Integrand size = 36, antiderivative size = 581 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {(c C-B d) x \sqrt {a+c x^4}}{\sqrt {c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {c^3+a d^2} x}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 d^{5/2} \sqrt {c^3+a d^2}}+\frac {\sqrt [4]{a} (c C-B d) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{3/4} d^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} \left (6 c^3 C-6 B c^2 d-2 \sqrt {a} c^{3/2} C d+3 \sqrt {a} B \sqrt {c} d^2+3 A c d^2-a C d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 c^{5/4} d^2 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}+\frac {\left (c^{3/2}+\sqrt {a} d\right ) \left (c^2 C-B c d+A d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (c^{3/2}-\sqrt {a} d\right )^2}{4 \sqrt {a} c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d^3 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}} \] Output:
1/3*C*x*(c*x^4+a)^(1/2)/c/d-(-B*d+C*c)*x*(c*x^4+a)^(1/2)/c^(1/2)/d^2/(a^(1 /2)+c^(1/2)*x^2)-1/2*c^(1/2)*(A*d^2-B*c*d+C*c^2)*arctan((a*d^2+c^3)^(1/2)* x/c^(1/2)/d^(1/2)/(c*x^4+a)^(1/2))/d^(5/2)/(a*d^2+c^3)^(1/2)+a^(1/4)*(-B*d +C*c)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Elli pticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/c^(3/4)/d^2/(c*x^4+a)^ (1/2)-1/6*a^(1/4)*(6*C*c^3-6*B*c^2*d-2*a^(1/2)*c^(3/2)*C*d+3*a^(1/2)*B*c^( 1/2)*d^2+3*A*c*d^2-C*a*d^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1 /2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2)) /c^(5/4)/d^2/(c^(3/2)-a^(1/2)*d)/(c*x^4+a)^(1/2)+1/4*(c^(3/2)+a^(1/2)*d)*( A*d^2-B*c*d+C*c^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^ 2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(3/2)-a^(1/2) *d)^2/a^(1/2)/c^(3/2)/d,1/2*2^(1/2))/a^(1/4)/c^(1/4)/d^3/(c^(3/2)-a^(1/2)* d)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.11 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} C d^2 x+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c C d^2 x^5-3 \sqrt {a} \sqrt {c} d (c C-B d) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (-3 i c^3 C+3 i B c^2 d+3 \sqrt {a} c^{3/2} C d-3 \sqrt {a} B \sqrt {c} d^2-3 i A c d^2+i a C d^2\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i c^3 C \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i B c^2 d \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i A c d^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^3 \sqrt {a+c x^4}} \] Input:
Integrate[(x^2*(A + B*x^2 + C*x^4))/((c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
(a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*C*d^2*x + Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*C*d^2*x ^5 - 3*Sqrt[a]*Sqrt[c]*d*(c*C - B*d)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSi nh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + ((-3*I)*c^3*C + (3*I)*B*c^2*d + 3*S qrt[a]*c^(3/2)*C*d - 3*Sqrt[a]*B*Sqrt[c]*d^2 - (3*I)*A*c*d^2 + I*a*C*d^2)* Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (3*I)*c^3*C*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*A rcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*B*c^2*d*Sqrt[1 + (c*x^4)/ a]*EllipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a] ]*x], -1] + (3*I)*A*c*d^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*d)/ c^(3/2), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(3*Sqrt[(I*Sqrt[c])/ Sqrt[a]]*c*d^3*Sqrt[a + c*x^4])
Time = 1.26 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2237, 25, 2233, 27, 1510, 2227, 27, 761, 2221}
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.
