\(\int \frac {(a+\frac {b}{x^2}) (c+\frac {d}{x^2})^{3/2}}{x^5} \, dx\) [158]

Optimal result

Integrand size = 22, antiderivative size = 74 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=-\frac {c (b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^3}+\frac {(2 b c-a d) \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^3} \] Output:

-1/5*c*(-a*d+b*c)*(c+d/x^2)^(5/2)/d^3+1/7*(-a*d+2*b*c)*(c+d/x^2)^(7/2)/d^3 
-1/9*b*(c+d/x^2)^(9/2)/d^3
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (d+c x^2\right )^2 \left (9 a d x^2 \left (-5 d+2 c x^2\right )+b \left (-35 d^2+20 c d x^2-8 c^2 x^4\right )\right )}{315 d^3 x^8} \] Input:

Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x]
 

Output:

(Sqrt[c + d/x^2]*(d + c*x^2)^2*(9*a*d*x^2*(-5*d + 2*c*x^2) + b*(-35*d^2 + 
20*c*d*x^2 - 8*c^2*x^4)))/(315*d^3*x^8)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05, 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.

\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle -\frac {1}{2} \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2}d\frac {1}{x^2}\)

\(\Big \downarrow \) 86

\(\displaystyle -\frac {1}{2} \int \left (\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{d^2}+\frac {(a d-2 b c) \left (c+\frac {d}{x^2}\right )^{5/2}}{d^2}+\frac {c (b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{d^2}\right )d\frac {1}{x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {2 \left (c+\frac {d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^3}-\frac {2 c \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^3}-\frac {2 b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^3}\right )\)

Input:

Int[((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x]
 

Output:

((-2*c*(b*c - a*d)*(c + d/x^2)^(5/2))/(5*d^3) + (2*(2*b*c - a*d)*(c + d/x^ 
2)^(7/2))/(7*d^3) - (2*b*(c + d/x^2)^(9/2))/(9*d^3))/2
 

Defintions of rubi rules used

rule 86

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]))))
 

rule 948

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]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.95

method	result	size
gosper	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (18 a c d \,x^{4}-8 b \,c^{2} x^{4}-45 a \,d^{2} x^{2}+20 b c d \,x^{2}-35 b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{315 d^{3} x^{6}}\)	\(70\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (18 a c d \,x^{4}-8 b \,c^{2} x^{4}-45 a \,d^{2} x^{2}+20 b c d \,x^{2}-35 b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{315 d^{3} x^{6}}\)	\(70\)
orering	\(\frac {\left (18 a c d \,x^{4}-8 b \,c^{2} x^{4}-45 a \,d^{2} x^{2}+20 b c d \,x^{2}-35 b \,d^{2}\right ) \left (c \,x^{2}+d \right ) \left (a +\frac {b}{x^{2}}\right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{315 d^{3} \left (a \,x^{2}+b \right ) x^{4}}\)	\(82\)
risch	\(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (18 a \,c^{3} d \,x^{8}-8 b \,c^{4} x^{8}-9 a \,c^{2} d^{2} x^{6}+4 b \,c^{3} d \,x^{6}-72 a c \,d^{3} x^{4}-3 b \,c^{2} d^{2} x^{4}-45 a \,d^{4} x^{2}-50 b c \,d^{3} x^{2}-35 b \,d^{4}\right )}{315 x^{8} d^{3}}\)	\(111\)
trager	\(\frac {\left (18 a \,c^{3} d \,x^{8}-8 b \,c^{4} x^{8}-9 a \,c^{2} d^{2} x^{6}+4 b \,c^{3} d \,x^{6}-72 a c \,d^{3} x^{4}-3 b \,c^{2} d^{2} x^{4}-45 a \,d^{4} x^{2}-50 b c \,d^{3} x^{2}-35 b \,d^{4}\right ) \sqrt {-\frac {-c \,x^{2}-d}{x^{2}}}}{315 x^{8} d^{3}}\)	\(115\)

Input:

int((a+b/x^2)*(c+d/x^2)^(3/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

1/315*((c*x^2+d)/x^2)^(3/2)*(18*a*c*d*x^4-8*b*c^2*x^4-45*a*d^2*x^2+20*b*c* 
d*x^2-35*b*d^2)*(c*x^2+d)/d^3/x^6
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=-\frac {{\left (2 \, {\left (4 \, b c^{4} - 9 \, a c^{3} d\right )} x^{8} - {\left (4 \, b c^{3} d - 9 \, a c^{2} d^{2}\right )} x^{6} + 35 \, b d^{4} + 3 \, {\left (b c^{2} d^{2} + 24 \, a c d^{3}\right )} x^{4} + 5 \, {\left (10 \, b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{315 \, d^{3} x^{8}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)/x^5,x, algorithm="fricas")
 

