\(\int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx\) [167]

Optimal result

Integrand size = 29, antiderivative size = 52 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\sqrt [4]{\frac {c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac {c}{a}} \sqrt {-a+c x^4}} \]

[Out]

EllipticE((c/a)^(1/4)*x,I)*(1-c*x^4/a)^(1/2)/(c/a)^(1/4)/(c*x^4-a)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1214, 1213, 435} \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\sqrt [4]{\frac {c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac {c}{a}} \sqrt {c x^4-a}} \]

[In]

Int[(1 + Sqrt[c/a]*x^2)/Sqrt[-a + c*x^4],x]

[Out]

(Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c/a)^(1/4)*x], -1])/((c/a)^(1/4)*Sqrt[-a + c*x^4])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 1214

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{\sqrt {-a+c x^4}} \\ & = \frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {1+\sqrt {\frac {c}{a}} x^2}}{\sqrt {1-\sqrt {\frac {c}{a}} x^2}} \, dx}{\sqrt {-a+c x^4}} \\ & = \frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac {c}{a}} x\right )\right |-1\right )}{\sqrt [4]{\frac {c}{a}} \sqrt {-a+c x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.63 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {\sqrt {1-\frac {c x^4}{a}} \left (3 x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+\sqrt {\frac {c}{a}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {-a+c x^4}} \]

[In]

Integrate[(1 + Sqrt[c/a]*x^2)/Sqrt[-a + c*x^4],x]

[Out]

(Sqrt[1 - (c*x^4)/a]*(3*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + Sqrt[c/a]*x^3*Hypergeometric2F1[1/2, 3
/4, 7/4, (c*x^4)/a]))/(3*Sqrt[-a + c*x^4])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (44 ) = 88\).

Time = 1.85 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.17

method	result	size
default	\(\frac {\sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\sqrt {\frac {c}{a}}\, \sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}\, \sqrt {c}}\)	\(165\)
elliptic	\(\frac {\left (1+x^{2} \sqrt {\frac {c}{a}}\right ) a \sqrt {-\frac {\left (-c \,x^{4}+a \right ) c}{a}}\, \left (\frac {\sqrt {c}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {\frac {c^{2} x^{4}}{a}-c}}+\frac {\sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\right )}{c \,x^{2} \sqrt {c \,x^{4}-a}+a \sqrt {-\frac {\left (-c \,x^{4}+a \right ) c}{a}}}\)	\(231\)

[In]

int((1+x^2*(c/a)^(1/2))/(c*x^4-a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/(-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)*E
llipticF(x*(-1/a^(1/2)*c^(1/2))^(1/2),I)+(c/a)^(1/2)*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)-E
llipticE(x*(-1/a^(1/2)*c^(1/2))^(1/2),I))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (43) = 86\).

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.52 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {2 \, a \sqrt {c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} \sqrt {\frac {c}{a}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - \sqrt {2} {\left (\sqrt {2} a x \sqrt {\frac {a}{c}} \sqrt {\frac {c}{a}} + \sqrt {2} c x \sqrt {\frac {a}{c}}\right )} \sqrt {c} \left (\frac {a}{c}\right )^{\frac {1}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, \sqrt {c x^{4} - a} a \sqrt {\frac {c}{a}}}{2 \, a c x} \]

[In]

integrate((1+x^2*(c/a)^(1/2))/(c*x^4-a)^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*a*sqrt(c)*x*(a/c)^(3/4)*sqrt(c/a)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - sqrt(2)*(sqrt(2)*a*x*sqrt(a/c
)*sqrt(c/a) + sqrt(2)*c*x*sqrt(a/c))*sqrt(c)*(a/c)^(1/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) + 2*sqrt(c*x^4
- a)*a*sqrt(c/a))/(a*c*x)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (37) = 74\).

Time = 0.87 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.46 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=- \frac {i x^{3} \sqrt {\frac {c}{a}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} - \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate((1+x**2*(c/a)**(1/2))/(c*x**4-a)**(1/2),x)

[Out]

-I*x**3*sqrt(c/a)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4/a)/(4*sqrt(a)*gamma(7/4)) - I*x*gamma(1/4)*hyper
((1/4, 1/2), (5/4,), c*x**4/a)/(4*sqrt(a)*gamma(5/4))

Maxima [F]

\[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\int { \frac {x^{2} \sqrt {\frac {c}{a}} + 1}{\sqrt {c x^{4} - a}} \,d x } \]

[In]

integrate((1+x^2*(c/a)^(1/2))/(c*x^4-a)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a), x)

Giac [F]

\[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\int { \frac {x^{2} \sqrt {\frac {c}{a}} + 1}{\sqrt {c x^{4} - a}} \,d x } \]

[In]

integrate((1+x^2*(c/a)^(1/2))/(c*x^4-a)^(1/2),x, algorithm="giac")

[Out]

integrate((x^2*sqrt(c/a) + 1)/sqrt(c*x^4 - a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\sqrt {-a+c x^4}} \, dx=\int \frac {x^2\,\sqrt {\frac {c}{a}}+1}{\sqrt {c\,x^4-a}} \,d x \]

[In]

int((x^2*(c/a)^(1/2) + 1)/(c*x^4 - a)^(1/2),x)

[Out]

int((x^2*(c/a)^(1/2) + 1)/(c*x^4 - a)^(1/2), x)