Integrand size = 20, antiderivative size = 59 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {a \sqrt {c+\frac {d}{x^2}} x^2}{2 c}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}} \] Output:
1/2*a*(c+d/x^2)^(1/2)*x^2/c+1/2*(-a*d+2*b*c)*arctanh((c+d/x^2)^(1/2)/c^(1/ 2))/c^(3/2)
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {a \sqrt {c} x \left (d+c x^2\right )+(-2 b c+a d) \sqrt {d+c x^2} \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )}{2 c^{3/2} \sqrt {c+\frac {d}{x^2}} x} \] Input:
Integrate[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]
Output:
(a*Sqrt[c]*x*(d + c*x^2) + (-2*b*c + a*d)*Sqrt[d + c*x^2]*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]])/(2*c^(3/2)*Sqrt[c + d/x^2]*x)
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {948, 87, 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.
\(\displaystyle \int \frac {x \left (a+\frac {b}{x^2}\right )}{\sqrt {c+\frac {d}{x^2}}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\frac {1}{2} \int \frac {\left (a+\frac {b}{x^2}\right ) x^4}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{c}-\frac {(2 b c-a d) \int \frac {x^2}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}}{2 c}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{c}-\frac {(2 b c-a d) \int \frac {1}{\frac {1}{d x^4}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{c d}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{c}\right )\) |
Input:
Int[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]
Output:
((a*Sqrt[c + d/x^2]*x^2)/c + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c ]])/c^(3/2))/2
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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39
method | result | size |
risch | \(\frac {a \left (c \,x^{2}+d \right )}{2 c \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}-\frac {\left (a d -2 c b \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) \sqrt {c \,x^{2}+d}}{2 c^{\frac {3}{2}} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}\) | \(82\) |
default | \(-\frac {\sqrt {c \,x^{2}+d}\, \left (-c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, a x +\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a c d -2 b \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) c^{2}\right )}{2 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \,c^{\frac {5}{2}}}\) | \(90\) |
Input:
int((a+b/x^2)*x/(c+d/x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*a/c*(c*x^2+d)/((c*x^2+d)/x^2)^(1/2)-1/2*(a*d-2*b*c)/c^(3/2)*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 = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\left [\frac {2 \, a c x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - {\left (2 \, b c - a d\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right )}{4 \, c^{2}}, \frac {a c x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - {\left (2 \, b c - a d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{2 \, c^{2}}\right ] \] Input:
integrate((a+b/x^2)*x/(c+d/x^2)^(1/2),x, algorithm="fricas")
Output:
[1/4*(2*a*c*x^2*sqrt((c*x^2 + d)/x^2) - (2*b*c - a*d)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d))/c^2, 1/2*(a*c*x^2*sqrt((c*x^2 + d)/x^2) - (2*b*c - a*d)*sqrt(-c)*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x ^2)/(c*x^2 + d)))/c^2]
Time = 29.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {a \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2 c} - \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 c^{\frac {3}{2}}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{\sqrt {c}} \] Input:
integrate((a+b/x**2)*x/(c+d/x**2)**(1/2),x)
Output:
a*sqrt(d)*x*sqrt(c*x**2/d + 1)/(2*c) - a*d*asinh(sqrt(c)*x/sqrt(d))/(2*c** (3/2)) + b*asinh(sqrt(c)*x/sqrt(d))/sqrt(c)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (47) = 94\).
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} d}{{\left (c + \frac {d}{x^{2}}\right )} c - c^{2}} + \frac {d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {b \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{2 \, \sqrt {c}} \] Input:
integrate((a+b/x^2)*x/(c+d/x^2)^(1/2),x, algorithm="maxima")
Output:
1/4*a*(2*sqrt(c + d/x^2)*d/((c + d/x^2)*c - c^2) + d*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/c^(3/2)) - 1/2*b*log((sqrt(c + d/x ^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/sqrt(c)
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {\sqrt {c x^{2} + d} a x}{2 \, c \mathrm {sgn}\left (x\right )} + \frac {{\left (2 \, b c \log \left ({\left | d \right |}\right ) - a d \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{4 \, c^{\frac {3}{2}}} - \frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate((a+b/x^2)*x/(c+d/x^2)^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(c*x^2 + d)*a*x/(c*sgn(x)) + 1/4*(2*b*c*log(abs(d)) - a*d*log(abs( d)))*sgn(x)/c^(3/2) - 1/2*(2*b*c - a*d)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/(c^(3/2)*sgn(x))
Time = 4.73 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {a\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2\,c}-\frac {a\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,c^{3/2}} \] Input:
int((x*(a + b/x^2))/(c + d/x^2)^(1/2),x)
Output:
(b*atanh((c + d/x^2)^(1/2)/c^(1/2)))/c^(1/2) + (a*x^2*(c + d/x^2)^(1/2))/( 2*c) - (a*d*atanh((c + d/x^2)^(1/2)/c^(1/2)))/(2*c^(3/2))
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\sqrt {c+\frac {d}{x^2}}} \, dx=\frac {\sqrt {c \,x^{2}+d}\, a c x -\sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a d +2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b c}{2 c^{2}} \] Input:
int((a+b/x^2)*x/(c+d/x^2)^(1/2),x)
Output:
(sqrt(c*x**2 + d)*a*c*x - sqrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt( d))*a*d + 2*sqrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt(d))*b*c)/(2*c* *2)