Integrand size = 27, antiderivative size = 553 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {e^2 (15 c d+4 b e) x \sqrt {a+b x^2-c x^4}}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}-\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \left (\frac {2 c \left (15 c^2 d^3+15 a c d e^2+4 a b e^3\right )}{b-\sqrt {b^2+4 a c}}+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right )\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}} \]
-1/15*e^2*(4*b*e+15*c*d)*x*(-c*x^4+b*x^2+a)^(1/2)/c^2-1/5*e^3*x^3*(-c*x^4+ b*x^2+a)^(1/2)/c-1/60*e*(45*c^2*d^2+8*b^2*e^2+3*c*e*(3*a*e+10*b*d))*Ellipt icE(x*2^(1/2)*c^(1/2)/(b+(4*a*c+b^2)^(1/2))^(1/2),((b+(4*a*c+b^2)^(1/2))/( b-(4*a*c+b^2)^(1/2)))^(1/2))*(b-(4*a*c+b^2)^(1/2))*(1-2*c*x^2/(b-(4*a*c+b^ 2)^(1/2)))^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)*(1-2*c*x^2/(b+(4*a*c+b^2)^(1/ 2)))^(1/2)/c^(7/2)*2^(1/2)/(-c*x^4+b*x^2+a)^(1/2)+1/60*EllipticF(x*2^(1/2) *c^(1/2)/(b+(4*a*c+b^2)^(1/2))^(1/2),((b+(4*a*c+b^2)^(1/2))/(b-(4*a*c+b^2) ^(1/2)))^(1/2))*(e*(45*c^2*d^2+8*b^2*e^2+3*c*e*(3*a*e+10*b*d))+2*c*(4*a*b* e^3+15*a*c*d*e^2+15*c^2*d^3)/(b-(4*a*c+b^2)^(1/2)))*(b-(4*a*c+b^2)^(1/2))* (1-2*c*x^2/(b-(4*a*c+b^2)^(1/2)))^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)*(1-2*c *x^2/(b+(4*a*c+b^2)^(1/2)))^(1/2)/c^(7/2)*2^(1/2)/(-c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.72 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=\frac {-4 c \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} e^2 x \left (a+b x^2-c x^4\right ) \left (4 b e+3 c \left (5 d+e x^2\right )\right )-i \sqrt {2} \left (-b+\sqrt {b^2+4 a c}\right ) e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+i \sqrt {2} \left (-30 c^3 d^3+8 b^2 \left (-b+\sqrt {b^2+4 a c}\right ) e^3+15 c^2 d e \left (-3 b d+3 \sqrt {b^2+4 a c} d-2 a e\right )+c e^2 \left (-30 b^2 d+30 b \sqrt {b^2+4 a c} d-17 a b e+9 a \sqrt {b^2+4 a c} e\right )\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{60 c^3 \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {a+b x^2-c x^4}} \]
(-4*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*e^2*x*(a + b*x^2 - c*x^4)*(4*b*e + 3*c*(5*d + e*x^2)) - I*Sqrt[2]*(-b + Sqrt[b^2 + 4*a*c])*e*(45*c^2*d^2 + 8*b^2*e^2 + 3*c*e*(10*b*d + 3*a*e))*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2) /(b + Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(-b + Sq rt[b^2 + 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a *c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + I*Sqrt[2]*(- 30*c^3*d^3 + 8*b^2*(-b + Sqrt[b^2 + 4*a*c])*e^3 + 15*c^2*d*e*(-3*b*d + 3*S qrt[b^2 + 4*a*c]*d - 2*a*e) + c*e^2*(-30*b^2*d + 30*b*Sqrt[b^2 + 4*a*c]*d - 17*a*b*e + 9*a*Sqrt[b^2 + 4*a*c]*e))*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x ^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(60*c^3*Sq rt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])
Time = 1.01 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1518, 25, 2207, 25, 1514, 399, 321, 327}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx\) |
| \(\Big \downarrow \) 1518 |
| \(\displaystyle -\frac {\int -\frac {e^2 (15 c d+4 b e) x^4+3 e \left (5 c d^2+a e^2\right ) x^2+5 c d^3}{\sqrt {-c x^4+b x^2+a}}dx}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {e^2 (15 c d+4 b e) x^4+3 e \left (5 c d^2+a e^2\right ) x^2+5 c d^3}{\sqrt {-c x^4+b x^2+a}}dx}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {-\frac {\int -\frac {15 c^2 d^3+15 a c e^2 d+4 a b e^3+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) x^2}{\sqrt {-c x^4+b x^2+a}}dx}{3 c}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{3 c}}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {15 c^2 d^3+15 a c e^2 d+4 a b e^3+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) x^2}{\sqrt {-c x^4+b x^2+a}}dx}{3 c}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{3 c}}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1514 |
