt {e}\, c d}{2}\right ) d \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (a \sqrt {2}\, \left (\frac {\left (-c d \sqrt {e}+e^{\frac {3}{2}} b \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (\left (a c -\frac {b^{2}}{2}\right ) e^{\frac {3}{2}}+\frac {b \sqrt {e}\, c d}{2}\right ) d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\frac {\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\left (-2 b e +c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\sqrt {e \,x^{2}+d}\, \sqrt {e}\, c x \right )}{2}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, c^{2}}\) | \(445\) |
1/2*x*(e*x^2+d)^(1/2)/c-1/2/c*((2*b*e-c*d)/c*ln(x*e^(1/2)+(e*x^2+d)^(1/2)) /e^(1/2)+1/c*a*2^(1/2)/(-d^2*(4*a*c-b^2))^(1/2)*((2*a*c*d*e-b^2*d*e+b*c*d^ 2+(-d^2*(4*a*c-b^2))^(1/2)*b*e-(-d^2*(4*a*c-b^2))^(1/2)*c*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))-(-2*a*c*d*e+b^2*d*e-b*c*d^2+( -d^2*(4*a*c-b^2))^(1/2)*b*e-(-d^2*(4*a*c-b^2))^(1/2)*c*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. 3263 vs. \(2 (338) = 676\).
Time = 14.87 (sec) , antiderivative size = 6534, normalized size of antiderivative = 16.75 \[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
\[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x^{4} \sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {e x^{2} + d} x^{4}}{c x^{4} + b x^{2} + a} \,d x } \]
Exception generated. \[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^4 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x^4\,\sqrt {e\,x^2+d}}{c\,x^4+b\,x^2+a} \,d x \]