Integrand size = 38, antiderivative size = 179 \[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {\sqrt [4]{a} \sqrt {\frac {c}{a}} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} e \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (\sqrt {\frac {c}{a}} d+e\right ) \sqrt {1-\sqrt {\frac {c}{a}} x^2} \sqrt {1+\sqrt {\frac {c}{a}} x^2} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e \sqrt {a-c x^4}} \] Output:
-a^(1/4)*(c/a)^(1/2)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(1 /4)/e/(-c*x^4+a)^(1/2)+a^(1/4)*((c/a)^(1/2)*d+e)*(1-(c/a)^(1/2)*x^2)^(1/2) *(1+(c/a)^(1/2)*x^2)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2) /d,I)/c^(1/4)/d/e/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.37 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.82 \[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {i \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {\frac {c}{a}} d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-\left (\sqrt {\frac {c}{a}} d+e\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d e \sqrt {a-c x^4}} \] Input:
Integrate[(1 - Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a - c*x^4]),x]
Output:
(I*Sqrt[1 - (c*x^4)/a]*(Sqrt[c/a]*d*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqr t[a])]*x], -1] - (Sqrt[c/a]*d + e)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e *Sqrt[a - c*x^4])
Time = 0.49 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1787, 415, 289, 413, 412, 765, 762}
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.
| \(\displaystyle \int \frac {1-x^2 \sqrt {\frac {c}{a}}}{\sqrt {a-c x^4} \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1787 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \int \frac {\sqrt {1-\sqrt {\frac {c}{a}} x^2}}{\sqrt {a \sqrt {\frac {c}{a}} x^2+a} \left (e x^2+d\right )}dx}{\sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 415 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \left (\frac {\left (d \sqrt {\frac {c}{a}}+e\right ) \int \frac {1}{\sqrt {1-\sqrt {\frac {c}{a}} x^2} \sqrt {a \sqrt {\frac {c}{a}} x^2+a} \left (e x^2+d\right )}dx}{e}-\frac {\sqrt {\frac {c}{a}} \int \frac {1}{\sqrt {1-\sqrt {\frac {c}{a}} x^2} \sqrt {a \sqrt {\frac {c}{a}} x^2+a}}dx}{e}\right )}{\sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \left (\frac {\left (d \sqrt {\frac {c}{a}}+e\right ) \int \frac {1}{\sqrt {1-\sqrt {\frac {c}{a}} x^2} \sqrt {a \sqrt {\frac {c}{a}} x^2+a} \left (e x^2+d\right )}dx}{e}-\frac {\sqrt {\frac {c}{a}} \sqrt {a-c x^4} \int \frac {1}{\sqrt {a-c x^4}}dx}{e \sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}\right )}{\sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \left (\frac {\sqrt {x^2 \sqrt {\frac {c}{a}}+1} \left (d \sqrt {\frac {c}{a}}+e\right ) \int \frac {1}{\sqrt {1-\sqrt {\frac {c}{a}} x^2} \sqrt {\sqrt {\frac {c}{a}} x^2+1} \left (e x^2+d\right )}dx}{e \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}-\frac {\sqrt {\frac {c}{a}} \sqrt {a-c x^4} \int \frac {1}{\sqrt {a-c x^4}}dx}{e \sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}\right )}{\sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \left (\frac {\sqrt {x^2 \sqrt {\frac {c}{a}}+1} \left (d \sqrt {\frac {c}{a}}+e\right ) \operatorname {EllipticPi}\left (-\frac {e}{\sqrt {\frac {c}{a}} d},\arcsin \left (\sqrt [4]{\frac {c}{a}} x\right ),-1\right )}{d e \sqrt [4]{\frac {c}{a}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}-\frac {\sqrt {\frac {c}{a}} \sqrt {a-c x^4} \int \frac {1}{\sqrt {a-c x^4}}dx}{e \sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}\right )}{\sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \left (\frac {\sqrt {x^2 \sqrt {\frac {c}{a}}+1} \left (d \sqrt {\frac {c}{a}}+e\right ) \operatorname {EllipticPi}\left (-\frac {e}{\sqrt {\frac {c}{a}} d},\arcsin \left (\sqrt [4]{\frac {c}{a}} x\right ),-1\right )}{d e \sqrt [4]{\frac {c}{a}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}-\frac {\sqrt {\frac {c}{a}} \sqrt {1-\frac {c x^4}{a}} \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{e \sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}\right )}{\sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a} \left (\frac {\sqrt {x^2 \sqrt {\frac {c}{a}}+1} \left (d \sqrt {\frac {c}{a}}+e\right ) \operatorname {EllipticPi}\left (-\frac {e}{\sqrt {\frac {c}{a}} d},\arcsin \left (\sqrt [4]{\frac {c}{a}} x\right ),-1\right )}{d e \sqrt [4]{\frac {c}{a}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}-\frac {\sqrt [4]{a} \sqrt {\frac {c}{a}} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} e \sqrt {1-x^2 \sqrt {\frac {c}{a}}} \sqrt {a x^2 \sqrt {\frac {c}{a}}+a}}\right )}{\sqrt {a-c x^4}}\) |
