\(\int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} (1-c^2 x^2)^5} \, dx\) [165]

Optimal result

Integrand size = 41, antiderivative size = 116 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {5 x}{9 (-1+c x)^{9/2} (1+c x)^{9/2}}-\frac {4 x}{63 (-1+c x)^{7/2} (1+c x)^{7/2}}+\frac {8 x}{105 (-1+c x)^{5/2} (1+c x)^{5/2}}-\frac {32 x}{315 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {64 x}{315 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

5/9*x/(c*x-1)^(9/2)/(c*x+1)^(9/2)-4/63*x/(c*x-1)^(7/2)/(c*x+1)^(7/2)+8/105 
*x/(c*x-1)^(5/2)/(c*x+1)^(5/2)-32/315*x/(c*x-1)^(3/2)/(c*x+1)^(3/2)+64/315 
*x/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.49 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {x \left (315-420 c^2 x^2+504 c^4 x^4-288 c^6 x^6+64 c^8 x^8\right )}{315 (-1+c x)^{9/2} (1+c x)^{9/2}} \] Input:

Integrate[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^5),x 
]
 

Output:

(x*(315 - 420*c^2*x^2 + 504*c^4*x^4 - 288*c^6*x^6 + 64*c^8*x^8))/(315*(-1 
+ c*x)^(9/2)*(1 + c*x)^(9/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2003, 35, 645, 42, 42, 42, 41}

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.

\(\displaystyle \int \frac {4 c^2 x^2+1}{\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^5} \, dx\)

\(\Big \downarrow \) 2003

\(\displaystyle \int \frac {4 c^2 x^2+1}{(-c x-1)^5 (c x-1)^{11/2} \sqrt {c x+1}}dx\)

\(\Big \downarrow \) 35

\(\displaystyle -\int \frac {4 c^2 x^2+1}{(c x-1)^{11/2} (c x+1)^{11/2}}dx\)

\(\Big \downarrow \) 645

\(\displaystyle \frac {4}{9} \int \frac {1}{(c x-1)^{9/2} (c x+1)^{9/2}}dx+\frac {5 x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle \frac {4}{9} \left (-\frac {6}{7} \int \frac {1}{(c x-1)^{7/2} (c x+1)^{7/2}}dx-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\right )+\frac {5 x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle \frac {4}{9} \left (-\frac {6}{7} \left (-\frac {4}{5} \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}\right )-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\right )+\frac {5 x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle \frac {4}{9} \left (-\frac {6}{7} \left (-\frac {4}{5} \left (-\frac {2}{3} \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}\right )-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\right )+\frac {5 x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\)

\(\Big \downarrow \) 41

\(\displaystyle \frac {5 x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}+\frac {4}{9} \left (-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}-\frac {6}{7} \left (-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}-\frac {4}{5} \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )\right )\)

Input:

Int[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^5),x]
 

Output:

(5*x)/(9*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2)) + (4*(-1/7*x/((-1 + c*x)^(7/2)* 
(1 + c*x)^(7/2)) - (6*(-1/5*x/((-1 + c*x)^(5/2)*(1 + c*x)^(5/2)) - (4*(-1/ 
3*x/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (2*x)/(3*Sqrt[-1 + c*x]*Sqrt[1 + 
c*x])))/5))/7))/9
 

Defintions of rubi rules used

rule 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- 
x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 
3)/(2*a*c*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]
 

rule 645

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2), x_Symbol] :> Simp[(b*c*e - a*d*f)*x*(c + d*x)^(m + 1)*((e + f*x)^(n + 
1)/(2*c*d*e*f*(m + 1))), x] - Simp[(b*c*e - a*d*f*(2*m + 3))/(2*c*d*e*f*(m 
+ 1))   Int[(c + d*x)^(m + 1)*(e + f*x)^(n + 1), x], x] /; FreeQ[{a, b, c, 
d, e, f, m, n}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && LtQ[m, -1]
 

rule 2003

Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : 
> Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} 
, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && 
  !IntegerQ[n]))
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.54

method	result	size
gosper	\(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, x \left (64 c^{8} x^{8}-288 c^{6} x^{6}+504 c^{4} x^{4}-420 c^{2} x^{2}+315\right )}{315 \left (c^{2} x^{2}-1\right )^{5}}\)	\(63\)
default	\(\frac {\left (64 c^{8} x^{8}-288 c^{6} x^{6}+504 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) x}{315 \left (c^{2} x^{2}-1\right )^{4} \sqrt {c x +1}\, \sqrt {c x -1}}\)	\(63\)
orering	\(-\frac {x \left (64 c^{8} x^{8}-288 c^{6} x^{6}+504 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{315 \left (-c^{2} x^{2}+1\right )^{5}}\)	\(64\)

Input:

int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^5,x,method=_RET 
URNVERBOSE)
 

