Integrand size = 25, antiderivative size = 414 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\frac {x \left (b^2 c^2+a^2 d^2+b d (b c+a d) x^2\right )}{3 a c (b c-a d)^2 \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}}+\frac {d \left (b^2 c^2+9 a b c d-2 a^2 d^2\right ) x}{3 a c^2 (b c-a d)^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {2 \sqrt {b} (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) \left (c+d x^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 (b c-a d)^4 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {\sqrt {b} d \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \left (c+d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} c^2 (b c-a d)^4 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:
1/3*x*(b^2*c^2+a^2*d^2+b*d*(a*d+b*c)*x^2)/a/c/(-a*d+b*c)^2/(a*c+(a*d+b*c)* x^2+b*d*x^4)^(3/2)+1/3*d*(-2*a^2*d^2+9*a*b*c*d+b^2*c^2)*x/a/c^2/(-a*d+b*c) ^3/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+2/3*b^(1/2)*(a*d+b*c)*(a^2*d^2-6*a*b* c*d+b^2*c^2)*(d*x^2+c)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a* d/b/c)^(1/2))/a^(3/2)/c^2/(-a*d+b*c)^4/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(a* c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)-1/3*b^(1/2)*d*(a^2*d^2-18*a*b*c*d+b^2*c^2)* (d*x^2+c)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^( 1/2)/c^2/(-a*d+b*c)^4/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(a*c+(a*d+b*c)*x^2+b *d*x^4)^(1/2)
Result contains complex when optimal does not.
Time = 7.58 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\frac {i \left (i \sqrt {\frac {b}{a}} x \left (a^2 c d^3 (b c-a d) \left (a+b x^2\right )^2-2 a^2 d^3 (-5 b c+a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )+a b^3 c^2 (-b c+a d) \left (c+d x^2\right )^2-2 b^3 c^2 (b c-5 a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )+b c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (2 \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\left (2 b^3 c^3-11 a b^2 c^2 d+8 a^2 b c d^2+a^3 d^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c-a d)^4 \left (\left (a+b x^2\right ) \left (c+d x^2\right )\right )^{3/2}} \] Input:
Integrate[(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(-5/2),x]
Output:
((I/3)*(I*Sqrt[b/a]*x*(a^2*c*d^3*(b*c - a*d)*(a + b*x^2)^2 - 2*a^2*d^3*(-5 *b*c + a*d)*(a + b*x^2)^2*(c + d*x^2) + a*b^3*c^2*(-(b*c) + a*d)*(c + d*x^ 2)^2 - 2*b^3*c^2*(b*c - 5*a*d)*(a + b*x^2)*(c + d*x^2)^2) + b*c*(a + b*x^2 )*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*(2*(b^3*c^3 - 5*a*b^ 2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d) /(b*c)] - (2*b^3*c^3 - 11*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*EllipticF [I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(a^2*Sqrt[b/a]*c^2*(b*c - a*d)^4* ((a + b*x^2)*(c + d*x^2))^(3/2))
Time = 1.58 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.92, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1405, 25, 1492, 27, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {1}{\left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}-\frac {\int -\frac {3 b d (b c+a d) x^2+2 \left (b^2 c^2-3 a b d c+a^2 d^2\right )}{\left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}dx}{3 a c (b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 b d (b c+a d) x^2+2 \left (b^2 c^2-3 a b d c+a^2 d^2\right )}{\left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}dx}{3 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {x \left (2 a^4 d^4-9 a^3 b c d^3+2 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2-9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {\int \frac {b d \left (2 (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2-18 a b d c+a^2 d^2\right )\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \left (2 a^4 d^4-9 a^3 b c d^3+2 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2-9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \int \frac {2 (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2-18 a b d c+a^2 d^2\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {x \left (2 a^4 d^4-9 a^3 b c d^3+2 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2-9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \left (\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {c} \left (a^2 d^2-18 a b c d+b^2 c^2\right )+\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{\sqrt {b} \sqrt {d}}\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {2 \sqrt {a} \sqrt {c} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \left (2 a^4 d^4-9 a^3 b c d^3+2 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2-9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \left (\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {c} \left (a^2 d^2-18 a b c d+b^2 c^2\right )+\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{\sqrt {b} \sqrt {d}}\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {x \left (2 a^4 d^4-9 a^3 b c d^3+2 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2-9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c} \left (a^2 d^2-18 a b c d+b^2 c^2\right )+\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{\sqrt {b} \sqrt {d}}\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}+\frac {\frac {x \left (2 a^4 d^4-9 a^3 b c d^3+2 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2-9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c} \left (a^2 d^2-18 a b c d+b^2 c^2\right )+\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{\sqrt {b} \sqrt {d}}\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}\right )}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}\) |
Input:
Int[(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(-5/2),x]
Output:
(x*(b^2*c^2 + a^2*d^2 + b*d*(b*c + a*d)*x^2))/(3*a*c*(b*c - a*d)^2*(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(3/2)) + ((x*(2*b^4*c^4 - 9*a*b^3*c^3*d - 2*a^2 *b^2*c^2*d^2 - 9*a^3*b*c*d^3 + 2*a^4*d^4 + 2*b*d*(b*c + a*d)*(b^2*c^2 - 6* a*b*c*d + a^2*d^2)*x^2))/(a*c*(b*c - a*d)^2*Sqrt[a*c + (b*c + a*d)*x^2 + b *d*x^4]) - (b*d*((-2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*(-((x*Sqr t[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2 )) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*Elli pticE[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(S qrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a* d)*x^2 + b*d*x^4])))/(Sqrt[b]*Sqrt[d]) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] *(b^2*c^2 - 18*a*b*c*d + a^2*d^2) + (2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2))/(Sqrt[b]*Sqrt[d]))*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[ (a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^ 2]*EllipticF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])))/(a*c*(b*c - a*d)^2))/(3*a*c*(b*c - a*d)^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs. \(2(393)=786\).
