Integrand size = 22, antiderivative size = 59 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=-a \sqrt {c+\frac {d}{x^2}}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d}+a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right ) \] Output:
-a*(c+d/x^2)^(1/2)-1/3*b*(c+d/x^2)^(3/2)/d+a*c^(1/2)*arctanh((c+d/x^2)^(1/ 2)/c^(1/2))
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=-\frac {\sqrt {c+\frac {d}{x^2}} \left (3 a d x^2+b \left (d+c x^2\right )+\frac {3 a \sqrt {c} d x^3 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )}{\sqrt {d+c x^2}}\right )}{3 d x^2} \] Input:
Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x,x]
Output:
-1/3*(Sqrt[c + d/x^2]*(3*a*d*x^2 + b*(d + c*x^2) + (3*a*Sqrt[c]*d*x^3*Log[ -(Sqrt[c]*x) + Sqrt[d + c*x^2]])/Sqrt[d + c*x^2]))/(d*x^2)
Time = 0.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 90, 60, 73, 221}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2d\frac {1}{x^2}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{2} \left (-a \int \sqrt {c+\frac {d}{x^2}} x^2d\frac {1}{x^2}-\frac {2 b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \left (c \int \frac {x^2}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}+2 \sqrt {c+\frac {d}{x^2}}\right )-\frac {2 b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-a \left (\frac {2 c \int \frac {1}{\frac {1}{d x^4}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{d}+2 \sqrt {c+\frac {d}{x^2}}\right )-\frac {2 b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-a \left (2 \sqrt {c+\frac {d}{x^2}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )\right )-\frac {2 b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d}\right )\) |
Input:
Int[((a + b/x^2)*Sqrt[c + d/x^2])/x,x]
Output:
((-2*b*(c + d/x^2)^(3/2))/(3*d) - a*(2*Sqrt[c + d/x^2] - 2*Sqrt[c]*ArcTanh [Sqrt[c + d/x^2]/Sqrt[c]]))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.42
method | result | size |
risch | \(-\frac {\left (3 a d \,x^{2}+b c \,x^{2}+b d \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{3 x^{2} d}+\frac {a \sqrt {c}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{\sqrt {c \,x^{2}+d}}\) | \(84\) |
default | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (3 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, a \,x^{4}-3 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,x^{2}+3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a c d \,x^{3}-\sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \right )}{3 x^{2} \sqrt {c \,x^{2}+d}\, d \sqrt {c}}\) | \(110\) |
Input:
int((a+b/x^2)*(c+d/x^2)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/3*(3*a*d*x^2+b*c*x^2+b*d)/x^2/d*((c*x^2+d)/x^2)^(1/2)+a*c^(1/2)*ln(c^(1 /2)*x+(c*x^2+d)^(1/2))*((c*x^2+d)/x^2)^(1/2)*x/(c*x^2+d)^(1/2)
Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.81 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=\left [\frac {3 \, a \sqrt {c} d x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left ({\left (b c + 3 \, a d\right )} x^{2} + b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, d x^{2}}, -\frac {3 \, a \sqrt {-c} d x^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left ({\left (b c + 3 \, a d\right )} x^{2} + b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3 \, d x^{2}}\right ] \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x,x, algorithm="fricas")
Output:
[1/6*(3*a*sqrt(c)*d*x^2*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) - 2*((b*c + 3*a*d)*x^2 + b*d)*sqrt((c*x^2 + d)/x^2))/(d*x^2), -1/3*( 3*a*sqrt(-c)*d*x^2*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + ((b*c + 3*a*d)*x^2 + b*d)*sqrt((c*x^2 + d)/x^2))/(d*x^2)]
Time = 9.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=- \frac {a \left (\begin {cases} \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {c + \frac {d}{x^{2}}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {c + \frac {d}{x^{2}}} & \text {for}\: d \neq 0 \\- \sqrt {c} \log {\left (x^{2} \right )} & \text {otherwise} \end {cases}\right )}{2} + \frac {b \left (\begin {cases} - \frac {\sqrt {c}}{x^{2}} & \text {for}\: d = 0 \\- \frac {2 \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right )}{2} \] Input:
integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x,x)
Output:
-a*Piecewise((2*c*atan(sqrt(c + d/x**2)/sqrt(-c))/sqrt(-c) + 2*sqrt(c + d/ x**2), Ne(d, 0)), (-sqrt(c)*log(x**2), True))/2 + b*Piecewise((-sqrt(c)/x* *2, Eq(d, 0)), (-2*(c + d/x**2)**(3/2)/(3*d), True))/2
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=-\frac {1}{2} \, {\left (\sqrt {c} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) + 2 \, \sqrt {c + \frac {d}{x^{2}}}\right )} a - \frac {b {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{3 \, d} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x,x, algorithm="maxima")
Output:
-1/2*(sqrt(c)*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c))) + 2*sqrt(c + d/x^2))*a - 1/3*b*(c + d/x^2)^(3/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (47) = 94\).
Time = 0.36 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.76 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=-\frac {1}{2} \, a \sqrt {c} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a \sqrt {c} d \mathrm {sgn}\left (x\right ) - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a \sqrt {c} d^{2} \mathrm {sgn}\left (x\right ) + b c^{\frac {3}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 3 \, a \sqrt {c} d^{3} \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{3}} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x,x, algorithm="giac")
Output:
-1/2*a*sqrt(c)*log((sqrt(c)*x - sqrt(c*x^2 + d))^2)*sgn(x) + 2/3*(3*(sqrt( c)*x - sqrt(c*x^2 + d))^4*b*c^(3/2)*sgn(x) + 3*(sqrt(c)*x - sqrt(c*x^2 + d ))^4*a*sqrt(c)*d*sgn(x) - 6*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*sqrt(c)*d^2* sgn(x) + b*c^(3/2)*d^2*sgn(x) + 3*a*sqrt(c)*d^3*sgn(x))/((sqrt(c)*x - sqrt (c*x^2 + d))^2 - d)^3
Time = 4.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-a\,\sqrt {c+\frac {d}{x^2}}-\frac {b\,\sqrt {c+\frac {d}{x^2}}\,\left (c\,x^2+d\right )}{3\,d\,x^2} \] Input:
int(((a + b/x^2)*(c + d/x^2)^(1/2))/x,x)
Output:
a*c^(1/2)*atanh((c + d/x^2)^(1/2)/c^(1/2)) - a*(c + d/x^2)^(1/2) - (b*(c + d/x^2)^(1/2)*(d + c*x^2))/(3*d*x^2)
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x} \, dx=\frac {-3 \sqrt {c \,x^{2}+d}\, a d \,x^{2}-\sqrt {c \,x^{2}+d}\, b c \,x^{2}-\sqrt {c \,x^{2}+d}\, b d +3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a d \,x^{3}+\sqrt {c}\, a d \,x^{3}-\sqrt {c}\, b c \,x^{3}}{3 d \,x^{3}} \] Input:
int((a+b/x^2)*(c+d/x^2)^(1/2)/x,x)
Output:
( - 3*sqrt(c*x**2 + d)*a*d*x**2 - sqrt(c*x**2 + d)*b*c*x**2 - sqrt(c*x**2 + d)*b*d + 3*sqrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt(d))*a*d*x**3 + sqrt(c)*a*d*x**3 - sqrt(c)*b*c*x**3)/(3*d*x**3)