Integrand size = 15, antiderivative size = 224 \[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=-\frac {\sqrt {a+c x^4}}{x}+\frac {2 \sqrt {c} x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}-\frac {2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt {a+c x^4}} \] Output:
-(c*x^4+a)^(1/2)/x+2*c^(1/2)*x*(c*x^4+a)^(1/2)/(a^(1/2)+c^(1/2)*x^2)-2*a^( 1/4)*c^(1/4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/ 2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/(c*x^4+a)^(1/2) +a^(1/4)*c^(1/4)*(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*x^4+a)^ (1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.40 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=-\frac {\sqrt {a+c x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\frac {c x^4}{a}\right )}{x \sqrt {1+\frac {c x^4}{a}}} \] Input:
Integrate[Sqrt[a + c*x^4]/x^2,x]
Output:
-((Sqrt[a + c*x^4]*Hypergeometric2F1[-1/2, -1/4, 3/4, -((c*x^4)/a)])/(x*Sq rt[1 + (c*x^4)/a]))
Time = 0.49 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {809, 834, 27, 761, 1510}
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.
\(\displaystyle \int \frac {\sqrt {a+c x^4}}{x^2} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle 2 c \int \frac {x^2}{\sqrt {c x^4+a}}dx-\frac {\sqrt {a+c x^4}}{x}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle 2 c \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{\sqrt {c}}\right )-\frac {\sqrt {a+c x^4}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 c \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )-\frac {\sqrt {a+c x^4}}{x}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle 2 c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {a+c x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )-\frac {\sqrt {a+c x^4}}{x}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle 2 c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {a+c x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}}{\sqrt {c}}\right )-\frac {\sqrt {a+c x^4}}{x}\) |
Input:
Int[Sqrt[a + c*x^4]/x^2,x]
Output:
-(Sqrt[a + c*x^4]/x) + 2*c*(-((-((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]))/Sqrt[c]) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqr t[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c ^(3/4)*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[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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], 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]
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {\sqrt {c \,x^{4}+a}}{x}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(112\) |
risch | \(-\frac {\sqrt {c \,x^{4}+a}}{x}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(112\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+a}}{x}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(112\) |
Input:
int((c*x^4+a)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-(c*x^4+a)^(1/2)/x+2*I*c^(1/2)*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1 /2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*(El lipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/ 2),I))
\[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{x^{2}} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/x^2,x, algorithm="fricas")
Output:
integral(sqrt(c*x^4 + a)/x^2, x)
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=\frac {\sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((c*x**4+a)**(1/2)/x**2,x)
Output:
sqrt(a)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), c*x**4*exp_polar(I*pi)/a)/ (4*x*gamma(3/4))
\[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{x^{2}} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(c*x^4 + a)/x^2, x)
\[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{x^{2}} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/x^2,x, algorithm="giac")
Output:
integrate(sqrt(c*x^4 + a)/x^2, x)
Time = 0.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=\frac {\sqrt {c\,x^4+a}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},-\frac {1}{4};\ \frac {3}{4};\ -\frac {a}{c\,x^4}\right )}{x\,\sqrt {\frac {a}{c\,x^4}+1}} \] Input:
int((a + c*x^4)^(1/2)/x^2,x)
Output:
((a + c*x^4)^(1/2)*hypergeom([-1/2, -1/4], 3/4, -a/(c*x^4)))/(x*(a/(c*x^4) + 1)^(1/2))
\[ \int \frac {\sqrt {a+c x^4}}{x^2} \, dx=\frac {\sqrt {c \,x^{4}+a}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{6}+a \,x^{2}}d x \right ) a x}{x} \] Input:
int((c*x^4+a)^(1/2)/x^2,x)
Output:
(sqrt(a + c*x**4) + 2*int(sqrt(a + c*x**4)/(a*x**2 + c*x**6),x)*a*x)/x