Integrand size = 20, antiderivative size = 123 \[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}} \] Output:
a^(3/4)*e*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/(-c*x^4 +a)^(1/2)+a^(1/4)*(d-a^(1/2)*e/c^(1/2))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4 )*x/a^(1/4),I)/c^(1/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.63 \[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\frac {\sqrt {1-\frac {c x^4}{a}} \left (3 d x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {a-c x^4}} \] Input:
Integrate[(d + e*x^2)/Sqrt[a - c*x^4],x]
Output:
(Sqrt[1 - (c*x^4)/a]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[a - c*x^4])
Time = 0.51 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1513, 27, 765, 762, 1390, 1389, 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.
| \(\displaystyle \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {\sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\) |
Input:
Int[(d + e*x^2)/Sqrt[a - c*x^4],x]
Output:
(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])
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)/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]
Time = 0.68 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(\frac {d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(154\) |
| elliptic | \(\frac {d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(154\) |
Input:
int((e*x^2+d)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
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)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I )-e*a^(1/2)/(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)/c^(1/2)*(EllipticF(x*(1/a^(1/ 2)*c^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I))
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=-\frac {a \sqrt {-c} e x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (c d + a e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {-c x^{4} + a} a e}{a c x} \] Input:
integrate((e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-(a*sqrt(-c)*e*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - (c*d + a*e)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) + sqrt (-c*x^4 + a)*a*e)/(a*c*x)
Time = 0.85 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67 \[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)/(-c*x**4+a)**(1/2),x)
Output:
d*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sq rt(a)*gamma(5/4)) + e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4*exp _polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4))
\[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)/sqrt(-c*x^4 + a), x)
\[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)/sqrt(-c*x^4 + a), x)
Timed out. \[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\int \frac {e\,x^2+d}{\sqrt {a-c\,x^4}} \,d x \] Input:
int((d + e*x^2)/(a - c*x^4)^(1/2),x)
Output:
int((d + e*x^2)/(a - c*x^4)^(1/2), x)
\[ \int \frac {d+e x^2}{\sqrt {a-c x^4}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) d +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) e \] Input:
int((e*x^2+d)/(-c*x^4+a)^(1/2),x)
Output:
int(sqrt(a - c*x**4)/(a - c*x**4),x)*d + int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*e