Integrand size = 22, antiderivative size = 278 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {c x \left (d-e x^2\right )}{2 a \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \left (c d^2-a e^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)+1/2*c^(1/4)*e*(1-c*x^ 4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/(-a*e^2+c*d^2)/(-c*x^4+a )^(1/2)+1/2*c^(1/4)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/ 4)/(c^(1/2)*d+a^(1/2)*e)/(-c*x^4+a)^(1/2)-a^(1/4)*e^2*(1-c*x^4/a)^(1/2)*El lipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1/4)/d/(-a*e^2+c*d^2 )/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2 x+\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d e x^3+i \sqrt {a} \sqrt {c} d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {c} d \left (-\sqrt {c} d+\sqrt {a} e\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-2 i a e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d \left (-c d^2+a e^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[1/((d + e*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(-(Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*x) + Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*e*x^3 + I*Sqrt[a]*Sqrt[c]*d*e*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqr t[c]/Sqrt[a])]*x], -1] - I*Sqrt[c]*d*(-(Sqrt[c]*d) + Sqrt[a]*e)*Sqrt[1 - ( c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (2*I)*a*e ^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sq rt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*(-(c*d^2) + a*e^2)*Sqrt[a - c*x^4])
Time = 0.98 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {1550, 25, 27, 1493, 25, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
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.
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1550 |
| \(\displaystyle -\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}-\frac {\int -\frac {c \left (d-e x^2\right )}{\left (a-c x^4\right )^{3/2}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c \left (d-e x^2\right )}{\left (a-c x^4\right )^{3/2}}dx}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {d-e x^2}{\left (a-c x^4\right )^{3/2}}dx}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 1493 |
| \(\displaystyle \frac {c \left (\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}-\frac {\int -\frac {e x^2+d}{\sqrt {a-c x^4}}dx}{2 a}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {e x^2+d}{\sqrt {a-c x^4}}dx}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {c \left (\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {\sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {c \left (\frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {c \left (\frac {\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {c \left (\frac {\frac {e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {c \left (\frac {\frac {\sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {c \left (\frac {\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{c d^2-a e^2}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {c \left (\frac {\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {e^2 \sqrt {1-\frac {c x^4}{a}} \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {c \left (\frac {\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (d-e x^2\right )}{2 a \sqrt {a-c x^4}}\right )}{c d^2-a e^2}-\frac {\sqrt [4]{a} e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4} \left (c d^2-a e^2\right )}\) |
Input:
Int[1/((d + e*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(c*((x*(d - e*x^2))/(2*a*Sqrt[a - c*x^4]) + ((a^(3/4)*e*Sqrt[1 - (c*x^4)/a ]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(d - (Sqrt[a]*e)/Sqrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c ^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]))/(2*a)))/(c*d^2 - a*e^2 ) - (a^(1/4)*e^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)) , ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*(c*d^2 - a*e^2)*Sqrt[a - c* x^4])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x )*(d + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) ) Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && Integer Q[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[(c *d^2 + a*e^2)^(p + 1/2)/e^(2*p + 1) Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[(c*d^2 + a*e^2)^(p + 1/2) Int[(a + c*x^4)^p*ExpandToSum[((c *d^2 + a*e^2)^(-p - 1/2) - e^(-2*p - 1)*(a + c*x^4)^(-p - 1/2))/(d + e*x^2) , x], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
Time = 0.47 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.55
| method | result | size |
| default | \(\frac {2 c \left (\frac {e \,x^{3}}{4 a \left (a \,e^{2}-c \,d^{2}\right )}-\frac {d x}{4 a \left (a \,e^{2}-c \,d^{2}\right )}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(431\) |
| elliptic | \(\frac {2 c \left (\frac {e \,x^{3}}{4 a \left (a \,e^{2}-c \,d^{2}\right )}-\frac {d x}{4 a \left (a \,e^{2}-c \,d^{2}\right )}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(431\) |
Input:
int(1/(e*x^2+d)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/4/a*e/(a*e^2-c*d^2)*x^3-1/4*d/a/(a*e^2-c*d^2)*x)/(-(x^4-1/c*a)*c)^( 1/2)-1/2*c/a*d/(a*e^2-c*d^2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2 )*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x* (1/a^(1/2)*c^(1/2))^(1/2),I)+1/2*c^(1/2)/a^(1/2)*e/(a*e^2-c*d^2)/(1/a^(1/2 )*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2) ^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-1/2*c^(1/ 2)/a^(1/2)*e/(a*e^2-c*d^2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)* x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x*(1 /a^(1/2)*c^(1/2))^(1/2),I)+1/(a*e^2-c*d^2)*e^2/d/(1/a^(1/2)*c^(1/2))^(1/2) *(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a )^(1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-1/a^ (1/2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))
\[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c*x^4 + a)/(c^2*e*x^10 + c^2*d*x^8 - 2*a*c*e*x^6 - 2*a*c*d* x^4 + a^2*e*x^2 + a^2*d), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a - c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )}\, dx \] Input:
integrate(1/(e*x**2+d)/(-c*x**4+a)**(3/2),x)
Output:
Integral(1/((a - c*x**4)**(3/2)*(d + e*x**2)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{3/2}\,\left (e\,x^2+d\right )} \,d x \] Input:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)),x)
Output:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \] Input:
int(1/(e*x^2+d)/(-c*x^4+a)^(3/2),x)
Output:
int(sqrt(a - c*x**4)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)