Integrand size = 32, antiderivative size = 208 \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {a-c x^4}}{a c x}-\frac {A \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}+\frac {A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} \sqrt {a-c x^4}} \] Output:
-A*(-c*x^4+a)^(1/2)/a/c/x-A*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4), I)/a^(1/4)/c^(3/4)/(-c*x^4+a)^(1/2)+A*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)* x/a^(1/4),I)/a^(1/4)/c^(3/4)/(-c*x^4+a)^(1/2)+a^(1/4)*(-A*d+B*c)*(1-c*x^4/ a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I)/c^(9/4)/(-c*x^ 4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.55 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.53 \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c+A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 x^4+i \sqrt {a} A c^{3/2} x \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {a} A c^{3/2} x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-i a B c x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a A d x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 x \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2)/(x^2*(c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
(-(a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c) + A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*x^4 + I*Sqrt[a]*A*c^(3/2)*x*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[ c]/Sqrt[a])]*x], -1] - I*Sqrt[a]*A*c^(3/2)*x*Sqrt[1 - (c*x^4)/a]*EllipticF [I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*a*B*c*x*Sqrt[1 - (c*x^4)/a ]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x] , -1] + I*a*A*d*x*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I *ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2 *x*Sqrt[a - c*x^4])
Time = 0.48 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2249, 2009}
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.
| \(\displaystyle \int \frac {A+B x^2}{x^2 \sqrt {a-c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {B c-A d}{c \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {A}{c x^2 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} \sqrt {a-c x^4}}+\frac {A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}-\frac {A \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}-\frac {A \sqrt {a-c x^4}}{a c x}\) |
Input:
Int[(A + B*x^2)/(x^2*(c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
-((A*Sqrt[a - c*x^4])/(a*c*x)) - (A*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[( c^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*c^(3/4)*Sqrt[a - c*x^4]) + (A*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*c^(3/4)*Sq rt[a - c*x^4]) + (a^(1/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sq rt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(9/4)*Sqrt[a - c*x ^4])
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & & PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 1.62 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(\frac {A \left (-\frac {\sqrt {-c \,x^{4}+a}}{a x}+\frac {\sqrt {c}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(214\) |
| risch | \(-\frac {A \sqrt {-c \,x^{4}+a}}{a c x}-\frac {-\frac {A \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (A d -B c \right ) a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{a c}\) | \(223\) |
| elliptic | \(-\frac {A \sqrt {-c \,x^{4}+a}}{a c x}+\frac {A \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {A \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) A}{c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) B}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(355\) |
Input:
int((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
A/c*(-1/a*(-c*x^4+a)^(1/2)/x+c^(1/2)/a^(1/2)/(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)*(E llipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2), I)))-(A*d-B*c)/c^2/(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)*d/c^(3/2),(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/ 2))
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c*x^4 + a)*(B*x^2 + A)/(c*d*x^8 + c^2*x^6 - a*d*x^4 - a*c* x^2), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2}}{x^{2} \sqrt {a - c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/x**2/(d*x**2+c)/(-c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2)/(x**2*sqrt(a - c*x**4)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(d*x^2 + c)*x^2), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(d*x^2 + c)*x^2), x)
Timed out. \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {B\,x^2+A}{x^2\,\sqrt {a-c\,x^4}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2)/(x^2*(a - c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2)/(x^2*(a - c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \] Input:
int((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(1/2),x)
Output:
int(sqrt(a - c*x**4)/(a*c*x**2 + a*d*x**4 - c**2*x**6 - c*d*x**8),x)*a + i nt(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*b