Integrand size = 22, antiderivative size = 93 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}+\frac {(3 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^2 x}-\frac {c (3 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 d^{5/2}} \]
-1/8*c*(-4*a*d+3*b*c)*arctanh(d^(1/2)/x/(c+d/x^2)^(1/2))/d^(5/2)-1/4*b*(c+ d/x^2)^(1/2)/d/x^3+1/8*(-4*a*d+3*b*c)*(c+d/x^2)^(1/2)/d^2/x
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.09 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=\frac {-\sqrt {d} \left (d+c x^2\right ) \left (2 b d-3 b c x^2+4 a d x^2\right )-c (3 b c-4 a d) x^4 \sqrt {d+c x^2} \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {c+\frac {d}{x^2}} x^5} \]
(-(Sqrt[d]*(d + c*x^2)*(2*b*d - 3*b*c*x^2 + 4*a*d*x^2)) - c*(3*b*c - 4*a*d )*x^4*Sqrt[d + c*x^2]*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/(8*d^(5/2)*Sqrt[c + d/x^2]*x^5)
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {959, 858, 262, 224, 219}
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.
\(\displaystyle \int \frac {a+\frac {b}{x^2}}{x^4 \sqrt {c+\frac {d}{x^2}}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle -\frac {(3 b c-4 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}} x^4}dx}{4 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {(3 b c-4 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}} x^2}d\frac {1}{x}}{4 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {(3 b c-4 a d) \left (\frac {\sqrt {c+\frac {d}{x^2}}}{2 d x}-\frac {c \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}}{2 d}\right )}{4 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {(3 b c-4 a d) \left (\frac {\sqrt {c+\frac {d}{x^2}}}{2 d x}-\frac {c \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}}{2 d}\right )}{4 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(3 b c-4 a d) \left (\frac {\sqrt {c+\frac {d}{x^2}}}{2 d x}-\frac {c \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 d^{3/2}}\right )}{4 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}\) |
-1/4*(b*Sqrt[c + d/x^2])/(d*x^3) + ((3*b*c - 4*a*d)*(Sqrt[c + d/x^2]/(2*d* x) - (c*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*d^(3/2))))/(4*d)
3.10.72.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20
method | result | size |
risch | \(-\frac {\left (c \,x^{2}+d \right ) \left (4 a d \,x^{2}-3 c b \,x^{2}+2 b d \right )}{8 d^{2} x^{5} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}+\frac {c \left (4 a d -3 b c \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) \sqrt {c \,x^{2}+d}}{8 d^{\frac {5}{2}} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}\) | \(112\) |
default | \(-\frac {\sqrt {c \,x^{2}+d}\, \left (-4 \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) a c \,d^{2} x^{4}+3 \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b \,c^{2} d \,x^{4}+4 d^{\frac {5}{2}} \sqrt {c \,x^{2}+d}\, a \,x^{2}-3 d^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, b c \,x^{2}+2 d^{\frac {5}{2}} \sqrt {c \,x^{2}+d}\, b \right )}{8 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x^{5} d^{\frac {7}{2}}}\) | \(146\) |
-1/8*(c*x^2+d)*(4*a*d*x^2-3*b*c*x^2+2*b*d)/d^2/x^5/((c*x^2+d)/x^2)^(1/2)+1 /8*c*(4*a*d-3*b*c)/d^(5/2)*ln((2*d+2*d^(1/2)*(c*x^2+d)^(1/2))/x)/((c*x^2+d )/x^2)^(1/2)/x*(c*x^2+d)^(1/2)
Time = 0.41 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.16 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=\left [-\frac {{\left (3 \, b c^{2} - 4 \, a c d\right )} \sqrt {d} x^{3} \log \left (-\frac {c x^{2} + 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (2 \, b d^{2} - {\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, d^{3} x^{3}}, \frac {{\left (3 \, b c^{2} - 4 \, a c d\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (2 \, b d^{2} - {\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, d^{3} x^{3}}\right ] \]
[-1/16*((3*b*c^2 - 4*a*c*d)*sqrt(d)*x^3*log(-(c*x^2 + 2*sqrt(d)*x*sqrt((c* x^2 + d)/x^2) + 2*d)/x^2) + 2*(2*b*d^2 - (3*b*c*d - 4*a*d^2)*x^2)*sqrt((c* x^2 + d)/x^2))/(d^3*x^3), 1/8*((3*b*c^2 - 4*a*c*d)*sqrt(-d)*x^3*arctan(sqr t(-d)*x*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - (2*b*d^2 - (3*b*c*d - 4*a*d^2 )*x^2)*sqrt((c*x^2 + d)/x^2))/(d^3*x^3)]
Time = 4.10 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.61 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=- \frac {a \sqrt {c} \sqrt {1 + \frac {d}{c x^{2}}}}{2 d x} + \frac {a c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 d^{\frac {3}{2}}} + \frac {3 b c^{\frac {3}{2}}}{8 d^{2} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b \sqrt {c}}{8 d x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{8 d^{\frac {5}{2}}} - \frac {b}{4 \sqrt {c} x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} \]
-a*sqrt(c)*sqrt(1 + d/(c*x**2))/(2*d*x) + a*c*asinh(sqrt(d)/(sqrt(c)*x))/( 2*d**(3/2)) + 3*b*c**(3/2)/(8*d**2*x*sqrt(1 + d/(c*x**2))) + b*sqrt(c)/(8* d*x**3*sqrt(1 + d/(c*x**2))) - 3*b*c**2*asinh(sqrt(d)/(sqrt(c)*x))/(8*d**( 5/2)) - b/(4*sqrt(c)*x**5*sqrt(1 + d/(c*x**2)))
Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (77) = 154\).
Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.15 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c x}{{\left (c + \frac {d}{x^{2}}\right )} d x^{2} - d^{2}} + \frac {c \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}}\right )} a + \frac {1}{16} \, b {\left (\frac {3 \, c^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{3} - 5 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} d^{2} x^{4} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} d^{3} x^{2} + d^{4}}\right )} \]
-1/4*(2*sqrt(c + d/x^2)*c*x/((c + d/x^2)*d*x^2 - d^2) + c*log((sqrt(c + d/ x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/d^(3/2))*a + 1/16*b*(3*c^ 2*log((sqrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/d^(5/2) + 2*(3*(c + d/x^2)^(3/2)*c^2*x^3 - 5*sqrt(c + d/x^2)*c^2*d*x)/((c + d/x^2 )^2*d^2*x^4 - 2*(c + d/x^2)*d^3*x^2 + d^4))
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.34 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=\frac {\frac {{\left (3 \, b c^{3} - 4 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d} d^{2}} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{3} - 4 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{2} d - 5 \, \sqrt {c x^{2} + d} b c^{3} d + 4 \, \sqrt {c x^{2} + d} a c^{2} d^{2}}{c^{2} d^{2} x^{4}}}{8 \, c \mathrm {sgn}\left (x\right )} \]
1/8*((3*b*c^3 - 4*a*c^2*d)*arctan(sqrt(c*x^2 + d)/sqrt(-d))/(sqrt(-d)*d^2) + (3*(c*x^2 + d)^(3/2)*b*c^3 - 4*(c*x^2 + d)^(3/2)*a*c^2*d - 5*sqrt(c*x^2 + d)*b*c^3*d + 4*sqrt(c*x^2 + d)*a*c^2*d^2)/(c^2*d^2*x^4))/(c*sgn(x))
Timed out. \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx=\int \frac {a+\frac {b}{x^2}}{x^4\,\sqrt {c+\frac {d}{x^2}}} \,d x \]