Integrand size = 28, antiderivative size = 59 \[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=-\frac {\sqrt {c+\frac {d}{x^2}}}{4 a d x}+\frac {c \arctan \left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{4 a d^{3/2}} \] Output:
-1/4*(c+d/x^2)^(1/2)/a/d/x+1/4*c*arctan(d^(1/2)/(c+d/x^2)^(1/2)/x)/a/d^(3/ 2)
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=-\frac {\frac {\sqrt {c+\frac {d}{x^2}}}{d}+\frac {c \sqrt {c+\frac {d}{x^2}} x^2 \arctan \left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )}{d^{3/2} \sqrt {d+c x^2}}}{4 a x} \] Input:
Integrate[Sqrt[c + d/x^2]/((a*c + (2*a*d)/x^2)*x^4),x]
Output:
-1/4*(Sqrt[c + d/x^2]/d + (c*Sqrt[c + d/x^2]*x^2*ArcTan[Sqrt[d + c*x^2]/Sq rt[d]])/(d^(3/2)*Sqrt[d + c*x^2]))/(a*x)
Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {997, 380, 27, 291, 218}
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.
\(\displaystyle \int \frac {\sqrt {c+\frac {d}{x^2}}}{x^4 \left (a c+\frac {2 a d}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 997 |
\(\displaystyle -\int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 380 |
\(\displaystyle \frac {\int \frac {c^2}{\sqrt {c+\frac {d}{x^2}} \left (c+\frac {2 d}{x^2}\right )}d\frac {1}{x}}{4 a d}-\frac {\sqrt {c+\frac {d}{x^2}}}{4 a d x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c^2 \int \frac {1}{\sqrt {c+\frac {d}{x^2}} \left (c+\frac {2 d}{x^2}\right )}d\frac {1}{x}}{4 a d}-\frac {\sqrt {c+\frac {d}{x^2}}}{4 a d x}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {c^2 \int \frac {1}{\frac {d c}{x^2}+c}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}}{4 a d}-\frac {\sqrt {c+\frac {d}{x^2}}}{4 a d x}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {c \arctan \left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{4 a d^{3/2}}-\frac {\sqrt {c+\frac {d}{x^2}}}{4 a d x}\) |
Input:
Int[Sqrt[c + d/x^2]/((a*c + (2*a*d)/x^2)*x^4),x]
Output:
-1/4*Sqrt[c + d/x^2]/(a*d*x) + (c*ArcTan[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(4* a*d^(3/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* (m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 *q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^(m + 2)), x], x, 1/ x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] && In tegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(175\) vs. \(2(47)=94\).
Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.98
method | result | size |
default | \(-\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (-2 \sqrt {c \,x^{2}+d}\, \sqrt {-d}\, c \,x^{2}-\ln \left (-\frac {2 c \left (\sqrt {-2 c d}\, x +\sqrt {-d}\, \sqrt {c \,x^{2}+d}+d \right )}{-c x +\sqrt {-2 c d}}\right ) c d \,x^{2}-\ln \left (-\frac {2 c \left (\sqrt {-2 c d}\, x -\sqrt {-d}\, \sqrt {c \,x^{2}+d}-d \right )}{c x +\sqrt {-2 c d}}\right ) c d \,x^{2}+2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-d}\right )}{8 x \sqrt {c \,x^{2}+d}\, a \,d^{2} \sqrt {-d}}\) | \(176\) |
risch | \(-\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{4 d x a}+\frac {\left (\frac {c \ln \left (\frac {-2 d +2 \sqrt {-2 c d}\, \left (x -\frac {\sqrt {-2 c d}}{c}\right )+2 \sqrt {-d}\, \sqrt {c \left (x -\frac {\sqrt {-2 c d}}{c}\right )^{2}+2 \sqrt {-2 c d}\, \left (x -\frac {\sqrt {-2 c d}}{c}\right )-d}}{x -\frac {\sqrt {-2 c d}}{c}}\right )}{8 d \sqrt {-d}}+\frac {c \ln \left (\frac {-2 d -2 \sqrt {-2 c d}\, \left (x +\frac {\sqrt {-2 c d}}{c}\right )+2 \sqrt {-d}\, \sqrt {c \left (x +\frac {\sqrt {-2 c d}}{c}\right )^{2}-2 \sqrt {-2 c d}\, \left (x +\frac {\sqrt {-2 c d}}{c}\right )-d}}{x +\frac {\sqrt {-2 c d}}{c}}\right )}{8 d \sqrt {-d}}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{a \sqrt {c \,x^{2}+d}}\) | \(258\) |