| \(\displaystyle \frac {\frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \int \frac {15 c^2 d^3+15 a c e^2 d+4 a b e^3+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) x^2}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{3 c \sqrt {a+b x^2-c x^4}}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{3 c}}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {\left (b-\sqrt {4 a c+b^2}\right ) \left (\frac {2 c \left (4 a b e^3+15 a c d e^2+15 c^2 d^3\right )}{b-\sqrt {4 a c+b^2}}+e \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{2 c}-\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{2 c}\right )}{3 c \sqrt {a+b x^2-c x^4}}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{3 c}}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \left (\frac {2 c \left (4 a b e^3+15 a c d e^2+15 c^2 d^3\right )}{b-\sqrt {4 a c+b^2}}+e \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2}}-\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{2 c}\right )}{3 c \sqrt {a+b x^2-c x^4}}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{3 c}}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \left (\frac {2 c \left (4 a b e^3+15 a c d e^2+15 c^2 d^3\right )}{b-\sqrt {4 a c+b^2}}+e \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2}}-\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2}}\right )}{3 c \sqrt {a+b x^2-c x^4}}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{3 c}}{5 c}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}\) |
-1/5*(e^3*x^3*Sqrt[a + b*x^2 - c*x^4])/c + (-1/3*(e^2*(15*c*d + 4*b*e)*x*S qrt[a + b*x^2 - c*x^4])/c + (Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*S qrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*(-1/2*((b - Sqrt[b^2 + 4*a*c])* Sqrt[b + Sqrt[b^2 + 4*a*c]]*e*(45*c^2*d^2 + 8*b^2*e^2 + 3*c*e*(10*b*d + 3* a*e))*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], ( b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*c^(3/2)) + ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*((2*c*(15*c^2*d^3 + 15*a*c *d*e^2 + 4*a*b*e^3))/(b - Sqrt[b^2 + 4*a*c]) + e*(45*c^2*d^2 + 8*b^2*e^2 + 3*c*e*(10*b*d + 3*a*e)))*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sq rt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(2*Sq rt[2]*c^(3/2))))/(3*c*Sqrt[a + b*x^2 - c*x^4]))/(5*c)
3.4.85.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt [1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[(d + e*x^2)/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Simp[e^q*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c*(4*p + 2*q + 1)) Int[(a + b*x^2 + c*x^4)^p*Expand ToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2* p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p + 1)) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 *n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) *x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
Time = 6.41 (sec) , antiderivative size = 489, normalized size of antiderivative = 0.88
| method | result | size |
| elliptic | \(-\frac {e^{3} x^{3} \sqrt {-c \,x^{4}+b \,x^{2}+a}}{5 c}-\frac {\left (3 d \,e^{2}+\frac {4 e^{3} b}{5 c}\right ) x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (d^{3}+\frac {a \left (3 d \,e^{2}+\frac {4 e^{3} b}{5 c}\right )}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {\left (3 d^{2} e +\frac {3 e^{3} a}{5 c}+\frac {2 b \left (3 d \,e^{2}+\frac {4 e^{3} b}{5 c}\right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\) | \(489\) |