\(\displaystyle \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\sqrt {a+c x^4} \left (c+d x^2\right )} \, dx\) |
\(\Big \downarrow \) 2237 |
\(\displaystyle \frac {\int -\frac {C \left (d x^2+c\right ) \left (3 c x^4+a\right )-3 c d x^2 \left (C x^4+B x^2+A\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 c d}+\frac {C x \sqrt {a+c x^4}}{3 c d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\int \frac {C \left (d x^2+c\right ) \left (3 c x^4+a\right )-3 c d x^2 \left (C x^4+B x^2+A\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 c d}\) |
\(\Big \downarrow \) 2233 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\int \frac {c \left (\sqrt {a} c \left (\sqrt {a} C d+3 \sqrt {c} (c C-B d)\right )-\left ((3 A c-a C) d^2+3 \sqrt {c} \left (c^{3/2}-\sqrt {a} d\right ) (c C-B d)\right ) x^2\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c d}-\frac {3 \sqrt {a} \sqrt {c} (c C-B d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{d}}{3 c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\int \frac {\sqrt {a} c \left (\sqrt {a} C d+3 \sqrt {c} (c C-B d)\right )-\left ((3 A c-a C) d^2+3 \sqrt {c} \left (c^{3/2}-\sqrt {a} d\right ) (c C-B d)\right ) x^2}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{d}-\frac {3 \sqrt {c} (c C-B d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{d}}{3 c d}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\int \frac {\sqrt {a} c \left (\sqrt {a} C d+3 \sqrt {c} (c C-B d)\right )-\left ((3 A c-a C) d^2+3 \sqrt {c} \left (c^{3/2}-\sqrt {a} d\right ) (c C-B d)\right ) x^2}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{d}-\frac {3 \sqrt {c} (c C-B d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{d}}{3 c d}\) |
\(\Big \downarrow \) 2227 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\frac {\sqrt {a} \left (3 \sqrt {a} B \sqrt {c} d^2-2 \sqrt {a} c^{3/2} C d-a C d^2+3 A c d^2-6 B c^2 d+6 c^3 C\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}-\frac {3 \sqrt {a} c^2 \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} (c C-B d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{d}}{3 c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\frac {\sqrt {a} \left (3 \sqrt {a} B \sqrt {c} d^2-2 \sqrt {a} c^{3/2} C d-a C d^2+3 A c d^2-6 B c^2 d+6 c^3 C\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}-\frac {3 c^2 \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} (c C-B d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{d}}{3 c d}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (3 \sqrt {a} B \sqrt {c} d^2-2 \sqrt {a} c^{3/2} C d-a C d^2+3 A c d^2-6 B c^2 d+6 c^3 C\right )}{2 \sqrt [4]{c} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {3 c^2 \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} (c C-B d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{d}}{3 c d}\) |
\(\Big \downarrow \) 2221 |
\(\displaystyle \frac {C x \sqrt {a+c x^4}}{3 c d}-\frac {\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (3 \sqrt {a} B \sqrt {c} d^2-2 \sqrt {a} c^{3/2} C d-a C d^2+3 A c d^2-6 B c^2 d+6 c^3 C\right )}{2 \sqrt [4]{c} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {3 c^2 \left (A d^2-B c d+c^2 C\right ) \left (\frac {\left (\sqrt {a} d+c^{3/2}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {c^{3/2}}{\sqrt {a}}-d\right )^2}{4 c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{5/4} d \sqrt {a+c x^4}}-\frac {\left (c^{3/2}-\sqrt {a} d\right ) \arctan \left (\frac {x \sqrt {a d^2+c^3}}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+c^3}}\right )}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} (c C-B d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{d}}{3 c d}\) |
Input:
Int[(x^2*(A + B*x^2 + C*x^4))/((c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
(C*x*Sqrt[a + c*x^4])/(3*c*d) - ((-3*Sqrt[c]*(c*C - B*d)*(-((x*Sqrt[a + c* x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4) ], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])))/d + ((a^(1/4)*(6*c^3*C - 6*B*c^2*d - 2*Sqrt[a]*c^(3/2)*C*d + 3*Sqrt[a]*B*Sqrt[c]*d^2 + 3*A*c*d^2 - a*C*d^2)*(Sq rt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF [2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(1/4)*(c^(3/2) - Sqrt[a]*d)*Sqr t[a + c*x^4]) - (3*c^2*(c^2*C - B*c*d + A*d^2)*(-1/2*((c^(3/2) - Sqrt[a]*d )*ArcTan[(Sqrt[c^3 + a*d^2]*x)/(Sqrt[c]*Sqrt[d]*Sqrt[a + c*x^4])])/(Sqrt[c ]*Sqrt[d]*Sqrt[c^3 + a*d^2]) + ((c^(3/2) + Sqrt[a]*d)*(Sqrt[a] + Sqrt[c]*x ^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(Sqrt[a]*( c^(3/2)/Sqrt[a] - d)^2)/(c^(3/2)*d), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/ (4*a^(1/4)*c^(5/4)*d*Sqrt[a + c*x^4])))/(c^(3/2) - Sqrt[a]*d))/d)/(3*c*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e ) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x ], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) - a*B*(e + d*q) )/(c*d^2 - a*e^2) Int[1/Sqrt[a + c*x^4], x], x] + Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2)) Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x] , x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff [P4x, x, 4]}, Simp[-C/(e*q) Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim p[1/(c*e) Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x ^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(Px_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> W ith[{q = Expon[Px, x]}, Simp[Coeff[Px, x, q]*x^(q - 5)*(Sqrt[a + c*x^4]/(c* e*(q - 3))), x] + Simp[1/(c*e*(q - 3)) Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]*x^(q - 6)*(d + e*x^2)*(a*(q - 5) + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; GtQ[q, 4]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x]
Result contains complex when optimal does not.