Input:
int((c+d/x^2)^(1/2)/(a*c+2*a*d/x^2)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/8*((c*x^2+d)/x^2)^(1/2)/x*(-2*(c*x^2+d)^(1/2)*(-d)^(1/2)*c*x^2-ln(-2*c* ((-2*c*d)^(1/2)*x+(-d)^(1/2)*(c*x^2+d)^(1/2)+d)/(-c*x+(-2*c*d)^(1/2)))*c*d *x^2-ln(-2*c*((-2*c*d)^(1/2)*x-(-d)^(1/2)*(c*x^2+d)^(1/2)-d)/(c*x+(-2*c*d) ^(1/2)))*c*d*x^2+2*(c*x^2+d)^(3/2)*(-d)^(1/2))/(c*x^2+d)^(1/2)/a/d^2/(-d)^ (1/2)
Time = 0.15 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.95 \[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=\left [-\frac {c \sqrt {-d} x \log \left (\frac {c^{2} x^{4} + 4 \, c \sqrt {-d} x^{3} \sqrt {\frac {c x^{2} + d}{x^{2}}} - 4 \, c d x^{2} - 4 \, d^{2}}{c^{2} x^{4} + 4 \, c d x^{2} + 4 \, d^{2}}\right ) + 4 \, d \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, a d^{2} x}, -\frac {c \sqrt {d} x \arctan \left (\frac {c \sqrt {d} x^{3} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, {\left (c d x^{2} + d^{2}\right )}}\right ) + 2 \, d \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, a d^{2} x}\right ] \] Input:
integrate((c+d/x^2)^(1/2)/(a*c+2*a*d/x^2)/x^4,x, algorithm="fricas")
Output:
[-1/16*(c*sqrt(-d)*x*log((c^2*x^4 + 4*c*sqrt(-d)*x^3*sqrt((c*x^2 + d)/x^2) - 4*c*d*x^2 - 4*d^2)/(c^2*x^4 + 4*c*d*x^2 + 4*d^2)) + 4*d*sqrt((c*x^2 + d )/x^2))/(a*d^2*x), -1/8*(c*sqrt(d)*x*arctan(1/2*c*sqrt(d)*x^3*sqrt((c*x^2 + d)/x^2)/(c*d*x^2 + d^2)) + 2*d*sqrt((c*x^2 + d)/x^2))/(a*d^2*x)]
\[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=\frac {\int \frac {\sqrt {c + \frac {d}{x^{2}}}}{c x^{4} + 2 d x^{2}}\, dx}{a} \] Input:
integrate((c+d/x**2)**(1/2)/(a*c+2*a*d/x**2)/x**4,x)
Output:
Integral(sqrt(c + d/x**2)/(c*x**4 + 2*d*x**2), x)/a
\[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=\int { \frac {\sqrt {c + \frac {d}{x^{2}}}}{{\left (a c + \frac {2 \, a d}{x^{2}}\right )} x^{4}} \,d x } \] Input:
integrate((c+d/x^2)^(1/2)/(a*c+2*a*d/x^2)/x^4,x, algorithm="maxima")
Output:
integrate(sqrt(c + d/x^2)/((a*c + 2*a*d/x^2)*x^4), x)
Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=-\frac {c \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {d}}\right ) \mathrm {sgn}\left (x\right )}{4 \, a d^{\frac {3}{2}}} - \frac {\sqrt {c x^{2} + d} \mathrm {sgn}\left (x\right )}{4 \, a d x^{2}} \] Input:
integrate((c+d/x^2)^(1/2)/(a*c+2*a*d/x^2)/x^4,x, algorithm="giac")
Output:
-1/4*c*arctan(sqrt(c*x^2 + d)/sqrt(d))*sgn(x)/(a*d^(3/2)) - 1/4*sqrt(c*x^2 + d)*sgn(x)/(a*d*x^2)
Timed out. \[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=\int \frac {\sqrt {c+\frac {d}{x^2}}}{x^4\,\left (a\,c+\frac {2\,a\,d}{x^2}\right )} \,d x \] Input:
int((c + d/x^2)^(1/2)/(x^4*(a*c + (2*a*d)/x^2)),x)
Output:
int((c + d/x^2)^(1/2)/(x^4*(a*c + (2*a*d)/x^2)), x)
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {c+\frac {d}{x^2}}}{\left (a c+\frac {2 a d}{x^2}\right ) x^4} \, dx=\frac {-2 \sqrt {d}\, \mathit {atan} \left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}\, \sqrt {2}+\sqrt {d}}\right ) c \,x^{2}-2 \sqrt {c \,x^{2}+d}\, d -\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}\, \sqrt {2}\, i +\sqrt {d}\, i}{\sqrt {d}}\right ) c i \,x^{2}+\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}\, \sqrt {2}\, i -\sqrt {d}\, i}{\sqrt {d}}\right ) c i \,x^{2}}{8 a \,d^{2} x^{2}} \] Input:
int((c+d/x^2)^(1/2)/(a*c+2*a*d/x^2)/x^4,x)
Output:
( - 2*sqrt(d)*atan((sqrt(c*x**2 + d) + sqrt(c)*x)/(sqrt(d)*sqrt(2) + sqrt( d)))*c*x**2 - 2*sqrt(c*x**2 + d)*d - sqrt(d)*log((sqrt(c*x**2 + d) + sqrt( c)*x - sqrt(d)*sqrt(2)*i + sqrt(d)*i)/sqrt(d))*c*i*x**2 + sqrt(d)*log((sqr t(c*x**2 + d) + sqrt(c)*x + sqrt(d)*sqrt(2)*i - sqrt(d)*i)/sqrt(d))*c*i*x* *2)/(8*a*d**2*x**2)