| risch | \(-\frac {e^{2} x \left (3 c \,x^{2} e +4 b e +15 c d \right ) \sqrt {-c \,x^{4}+b \,x^{2}+a}}{15 c^{2}}+\frac {\frac {15 c^{2} d^{3} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}+\frac {e^{3} a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}+\frac {15 a c d \,e^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {\left (9 e^{3} a c +8 b^{2} e^{3}+30 d \,e^{2} b c +45 d^{2} e \,c^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}}{15 c^{2}}\) | \(745\) |
| default | \(\text {Expression too large to display}\) | \(1195\) |
-1/5*e^3*x^3*(-c*x^4+b*x^2+a)^(1/2)/c-1/3*(3*d*e^2+4/5*e^3/c*b)/c*x*(-c*x^ 4+b*x^2+a)^(1/2)+1/4*(d^3+1/3*a/c*(3*d*e^2+4/5*e^3/c*b))*2^(1/2)/((-b+(4*a *c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+( 4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(-c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1 /2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c )^(1/2))-1/2*(3*d^2*e+3/5*e^3/c*a+2/3*b/c*(3*d*e^2+4/5*e^3/c*b))*a*2^(1/2) /((-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2) *(4+2*(b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(-c*x^4+b*x^2+a)^(1/2)/(b+(4*a*c+ b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2* (-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(4* a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left ({\left (45 \, a c^{3} d^{2} e + 30 \, a b c^{2} d e^{2} + {\left (8 \, a b^{2} c + 9 \, a^{2} c^{2}\right )} e^{3}\right )} \sqrt {-c} x \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + {\left (45 \, a b c^{2} d^{2} e + 30 \, a b^{2} c d e^{2} + {\left (8 \, a b^{3} + 9 \, a^{2} b c\right )} e^{3}\right )} \sqrt {-c} x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (15 \, c^{4} d^{3} + 45 \, a c^{3} d^{2} e + 15 \, {\left (2 \, a b c^{2} + a c^{3}\right )} d e^{2} + {\left (8 \, a b^{2} c + {\left (9 \, a^{2} + 4 \, a b\right )} c^{2}\right )} e^{3}\right )} \sqrt {-c} x \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - {\left (15 \, b c^{3} d^{3} - 45 \, a b c^{2} d^{2} e - 15 \, {\left (2 \, a b^{2} c - a b c^{2}\right )} d e^{2} - {\left (8 \, a b^{3} + {\left (9 \, a^{2} b - 4 \, a b^{2}\right )} c\right )} e^{3}\right )} \sqrt {-c} x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (3 \, a c^{3} e^{3} x^{4} + 45 \, a c^{3} d^{2} e + 30 \, a b c^{2} d e^{2} + {\left (8 \, a b^{2} c + 9 \, a^{2} c^{2}\right )} e^{3} + {\left (15 \, a c^{3} d e^{2} + 4 \, a b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c x^{4} + b x^{2} + a}}{30 \, a c^{4} x} \]
-1/30*(sqrt(1/2)*((45*a*c^3*d^2*e + 30*a*b*c^2*d*e^2 + (8*a*b^2*c + 9*a^2* c^2)*e^3)*sqrt(-c)*x*sqrt((b^2 + 4*a*c)/c^2) + (45*a*b*c^2*d^2*e + 30*a*b^ 2*c*d*e^2 + (8*a*b^3 + 9*a^2*b*c)*e^3)*sqrt(-c)*x)*sqrt((c*sqrt((b^2 + 4*a *c)/c^2) + b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 + 4*a*c)/c^ 2) + b)/c)/x), 1/2*(b*c*sqrt((b^2 + 4*a*c)/c^2) - b^2 - 2*a*c)/(a*c)) - sq rt(1/2)*((15*c^4*d^3 + 45*a*c^3*d^2*e + 15*(2*a*b*c^2 + a*c^3)*d*e^2 + (8* a*b^2*c + (9*a^2 + 4*a*b)*c^2)*e^3)*sqrt(-c)*x*sqrt((b^2 + 4*a*c)/c^2) - ( 15*b*c^3*d^3 - 45*a*b*c^2*d^2*e - 15*(2*a*b^2*c - a*b*c^2)*d*e^2 - (8*a*b^ 3 + (9*a^2*b - 4*a*b^2)*c)*e^3)*sqrt(-c)*x)*sqrt((c*sqrt((b^2 + 4*a*c)/c^2 ) + b)/c)*elliptic_f(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 + 4*a*c)/c^2) + b) /c)/x), 1/2*(b*c*sqrt((b^2 + 4*a*c)/c^2) - b^2 - 2*a*c)/(a*c)) + 2*(3*a*c^ 3*e^3*x^4 + 45*a*c^3*d^2*e + 30*a*b*c^2*d*e^2 + (8*a*b^2*c + 9*a^2*c^2)*e^ 3 + (15*a*c^3*d*e^2 + 4*a*b*c^2*e^3)*x^2)*sqrt(-c*x^4 + b*x^2 + a))/(a*c^4 *x)
\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=\int \frac {\left (d + e x^{2}\right )^{3}}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + b x^{2} + a}} \,d x } \]
\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + b x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]