Input:
Int[(1 - Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a - c*x^4]),x]
Output:
(Sqrt[1 - Sqrt[c/a]*x^2]*Sqrt[a + a*Sqrt[c/a]*x^2]*(-((a^(1/4)*Sqrt[c/a]*S qrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*e* Sqrt[1 - Sqrt[c/a]*x^2]*Sqrt[a + a*Sqrt[c/a]*x^2])) + ((Sqrt[c/a]*d + e)*S qrt[1 + Sqrt[c/a]*x^2]*EllipticPi[-(e/(Sqrt[c/a]*d)), ArcSin[(c/a)^(1/4)*x ], -1])/((c/a)^(1/4)*d*e*Sqrt[a + a*Sqrt[c/a]*x^2])))/Sqrt[a - c*x^4]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart [p]) Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* c + a*d, 0] && !IntegerQ[p]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[d/b Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[(b*c - a*d)/b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2 ]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]
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_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]) Int[(d + e*x^n)^(p + q) *(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p , q, r}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
Time = 0.92 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c}{a}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (\sqrt {\frac {c}{a}}\, d +e \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(186\) |
| elliptic | \(\frac {\left (1-\sqrt {\frac {c}{a}}\, x^{2}\right ) \sqrt {\frac {\left (-c \,x^{4}+a \right ) c}{a}}\, a \left (-\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{a e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-\frac {c^{2} x^{4}}{a}+c}}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{a e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-\frac {c^{2} x^{4}}{a}+c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{-c \,x^{2} \sqrt {-c \,x^{4}+a}+a \sqrt {\frac {\left (-c \,x^{4}+a \right ) c}{a}}}\) | \(346\) |
Input:
int((1-(c/a)^(1/2)*x^2)/(e*x^2+d)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE )
Output:
-(c/a)^(1/2)/e/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^ (1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1 /2),I)+((c/a)^(1/2)*d+e)/e/d/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2 ))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1 /2)/a^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/ a^(1/2))^(1/2))
\[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { -\frac {x^{2} \sqrt {\frac {c}{a}} - 1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((1-(c/a)^(1/2)*x^2)/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="fri cas")
Output:
integral((sqrt(-c*x^4 + a)*x^2*sqrt(c/a) - sqrt(-c*x^4 + a))/(c*e*x^6 + c* d*x^4 - a*e*x^2 - a*d), x)
\[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=- \int \frac {x^{2} \sqrt {\frac {c}{a}}}{d \sqrt {a - c x^{4}} + e x^{2} \sqrt {a - c x^{4}}}\, dx - \int \left (- \frac {1}{d \sqrt {a - c x^{4}} + e x^{2} \sqrt {a - c x^{4}}}\right )\, dx \] Input:
integrate((1-(c/a)**(1/2)*x**2)/(e*x**2+d)/(-c*x**4+a)**(1/2),x)
Output:
-Integral(x**2*sqrt(c/a)/(d*sqrt(a - c*x**4) + e*x**2*sqrt(a - c*x**4)), x ) - Integral(-1/(d*sqrt(a - c*x**4) + e*x**2*sqrt(a - c*x**4)), x)
\[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { -\frac {x^{2} \sqrt {\frac {c}{a}} - 1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((1-(c/a)^(1/2)*x^2)/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="max ima")
Output:
-integrate((x^2*sqrt(c/a) - 1)/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
\[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { -\frac {x^{2} \sqrt {\frac {c}{a}} - 1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((1-(c/a)^(1/2)*x^2)/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="gia c")
Output:
integrate(-(x^2*sqrt(c/a) - 1)/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=-\int \frac {x^2\,\sqrt {\frac {c}{a}}-1}{\sqrt {a-c\,x^4}\,\left (e\,x^2+d\right )} \,d x \] Input:
int(-(x^2*(c/a)^(1/2) - 1)/((a - c*x^4)^(1/2)*(d + e*x^2)),x)
Output:
-int((x^2*(c/a)^(1/2) - 1)/((a - c*x^4)^(1/2)*(d + e*x^2)), x)
\[ \int \frac {1-\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right )+\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a}{a} \] Input:
int((1-(c/a)^(1/2)*x^2)/(e*x^2+d)/(-c*x^4+a)^(1/2),x)
Output:
( - sqrt(c)*sqrt(a)*int((sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x) + int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x* *6),x)*a)/a