Time = 5.20 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\frac {A \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {C \,c^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i d \left (B d -C c \right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+C \,d^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )-\frac {B c d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{d^{3}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d^{3} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(546\) |
risch | \(\frac {C x \sqrt {c \,x^{4}+a}}{3 c d}+\frac {\frac {\frac {3 i d \left (B d -C c \right ) \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 C \,c^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 A c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {C a \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 d \,c^{2} B \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{d^{2}}-\frac {3 c \left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{3 c d}\) | \(560\) |
elliptic | \(\text {Expression too large to display}\) | \(965\) |
Input:
int(x^2*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/d^3*(A*d^2/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+ I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2 ))^(1/2),I)+C*c^2/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2 )*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a ^(1/2))^(1/2),I)+I*d*(B*d-C*c)*a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1 /2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)/c^( 1/2)*(EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1 /2))^(1/2),I))+C*d^2*(1/3/c*x*(c*x^4+a)^(1/2)-1/3*a/c/(I*c^(1/2)/a^(1/2))^ (1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x ^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I))-B*c*d/(I*c^(1/2)/a^( 1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2 )/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I))-(A*d^2-B*c*d+C *c^2)/d^3/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c ^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I*c^(1/2)/a^(1/2)) ^(1/2),I/c^(3/2)*a^(1/2)*d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I*c^(1/2)/a^(1/2))^ (1/2))
Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Timed out} \] Input:
integrate(x^2*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {x^{2} \left (A + B x^{2} + C x^{4}\right )}{\sqrt {a + c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate(x**2*(C*x**4+B*x**2+A)/(d*x**2+c)/(c*x**4+a)**(1/2),x)
Output:
Integral(x**2*(A + B*x**2 + C*x**4)/(sqrt(a + c*x**4)*(c + d*x**2)), x)
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{2}}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^2*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="maxi ma")
Output:
integrate((C*x^4 + B*x^2 + A)*x^2/(sqrt(c*x^4 + a)*(d*x^2 + c)), x)
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{2}}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^2*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="giac ")
Output:
integrate((C*x^4 + B*x^2 + A)*x^2/(sqrt(c*x^4 + a)*(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {x^2\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {c\,x^4+a}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((x^2*(A + B*x^2 + C*x^4))/((a + c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((x^2*(A + B*x^2 + C*x^4))/((a + c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {\sqrt {c \,x^{4}+a}\, x -\left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c +3 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{4}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b d -3 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{4}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) c^{2}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a d}{3 d} \] Input:
int(x^2*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x)
Output:
(sqrt(a + c*x**4)*x - int(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c *d*x**6),x)*a*c + 3*int((sqrt(a + c*x**4)*x**4)/(a*c + a*d*x**2 + c**2*x** 4 + c*d*x**6),x)*b*d - 3*int((sqrt(a + c*x**4)*x**4)/(a*c + a*d*x**2 + c** 2*x**4 + c*d*x**6),x)*c**2 + 2*int((sqrt(a + c*x**4)*x**2)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*a*d)/(3*d)