Integrand size = 22, antiderivative size = 72 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}} \] Output:
a^(1/4)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/ d,I)/c^(1/4)/d/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {i \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d \sqrt {a-c x^4}} \] Input:
Integrate[1/((d + e*x^2)*Sqrt[a - c*x^4]),x]
Output:
((-I)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh [Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*Sqrt[a - c* x^4])
Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt {a-c x^4} \left (d+e x^2\right )} \, dx\) |
\(\Big \downarrow \) 1543 |
\(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}\) |
\(\Big \downarrow \) 1542 |
\(\displaystyle \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}}\) |
Input:
Int[1/((d + e*x^2)*Sqrt[a - c*x^4]),x]
Output:
(a^(1/4)*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*Sqrt[a - c*x^4])
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]
Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(97\) |
elliptic | \(\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(97\) |
Input:
int(1/(e*x^2+d)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/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 ))
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c*x^4 + a)/(c*e*x^6 + c*d*x^4 - a*e*x^2 - a*d), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )}\, dx \] Input:
integrate(1/(e*x**2+d)/(-c*x**4+a)**(1/2),x)
Output:
Integral(1/(sqrt(a - c*x**4)*(d + e*x**2)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,\left (e\,x^2+d\right )} \,d x \] Input:
int(1/((a - c*x^4)^(1/2)*(d + e*x^2)),x)
Output:
int(1/((a - c*x^4)^(1/2)*(d + e*x^2)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \] Input:
int(1/(e*x^2+d)/(-c*x^4+a)^(1/2),x)
Output:
int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)