Integrand size = 20, antiderivative size = 86 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {2 b c-3 a d}{2 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^2}{2 c \sqrt {c+\frac {d}{x^2}}}+\frac {(2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{5/2}} \]
1/2*(-3*a*d+2*b*c)*arctanh((c+d/x^2)^(1/2)/c^(1/2))/c^(5/2)+1/2*(3*a*d-2*b *c)/c^2/(c+d/x^2)^(1/2)+1/2*a*x^2/c/(c+d/x^2)^(1/2)
Time = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\frac {6 a d \sqrt {d+c x^2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {d}-\sqrt {d+c x^2}}\right )+\sqrt {c} \left (-2 b c x+3 a d x+a c x^3+4 b \sqrt {c} \sqrt {d+c x^2} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {d}+\sqrt {d+c x^2}}\right )\right )}{2 c^{5/2} \sqrt {c+\frac {d}{x^2}} x} \]
(6*a*d*Sqrt[d + c*x^2]*ArcTanh[(Sqrt[c]*x)/(Sqrt[d] - Sqrt[d + c*x^2])] + Sqrt[c]*(-2*b*c*x + 3*a*d*x + a*c*x^3 + 4*b*Sqrt[c]*Sqrt[d + c*x^2]*ArcTan h[(Sqrt[c]*x)/(-Sqrt[d] + Sqrt[d + c*x^2])]))/(2*c^(5/2)*Sqrt[c + d/x^2]*x )
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 87, 61, 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.
\(\displaystyle \int \frac {x \left (a+\frac {b}{x^2}\right )}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\frac {1}{2} \int \frac {\left (a+\frac {b}{x^2}\right ) x^4}{\left (c+\frac {d}{x^2}\right )^{3/2}}d\frac {1}{x^2}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^2}{c \sqrt {c+\frac {d}{x^2}}}-\frac {(2 b c-3 a d) \int \frac {x^2}{\left (c+\frac {d}{x^2}\right )^{3/2}}d\frac {1}{x^2}}{2 c}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^2}{c \sqrt {c+\frac {d}{x^2}}}-\frac {(2 b c-3 a d) \left (\frac {\int \frac {x^2}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}}{c}+\frac {2}{c \sqrt {c+\frac {d}{x^2}}}\right )}{2 c}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^2}{c \sqrt {c+\frac {d}{x^2}}}-\frac {(2 b c-3 a d) \left (\frac {2 \int \frac {1}{\frac {1}{d x^4}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{c d}+\frac {2}{c \sqrt {c+\frac {d}{x^2}}}\right )}{2 c}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^2}{c \sqrt {c+\frac {d}{x^2}}}-\frac {(2 b c-3 a d) \left (\frac {2}{c \sqrt {c+\frac {d}{x^2}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{c^{3/2}}\right )}{2 c}\right )\) |
((a*x^2)/(c*Sqrt[c + d/x^2]) - ((2*b*c - 3*a*d)*(2/(c*Sqrt[c + d/x^2]) - ( 2*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/c^(3/2)))/(2*c))/2
3.10.74.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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*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 = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.34
method | result | size |
default | \(-\frac {\left (c \,x^{2}+d \right ) \left (-c^{\frac {5}{2}} a \,x^{3}-3 c^{\frac {3}{2}} a d x +2 c^{\frac {5}{2}} b x +3 \sqrt {c \,x^{2}+d}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a c d -2 \sqrt {c \,x^{2}+d}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b \,c^{2}\right )}{2 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} c^{\frac {7}{2}}}\) | \(115\) |
risch | \(\frac {a \left (c \,x^{2}+d \right )}{2 c^{2} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}-\frac {\left (\frac {a d x}{\sqrt {c \,x^{2}+d}}+\left (3 a c d -2 b \,c^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+d}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )}{c^{\frac {3}{2}}}\right )\right ) \sqrt {c \,x^{2}+d}}{2 c^{2} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}\) | \(121\) |
-1/2*(c*x^2+d)*(-c^(5/2)*a*x^3-3*c^(3/2)*a*d*x+2*c^(5/2)*b*x+3*(c*x^2+d)^( 1/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*a*c*d-2*(c*x^2+d)^(1/2)*ln(c^(1/2)*x+(c *x^2+d)^(1/2))*b*c^2)/((c*x^2+d)/x^2)^(3/2)/x^3/c^(7/2)
Time = 0.51 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.90 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\left [-\frac {{\left (2 \, b c d - 3 \, a d^{2} + {\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (a c^{2} x^{4} - {\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, {\left (c^{4} x^{2} + c^{3} d\right )}}, -\frac {{\left (2 \, b c d - 3 \, a d^{2} + {\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (a c^{2} x^{4} - {\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, {\left (c^{4} x^{2} + c^{3} d\right )}}\right ] \]
[-1/4*((2*b*c*d - 3*a*d^2 + (2*b*c^2 - 3*a*c*d)*x^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) - 2*(a*c^2*x^4 - (2*b*c^2 - 3*a *c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/(c^4*x^2 + c^3*d), -1/2*((2*b*c*d - 3*a* d^2 + (2*b*c^2 - 3*a*c*d)*x^2)*sqrt(-c)*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - (a*c^2*x^4 - (2*b*c^2 - 3*a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/(c^4*x^2 + c^3*d)]
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (73) = 146\).
