Integrand size = 27, antiderivative size = 163 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}-\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:
-(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x-2*a^(3/2)*b^(1/2)*(1-b^2*x^4/a^2)^(1/2 )*EllipticE(b^(1/2)*x/a^(1/2),I)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)+2*a^(3/2 )*b^(1/2)*(1-b^2*x^4/a^2)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),I)/(-b*x^2+a)^ (1/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.62 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\frac {b^2 x^4}{a^2}\right )}{x \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:
Integrate[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^2,x]
Output:
-((Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, (b^2*x^4)/a^2])/(x*Sqrt[1 - (b^2*x^4)/a^2]))
Time = 0.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {339, 344, 836, 27, 765, 762, 1390, 1389, 327}
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.
\(\displaystyle \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx\) |
\(\Big \downarrow \) 339 |
\(\displaystyle -2 b^2 \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 344 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \int \frac {x^2}{\sqrt {a^2-b^2 x^4}}dx}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {a \int \frac {b x^2+a}{a \sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {1}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 1390 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {b x^2+a}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {2 b^2 \sqrt {a^2-b^2 x^4} \left (\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x}\) |
Input:
Int[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^2,x]
Output:
-((Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x) - (2*b^2*Sqrt[a^2 - b^2*x^4]*((a^(3 /2)*Sqrt[1 - (b^2*x^4)/a^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(b ^(3/2)*Sqrt[a^2 - b^2*x^4]) - (a^(3/2)*Sqrt[1 - (b^2*x^4)/a^2]*EllipticF[A rcSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(b^(3/2)*Sqrt[a^2 - b^2*x^4])))/(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^p*((c + d*x^2)^p/(e*(m + 1))) , x] - Simp[4*b*d*(p/(e^4*(m + 1))) Int[(e*x)^(m + 4)*(a + b*x^2)^(p - 1) *(c + d*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c + a *d, 0] && GtQ[p, 0] && LtQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart[p]) Int[(e*x)^m*(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a , b, c, d, e, m, p}, x] && EqQ[b*c + a*d, 0] && !IntegerQ[p]
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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Time = 0.88 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81
method | result | size |
elliptic | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {\sqrt {-b^{2} x^{4}+a^{2}}}{x}+\frac {2 b a \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(132\) |
risch | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{x}+\frac {2 b a \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{\sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(140\) |
default | \(\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {\frac {b}{a}}\, b^{2} x^{4}+2 b a \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, x \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-2 b a \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, x \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )-\sqrt {\frac {b}{a}}\, a^{2}\right )}{\left (-b^{2} x^{4}+a^{2}\right ) x \sqrt {\frac {b}{a}}}\) | \(160\) |
Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
(-b^2*x^4+a^2)^(1/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)*(-(-b^2*x^4+a^2)^(1/ 2)/x+2*b*a/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+b*x^2/a)^(1/2)/(-b^2*x^4+a^2)^ (1/2)*(EllipticF(x*(b/a)^(1/2),I)-EllipticE(x*(b/a)^(1/2),I)))
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^2,x, algorithm="fricas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/x^2, x)
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=\int \frac {\sqrt {a - b x^{2}} \sqrt {a + b x^{2}}}{x^{2}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2)/x**2,x)
Output:
Integral(sqrt(a - b*x**2)*sqrt(a + b*x**2)/x**2, x)
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/x^2, x)
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^2,x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/x^2, x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {a-b\,x^2}}{x^2} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^2,x)
Output:
int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^2, x)
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2} \, dx=\frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}+2 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} x^{6}+a^{2} x^{2}}d x \right ) a^{2} x}{x} \] Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^2,x)
Output:
(sqrt(a + b*x**2)*sqrt(a - b*x**2) + 2*int((sqrt(a + b*x**2)*sqrt(a - b*x* *2))/(a**2*x**2 - b**2*x**6),x)*a**2*x)/x