Integrand size = 22, antiderivative size = 100 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=-\frac {c^2 (b c-a d)}{d^4 \sqrt {c+\frac {d}{x^2}}}-\frac {c (3 b c-2 a d) \sqrt {c+\frac {d}{x^2}}}{d^4}+\frac {(3 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^4} \]
1/3*(-a*d+3*b*c)*(c+d/x^2)^(3/2)/d^4-1/5*b*(c+d/x^2)^(5/2)/d^4-c^2*(-a*d+b *c)/d^4/(c+d/x^2)^(1/2)-c*(-2*a*d+3*b*c)*(c+d/x^2)^(1/2)/d^4
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.81 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=\frac {-5 a d x^2 \left (d^2-4 c d x^2-8 c^2 x^4\right )-3 b \left (d^3-2 c d^2 x^2+8 c^2 d x^4+16 c^3 x^6\right )}{15 d^4 \sqrt {c+\frac {d}{x^2}} x^6} \]
(-5*a*d*x^2*(d^2 - 4*c*d*x^2 - 8*c^2*x^4) - 3*b*(d^3 - 2*c*d^2*x^2 + 8*c^2 *d*x^4 + 16*c^3*x^6))/(15*d^4*Sqrt[c + d/x^2]*x^6)
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 86, 2009}
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.
\(\displaystyle \int \frac {a+\frac {b}{x^2}}{x^7 \left (c+\frac {d}{x^2}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\frac {1}{2} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^4}d\frac {1}{x^2}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {1}{2} \int \left (-\frac {(b c-a d) c^2}{d^3 \left (c+\frac {d}{x^2}\right )^{3/2}}+\frac {(3 b c-2 a d) c}{d^3 \sqrt {c+\frac {d}{x^2}}}+\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{d^3}+\frac {(a d-3 b c) \sqrt {c+\frac {d}{x^2}}}{d^3}\right )d\frac {1}{x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 c^2 (b c-a d)}{d^4 \sqrt {c+\frac {d}{x^2}}}+\frac {2 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-a d)}{3 d^4}-\frac {2 c \sqrt {c+\frac {d}{x^2}} (3 b c-2 a d)}{d^4}-\frac {2 b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^4}\right )\) |
((-2*c^2*(b*c - a*d))/(d^4*Sqrt[c + d/x^2]) - (2*c*(3*b*c - 2*a*d)*Sqrt[c + d/x^2])/d^4 + (2*(3*b*c - a*d)*(c + d/x^2)^(3/2))/(3*d^4) - (2*b*(c + d/ x^2)^(5/2))/(5*d^4))/2
3.10.78.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 94, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {\left (40 a \,c^{2} d \,x^{6}-48 b \,c^{3} x^{6}+20 a c \,d^{2} x^{4}-24 b \,c^{2} d \,x^{4}-5 a \,d^{3} x^{2}+6 b c \,d^{2} x^{2}-3 b \,d^{3}\right ) \left (c \,x^{2}+d \right )}{15 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{4} x^{8}}\) | \(94\) |
default | \(\frac {\left (40 a \,c^{2} d \,x^{6}-48 b \,c^{3} x^{6}+20 a c \,d^{2} x^{4}-24 b \,c^{2} d \,x^{4}-5 a \,d^{3} x^{2}+6 b c \,d^{2} x^{2}-3 b \,d^{3}\right ) \left (c \,x^{2}+d \right )}{15 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{4} x^{8}}\) | \(94\) |
risch | \(\frac {\left (c \,x^{2}+d \right ) \left (25 a c d \,x^{4}-33 b \,c^{2} x^{4}-5 a \,d^{2} x^{2}+9 b c d \,x^{2}-3 b \,d^{2}\right )}{15 d^{4} x^{6} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}+\frac {\left (a d -b c \right ) c^{2}}{d^{4} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}\) | \(99\) |
trager | \(\frac {\left (40 a \,c^{2} d \,x^{6}-48 b \,c^{3} x^{6}+20 a c \,d^{2} x^{4}-24 b \,c^{2} d \,x^{4}-5 a \,d^{3} x^{2}+6 b c \,d^{2} x^{2}-3 b \,d^{3}\right ) \sqrt {-\frac {-c \,x^{2}-d}{x^{2}}}}{15 x^{4} d^{4} \left (c \,x^{2}+d \right )}\) | \(100\) |
1/15*(40*a*c^2*d*x^6-48*b*c^3*x^6+20*a*c*d^2*x^4-24*b*c^2*d*x^4-5*a*d^3*x^ 2+6*b*c*d^2*x^2-3*b*d^3)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^4/x^8
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=-\frac {{\left (8 \, {\left (6 \, b c^{3} - 5 \, a c^{2} d\right )} x^{6} + 4 \, {\left (6 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{4} + 3 \, b d^{3} - {\left (6 \, b c d^{2} - 5 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, {\left (c d^{4} x^{6} + d^{5} x^{4}\right )}} \]