Time = 16.94 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.07 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=a \left (\frac {x^{3}}{2 c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 \sqrt {d} x}{2 c^{2} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {3 d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 c^{\frac {5}{2}}}\right ) + b \left (- \frac {2 c^{3} x^{2} \sqrt {1 + \frac {d}{c x^{2}}}}{2 c^{\frac {9}{2}} x^{2} + 2 c^{\frac {7}{2}} d} - \frac {c^{3} x^{2} \log {\left (\frac {d}{c x^{2}} \right )}}{2 c^{\frac {9}{2}} x^{2} + 2 c^{\frac {7}{2}} d} + \frac {2 c^{3} x^{2} \log {\left (\sqrt {1 + \frac {d}{c x^{2}}} + 1 \right )}}{2 c^{\frac {9}{2}} x^{2} + 2 c^{\frac {7}{2}} d} - \frac {c^{2} d \log {\left (\frac {d}{c x^{2}} \right )}}{2 c^{\frac {9}{2}} x^{2} + 2 c^{\frac {7}{2}} d} + \frac {2 c^{2} d \log {\left (\sqrt {1 + \frac {d}{c x^{2}}} + 1 \right )}}{2 c^{\frac {9}{2}} x^{2} + 2 c^{\frac {7}{2}} d}\right ) \]
a*(x**3/(2*c*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*sqrt(d)*x/(2*c**2*sqrt(c*x**2 /d + 1)) - 3*d*asinh(sqrt(c)*x/sqrt(d))/(2*c**(5/2))) + b*(-2*c**3*x**2*sq rt(1 + d/(c*x**2))/(2*c**(9/2)*x**2 + 2*c**(7/2)*d) - c**3*x**2*log(d/(c*x **2))/(2*c**(9/2)*x**2 + 2*c**(7/2)*d) + 2*c**3*x**2*log(sqrt(1 + d/(c*x** 2)) + 1)/(2*c**(9/2)*x**2 + 2*c**(7/2)*d) - c**2*d*log(d/(c*x**2))/(2*c**( 9/2)*x**2 + 2*c**(7/2)*d) + 2*c**2*d*log(sqrt(1 + d/(c*x**2)) + 1)/(2*c**( 9/2)*x**2 + 2*c**(7/2)*d))
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (70) = 140\).
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\frac {1}{4} \, a {\left (\frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )} d - 2 \, c d\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} - \sqrt {c + \frac {d}{x^{2}}} c^{3}} + \frac {3 \, d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} - \frac {1}{2} \, b {\left (\frac {\log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2}{\sqrt {c + \frac {d}{x^{2}}} c}\right )} \]
1/4*a*(2*(3*(c + d/x^2)*d - 2*c*d)/((c + d/x^2)^(3/2)*c^2 - sqrt(c + d/x^2 )*c^3) + 3*d*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/ c^(5/2)) - 1/2*b*(log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt( c)))/c^(3/2) + 2/(sqrt(c + d/x^2)*c))
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\frac {x {\left (\frac {a x^{2}}{c \mathrm {sgn}\left (x\right )} - \frac {2 \, b c^{2} \mathrm {sgn}\left (x\right ) - 3 \, a c d \mathrm {sgn}\left (x\right )}{c^{3}}\right )}}{2 \, \sqrt {c x^{2} + d}} + \frac {{\left (2 \, b c \log \left ({\left | d \right |}\right ) - 3 \, a d \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{4 \, c^{\frac {5}{2}}} - \frac {{\left (2 \, b c - 3 \, a d\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{2 \, c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]
1/2*x*(a*x^2/(c*sgn(x)) - (2*b*c^2*sgn(x) - 3*a*c*d*sgn(x))/c^3)/sqrt(c*x^ 2 + d) + 1/4*(2*b*c*log(abs(d)) - 3*a*d*log(abs(d)))*sgn(x)/c^(5/2) - 1/2* (2*b*c - 3*a*d)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/(c^(5/2)*sgn(x))
Time = 9.82 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {b}{c\,\sqrt {c+\frac {d}{x^2}}}+\frac {a\,x^2}{2\,c\,\sqrt {c+\frac {d}{x^2}}}-\frac {3\,a\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,c^{5/2}}+\frac {3\,a\,d}{2\,c^2\,\sqrt {c+\frac {d}{x^2}}} \]