Output:

-1/315*(2*(4*b*c^4 - 9*a*c^3*d)*x^8 - (4*b*c^3*d - 9*a*c^2*d^2)*x^6 + 35*b 
*d^4 + 3*(b*c^2*d^2 + 24*a*c*d^3)*x^4 + 5*(10*b*c*d^3 + 9*a*d^4)*x^2)*sqrt 
((c*x^2 + d)/x^2)/(d^3*x^8)
 

Sympy [A] (verification not implemented)

Time = 4.70 (sec) , antiderivative size = 258, normalized size of antiderivative = 3.49 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=- \frac {a c \left (\begin {cases} \frac {2 \left (- \frac {c \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{2 x^{4}} & \text {otherwise} \end {cases}\right )}{2} - \frac {a d \left (\begin {cases} \frac {2 \left (\frac {c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{3 x^{6}} & \text {otherwise} \end {cases}\right )}{2} - \frac {b c \left (\begin {cases} \frac {2 \left (\frac {c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{3 x^{6}} & \text {otherwise} \end {cases}\right )}{2} - \frac {b d \left (\begin {cases} \frac {2 \left (- \frac {c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{4 x^{8}} & \text {otherwise} \end {cases}\right )}{2} \] Input:

integrate((a+b/x**2)*(c+d/x**2)**(3/2)/x**5,x)
 

Output:

-a*c*Piecewise((2*(-c*(c + d/x**2)**(3/2)/3 + (c + d/x**2)**(5/2)/5)/d**2, 
 Ne(d, 0)), (sqrt(c)/(2*x**4), True))/2 - a*d*Piecewise((2*(c**2*(c + d/x* 
*2)**(3/2)/3 - 2*c*(c + d/x**2)**(5/2)/5 + (c + d/x**2)**(7/2)/7)/d**3, Ne 
(d, 0)), (sqrt(c)/(3*x**6), True))/2 - b*c*Piecewise((2*(c**2*(c + d/x**2) 
**(3/2)/3 - 2*c*(c + d/x**2)**(5/2)/5 + (c + d/x**2)**(7/2)/7)/d**3, Ne(d, 
 0)), (sqrt(c)/(3*x**6), True))/2 - b*d*Piecewise((2*(-c**3*(c + d/x**2)** 
(3/2)/3 + 3*c**2*(c + d/x**2)**(5/2)/5 - 3*c*(c + d/x**2)**(7/2)/7 + (c + 
d/x**2)**(9/2)/9)/d**4, Ne(d, 0)), (sqrt(c)/(4*x**8), True))/2
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{2}} - \frac {7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{2}}\right )} a - \frac {1}{315} \, {\left (\frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}}}{d^{3}} - \frac {90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c}{d^{3}} + \frac {63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{2}}{d^{3}}\right )} b \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)/x^5,x, algorithm="maxima")
 

Output:

-1/35*(5*(c + d/x^2)^(7/2)/d^2 - 7*(c + d/x^2)^(5/2)*c/d^2)*a - 1/315*(35* 
(c + d/x^2)^(9/2)/d^3 - 90*(c + d/x^2)^(7/2)*c/d^3 + 63*(c + d/x^2)^(5/2)* 
c^2/d^3)*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (62) = 124\).

Time = 1.75 (sec) , antiderivative size = 430, normalized size of antiderivative = 5.81 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=\frac {4 \, {\left (315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {7}{2}} d \mathrm {sgn}\left (x\right ) + 1260 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {9}{2}} d \mathrm {sgn}\left (x\right ) + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {7}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 1764 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {9}{2}} d^{2} \mathrm {sgn}\left (x\right ) - 819 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {7}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {9}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 441 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {7}{2}} d^{4} \mathrm {sgn}\left (x\right ) + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {9}{2}} d^{4} \mathrm {sgn}\left (x\right ) - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {7}{2}} d^{5} \mathrm {sgn}\left (x\right ) - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {9}{2}} d^{5} \mathrm {sgn}\left (x\right ) + 81 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {7}{2}} d^{6} \mathrm {sgn}\left (x\right ) + 4 \, b c^{\frac {9}{2}} d^{6} \mathrm {sgn}\left (x\right ) - 9 \, a c^{\frac {7}{2}} d^{7} \mathrm {sgn}\left (x\right )\right )}}{315 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{9}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)/x^5,x, algorithm="giac")
 