-1/15*(8*(6*b*c^3 - 5*a*c^2*d)*x^6 + 4*(6*b*c^2*d - 5*a*c*d^2)*x^4 + 3*b*d ^3 - (6*b*c*d^2 - 5*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2)/(c*d^4*x^6 + d^5*x^4 )
Time = 2.65 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.14 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=\begin {cases} \frac {2 \left (- \frac {b \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{10 d^{3}} + \frac {c^{2} \left (a d - b c\right )}{2 d^{3} \sqrt {c + \frac {d}{x^{2}}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} \left (a d - 3 b c\right )}{6 d^{3}} - \frac {\sqrt {c + \frac {d}{x^{2}}} \left (- 2 a c d + 3 b c^{2}\right )}{2 d^{3}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {- \frac {a}{3 x^{6}} - \frac {b}{4 x^{8}}}{2 c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(-b*(c + d/x**2)**(5/2)/(10*d**3) + c**2*(a*d - b*c)/(2*d**3* sqrt(c + d/x**2)) - (c + d/x**2)**(3/2)*(a*d - 3*b*c)/(6*d**3) - sqrt(c + d/x**2)*(-2*a*c*d + 3*b*c**2)/(2*d**3))/d, Ne(d, 0)), ((-a/(3*x**6) - b/(4 *x**8))/(2*c**(3/2)), True))
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.16 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=-\frac {1}{5} \, b {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{4}} - \frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{4}} + \frac {15 \, \sqrt {c + \frac {d}{x^{2}}} c^{2}}{d^{4}} + \frac {5 \, c^{3}}{\sqrt {c + \frac {d}{x^{2}}} d^{4}}\right )} - \frac {1}{3} \, a {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{d^{3}} - \frac {6 \, \sqrt {c + \frac {d}{x^{2}}} c}{d^{3}} - \frac {3 \, c^{2}}{\sqrt {c + \frac {d}{x^{2}}} d^{3}}\right )} \]
-1/5*b*((c + d/x^2)^(5/2)/d^4 - 5*(c + d/x^2)^(3/2)*c/d^4 + 15*sqrt(c + d/ x^2)*c^2/d^4 + 5*c^3/(sqrt(c + d/x^2)*d^4)) - 1/3*a*((c + d/x^2)^(3/2)/d^3 - 6*sqrt(c + d/x^2)*c/d^3 - 3*c^2/(sqrt(c + d/x^2)*d^3))
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (88) = 176\).
Time = 0.70 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.03 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=-\frac {{\left (b c^{3} - a c^{2} d\right )} x}{\sqrt {c x^{2} + d} d^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {5}{2}} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {3}{2}} d - 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {5}{2}} d + 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {3}{2}} d^{2} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {5}{2}} d^{2} - 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {3}{2}} d^{3} - 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {5}{2}} d^{3} + 110 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {3}{2}} d^{4} + 33 \, b c^{\frac {5}{2}} d^{4} - 25 \, a c^{\frac {3}{2}} d^{5}\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{5} d^{3} \mathrm {sgn}\left (x\right )} \]
-(b*c^3 - a*c^2*d)*x/(sqrt(c*x^2 + d)*d^4*sgn(x)) + 2/15*(15*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(5/2) - 15*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(3/2 )*d - 90*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(5/2)*d + 90*(sqrt(c)*x - sqr t(c*x^2 + d))^6*a*c^(3/2)*d^2 + 240*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(5 /2)*d^2 - 160*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(3/2)*d^3 - 150*(sqrt(c) *x - sqrt(c*x^2 + d))^2*b*c^(5/2)*d^3 + 110*(sqrt(c)*x - sqrt(c*x^2 + d))^ 2*a*c^(3/2)*d^4 + 33*b*c^(5/2)*d^4 - 25*a*c^(3/2)*d^5)/(((sqrt(c)*x - sqrt (c*x^2 + d))^2 - d)^5*d^3*sgn(x))
Time = 9.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx=-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (48\,b\,c^3\,x^6-40\,a\,c^2\,d\,x^6+24\,b\,c^2\,d\,x^4-20\,a\,c\,d^2\,x^4-6\,b\,c\,d^2\,x^2+5\,a\,d^3\,x^2+3\,b\,d^3\right )}{15\,d^4\,x^4\,\left (c\,x^2+d\right )} \]