Time = 0.62 (sec) , antiderivative size = 825, normalized size of antiderivative = 1.99
| method | result | size |
| default | \(\frac {\left (\frac {\left (a d +b c \right ) x^{3}}{3 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{3 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} d^{2}}\right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right )^{2}}-\frac {2 b d \left (-\frac {\left (a d +b c \right ) \left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) x^{3}}{3 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}-\frac {\left (2 a^{4} d^{4}-9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}-9 a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x}{6 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2} b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {\frac {2}{3} a^{2} d^{2}-2 a b c d +\frac {2}{3} b^{2} c^{2}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2} c^{2}}-\frac {2 a^{4} d^{4}-9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}-9 a \,b^{3} c^{3} d +2 b^{4} c^{4}}{3 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (\frac {4 b d \left (a^{3} d^{3}-5 a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2} a^{2} c^{2}}-\frac {2 b d \left (a d +b c \right ) \left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right )}{a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\) | \(825\) |
| elliptic | \(\frac {\left (\frac {\left (a d +b c \right ) x^{3}}{3 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{3 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} d^{2}}\right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right )^{2}}-\frac {2 b d \left (-\frac {\left (a d +b c \right ) \left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) x^{3}}{3 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}-\frac {\left (2 a^{4} d^{4}-9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}-9 a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x}{6 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2} b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {\frac {2}{3} a^{2} d^{2}-2 a b c d +\frac {2}{3} b^{2} c^{2}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2} c^{2}}-\frac {2 a^{4} d^{4}-9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}-9 a \,b^{3} c^{3} d +2 b^{4} c^{4}}{3 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (\frac {4 b d \left (a^{3} d^{3}-5 a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2} a^{2} c^{2}}-\frac {2 b d \left (a d +b c \right ) \left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right )}{a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\) | \(825\) |
Input:
int(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/3/a/c*(a*d+b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b/d*x^3+1/3*(a^2*d^2+b^2*c^ 2)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^2/d^2*x)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c )^(1/2)/(x^4+(a*d+b*c)/b/d*x^2+a*c/b/d)^2-2*b*d*(-1/3*(a*d+b*c)*(a^2*d^2-6 *a*b*c*d+b^2*c^2)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2*x^3-1/6*(2*a^4*d^4 -9*a^3*b*c*d^3-2*a^2*b^2*c^2*d^2-9*a*b^3*c^3*d+2*b^4*c^4)/a^2/c^2/(a^2*d^2 -2*a*b*c*d+b^2*c^2)^2/b/d*x)/((x^4+(a*d+b*c)/b/d*x^2+a*c/b/d)*b*d)^(1/2)+( 2/3*(a^2*d^2-3*a*b*c*d+b^2*c^2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/c^2-1/3*(2 *a^4*d^4-9*a^3*b*c*d^3-2*a^2*b^2*c^2*d^2-9*a*b^3*c^3*d+2*b^4*c^4)/a^2/c^2/ (a^2*d^2-2*a*b*c*d+b^2*c^2)^2)/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ (1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a* d+b*c)/c/b)^(1/2))-(4/3*b*d*(a^3*d^3-5*a^2*b*c*d^2-5*a*b^2*c^2*d+b^3*c^3)/ (a^2*d^2-2*a*b*c*d+b^2*c^2)^2/a^2/c^2-2*b*d*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b ^2*c^2)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2)*c/(-b/a)^(1/2)*(1+b*x^2/a)^ (1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x *(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+ b*c)/c/b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1461 vs. \(2 (393) = 786\).