Output:

4/315*(315*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(7/2)*sgn(x) + 840*(sqrt(c 
)*x - sqrt(c*x^2 + d))^12*b*c^(9/2)*sgn(x) - 315*(sqrt(c)*x - sqrt(c*x^2 + 
 d))^12*a*c^(7/2)*d*sgn(x) + 1260*(sqrt(c)*x - sqrt(c*x^2 + d))^10*b*c^(9/ 
2)*d*sgn(x) + 315*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(7/2)*d^2*sgn(x) + 
1764*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(9/2)*d^2*sgn(x) - 819*(sqrt(c)*x 
 - sqrt(c*x^2 + d))^8*a*c^(7/2)*d^3*sgn(x) + 504*(sqrt(c)*x - sqrt(c*x^2 + 
 d))^6*b*c^(9/2)*d^3*sgn(x) + 441*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(7/2 
)*d^4*sgn(x) + 144*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(9/2)*d^4*sgn(x) - 
9*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(7/2)*d^5*sgn(x) - 36*(sqrt(c)*x - s 
qrt(c*x^2 + d))^2*b*c^(9/2)*d^5*sgn(x) + 81*(sqrt(c)*x - sqrt(c*x^2 + d))^ 
2*a*c^(7/2)*d^6*sgn(x) + 4*b*c^(9/2)*d^6*sgn(x) - 9*a*c^(7/2)*d^7*sgn(x))/ 
((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^9
 

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=\frac {2\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{35\,d^2}-\frac {8\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3}-\frac {8\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{35\,x^4}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{7\,x^6}-\frac {10\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,x^6}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{35\,d\,x^2}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d\,x^4}+\frac {4\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^2\,x^2} \] Input:

int(((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x)
 

Output:

(2*a*c^3*(c + d/x^2)^(1/2))/(35*d^2) - (8*b*c^4*(c + d/x^2)^(1/2))/(315*d^ 
3) - (8*a*c*(c + d/x^2)^(1/2))/(35*x^4) - (a*d*(c + d/x^2)^(1/2))/(7*x^6) 
- (10*b*c*(c + d/x^2)^(1/2))/(63*x^6) - (b*d*(c + d/x^2)^(1/2))/(9*x^8) - 
(a*c^2*(c + d/x^2)^(1/2))/(35*d*x^2) - (b*c^2*(c + d/x^2)^(1/2))/(105*d*x^ 
4) + (4*b*c^3*(c + d/x^2)^(1/2))/(315*d^2*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx=\frac {18 \sqrt {c \,x^{2}+d}\, a \,c^{3} d \,x^{8}-9 \sqrt {c \,x^{2}+d}\, a \,c^{2} d^{2} x^{6}-72 \sqrt {c \,x^{2}+d}\, a c \,d^{3} x^{4}-45 \sqrt {c \,x^{2}+d}\, a \,d^{4} x^{2}-8 \sqrt {c \,x^{2}+d}\, b \,c^{4} x^{8}+4 \sqrt {c \,x^{2}+d}\, b \,c^{3} d \,x^{6}-3 \sqrt {c \,x^{2}+d}\, b \,c^{2} d^{2} x^{4}-50 \sqrt {c \,x^{2}+d}\, b c \,d^{3} x^{2}-35 \sqrt {c \,x^{2}+d}\, b \,d^{4}-18 \sqrt {c}\, a \,c^{3} d \,x^{9}+8 \sqrt {c}\, b \,c^{4} x^{9}}{315 d^{3} x^{9}} \] Input:

int((a+b/x^2)*(c+d/x^2)^(3/2)/x^5,x)
 

Output:

(18*sqrt(c*x**2 + d)*a*c**3*d*x**8 - 9*sqrt(c*x**2 + d)*a*c**2*d**2*x**6 - 
 72*sqrt(c*x**2 + d)*a*c*d**3*x**4 - 45*sqrt(c*x**2 + d)*a*d**4*x**2 - 8*s 
qrt(c*x**2 + d)*b*c**4*x**8 + 4*sqrt(c*x**2 + d)*b*c**3*d*x**6 - 3*sqrt(c* 
x**2 + d)*b*c**2*d**2*x**4 - 50*sqrt(c*x**2 + d)*b*c*d**3*x**2 - 35*sqrt(c 
*x**2 + d)*b*d**4 - 18*sqrt(c)*a*c**3*d*x**9 + 8*sqrt(c)*b*c**4*x**9)/(315 
*d**3*x**9)