Time = 0.18 (sec) , antiderivative size = 1461, normalized size of antiderivative = 3.53 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(5/2),x, algorithm="fricas")
Output:
-1/3*(2*(a^2*b^4*c^5 - 5*a^3*b^3*c^4*d - 5*a^4*b^2*c^3*d^2 + a^5*b*c^2*d^3 + (b^6*c^3*d^2 - 5*a*b^5*c^2*d^3 - 5*a^2*b^4*c*d^4 + a^3*b^3*d^5)*x^8 + 2 *(b^6*c^4*d - 4*a*b^5*c^3*d^2 - 10*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^4 *b^2*d^5)*x^6 + (b^6*c^5 - a*b^5*c^4*d - 24*a^2*b^4*c^3*d^2 - 24*a^3*b^3*c ^2*d^3 - a^4*b^2*c*d^4 + a^5*b*d^5)*x^4 + 2*(a*b^5*c^5 - 4*a^2*b^4*c^4*d - 10*a^3*b^3*c^3*d^2 - 4*a^4*b^2*c^2*d^3 + a^5*b*c*d^4)*x^2)*sqrt(a*c)*sqrt (-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*a^2*b^4*c^5 + (2*b ^6*c^3*d^2 + (a^2*b^4 - 10*a*b^5)*c^2*d^3 - 2*(9*a^3*b^3 + 5*a^2*b^4)*c*d^ 4 + (a^4*b^2 + 2*a^3*b^3)*d^5)*x^8 + 2*(2*b^6*c^4*d + (a^2*b^4 - 8*a*b^5)* c^3*d^2 - (17*a^3*b^3 + 20*a^2*b^4)*c^2*d^3 - (17*a^4*b^2 + 8*a^3*b^3)*c*d ^4 + (a^5*b + 2*a^4*b^2)*d^5)*x^6 + (a^4*b^2 - 10*a^3*b^3)*c^4*d - 2*(9*a^ 5*b + 5*a^4*b^2)*c^3*d^2 + (a^6 + 2*a^5*b)*c^2*d^3 + (2*b^6*c^5 + (a^2*b^4 - 2*a*b^5)*c^4*d - 2*(7*a^3*b^3 + 24*a^2*b^4)*c^3*d^2 - 2*(35*a^4*b^2 + 2 4*a^3*b^3)*c^2*d^3 - 2*(7*a^5*b + a^4*b^2)*c*d^4 + (a^6 + 2*a^5*b)*d^5)*x^ 4 + 2*(2*a*b^5*c^5 + (a^3*b^3 - 8*a^2*b^4)*c^4*d - (17*a^4*b^2 + 20*a^3*b^ 3)*c^3*d^2 - (17*a^5*b + 8*a^4*b^2)*c^2*d^3 + (a^6 + 2*a^5*b)*c*d^4)*x^2)* sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*(a*b ^5*c^3*d^2 - 5*a^2*b^4*c^2*d^3 - 5*a^3*b^3*c*d^4 + a^4*b^2*d^5)*x^7 + (4*a *b^5*c^4*d - 17*a^2*b^4*c^3*d^2 - 22*a^3*b^3*c^2*d^3 - 17*a^4*b^2*c*d^4 + 4*a^5*b*d^5)*x^5 + 2*(a*b^5*c^5 - 2*a^2*b^4*c^4*d - 11*a^3*b^3*c^3*d^2 ...
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (a c + b d x^{4} + x^{2} \left (a d + b c\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a*c+(a*d+b*c)*x**2+b*d*x**4)**(5/2),x)
Output:
Integral((a*c + b*d*x**4 + x**2*(a*d + b*c))**(-5/2), x)
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(5/2),x, algorithm="maxima")
Output:
integrate((b*d*x^4 + (b*c + a*d)*x^2 + a*c)^(-5/2), x)
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(5/2),x, algorithm="giac")
Output:
integrate((b*d*x^4 + (b*c + a*d)*x^2 + a*c)^(-5/2), x)
Timed out. \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c\right )}^{5/2}} \,d x \] Input:
int(1/(a*c + x^2*(a*d + b*c) + b*d*x^4)^(5/2),x)
Output:
int(1/(a*c + x^2*(a*d + b*c) + b*d*x^4)^(5/2), x)
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d^{3} x^{12}+3 a \,b^{2} d^{3} x^{10}+3 b^{3} c \,d^{2} x^{10}+3 a^{2} b \,d^{3} x^{8}+9 a \,b^{2} c \,d^{2} x^{8}+3 b^{3} c^{2} d \,x^{8}+a^{3} d^{3} x^{6}+9 a^{2} b c \,d^{2} x^{6}+9 a \,b^{2} c^{2} d \,x^{6}+b^{3} c^{3} x^{6}+3 a^{3} c \,d^{2} x^{4}+9 a^{2} b \,c^{2} d \,x^{4}+3 a \,b^{2} c^{3} x^{4}+3 a^{3} c^{2} d \,x^{2}+3 a^{2} b \,c^{3} x^{2}+a^{3} c^{3}}d x \] Input:
int(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(5/2),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3*c**3 + 3*a**3*c**2*d*x**2 + 3*a**3*c*d**2*x**4 + a**3*d**3*x**6 + 3*a**2*b*c**3*x**2 + 9*a**2*b*c**2*d *x**4 + 9*a**2*b*c*d**2*x**6 + 3*a**2*b*d**3*x**8 + 3*a*b**2*c**3*x**4 + 9 *a*b**2*c**2*d*x**6 + 9*a*b**2*c*d**2*x**8 + 3*a*b**2*d**3*x**10 + b**3*c* *3*x**6 + 3*b**3*c**2*d*x**8 + 3*b**3*c*d**2*x**10 + b**3*d**3*x**12),x)