Output:

1/315*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*(64*c^8*x^8-288*c^6*x^6+504*c^4*x^4-42 
0*c^2*x^2+315)/(c^2*x^2-1)^5
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.22 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {64 \, c^{10} x^{10} - 320 \, c^{8} x^{8} + 640 \, c^{6} x^{6} - 640 \, c^{4} x^{4} + 320 \, c^{2} x^{2} + {\left (64 \, c^{9} x^{9} - 288 \, c^{7} x^{7} + 504 \, c^{5} x^{5} - 420 \, c^{3} x^{3} + 315 \, c x\right )} \sqrt {c x + 1} \sqrt {c x - 1} - 64}{315 \, {\left (c^{11} x^{10} - 5 \, c^{9} x^{8} + 10 \, c^{7} x^{6} - 10 \, c^{5} x^{4} + 5 \, c^{3} x^{2} - c\right )}} \] Input:

integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^5,x, algo 
rithm="fricas")
 

Output:

1/315*(64*c^10*x^10 - 320*c^8*x^8 + 640*c^6*x^6 - 640*c^4*x^4 + 320*c^2*x^ 
2 + (64*c^9*x^9 - 288*c^7*x^7 + 504*c^5*x^5 - 420*c^3*x^3 + 315*c*x)*sqrt( 
c*x + 1)*sqrt(c*x - 1) - 64)/(c^11*x^10 - 5*c^9*x^8 + 10*c^7*x^6 - 10*c^5* 
x^4 + 5*c^3*x^2 - c)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=- \int \frac {4 c^{2} x^{2}}{c^{10} x^{10} \sqrt {c x - 1} \sqrt {c x + 1} - 5 c^{8} x^{8} \sqrt {c x - 1} \sqrt {c x + 1} + 10 c^{6} x^{6} \sqrt {c x - 1} \sqrt {c x + 1} - 10 c^{4} x^{4} \sqrt {c x - 1} \sqrt {c x + 1} + 5 c^{2} x^{2} \sqrt {c x - 1} \sqrt {c x + 1} - \sqrt {c x - 1} \sqrt {c x + 1}}\, dx - \int \frac {1}{c^{10} x^{10} \sqrt {c x - 1} \sqrt {c x + 1} - 5 c^{8} x^{8} \sqrt {c x - 1} \sqrt {c x + 1} + 10 c^{6} x^{6} \sqrt {c x - 1} \sqrt {c x + 1} - 10 c^{4} x^{4} \sqrt {c x - 1} \sqrt {c x + 1} + 5 c^{2} x^{2} \sqrt {c x - 1} \sqrt {c x + 1} - \sqrt {c x - 1} \sqrt {c x + 1}}\, dx \] Input:

integrate((4*c**2*x**2+1)/(c*x-1)**(1/2)/(c*x+1)**(1/2)/(-c**2*x**2+1)**5, 
x)
 

Output:

-Integral(4*c**2*x**2/(c**10*x**10*sqrt(c*x - 1)*sqrt(c*x + 1) - 5*c**8*x* 
*8*sqrt(c*x - 1)*sqrt(c*x + 1) + 10*c**6*x**6*sqrt(c*x - 1)*sqrt(c*x + 1) 
- 10*c**4*x**4*sqrt(c*x - 1)*sqrt(c*x + 1) + 5*c**2*x**2*sqrt(c*x - 1)*sqr 
t(c*x + 1) - sqrt(c*x - 1)*sqrt(c*x + 1)), x) - Integral(1/(c**10*x**10*sq 
rt(c*x - 1)*sqrt(c*x + 1) - 5*c**8*x**8*sqrt(c*x - 1)*sqrt(c*x + 1) + 10*c 
**6*x**6*sqrt(c*x - 1)*sqrt(c*x + 1) - 10*c**4*x**4*sqrt(c*x - 1)*sqrt(c*x 
 + 1) + 5*c**2*x**2*sqrt(c*x - 1)*sqrt(c*x + 1) - sqrt(c*x - 1)*sqrt(c*x + 
 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {64 \, x}{315 \, \sqrt {c^{2} x^{2} - 1}} - \frac {32 \, x}{315 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {3}{2}}} + \frac {8 \, x}{105 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {5}{2}}} - \frac {4 \, x}{63 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {7}{2}}} + \frac {5 \, x}{9 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {9}{2}}} \] Input:

integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^5,x, algo 
rithm="maxima")
 

Output:

64/315*x/sqrt(c^2*x^2 - 1) - 32/315*x/(c^2*x^2 - 1)^(3/2) + 8/105*x/(c^2*x 
^2 - 1)^(5/2) - 4/63*x/(c^2*x^2 - 1)^(7/2) + 5/9*x/(c^2*x^2 - 1)^(9/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (86) = 172\).

Time = 0.23 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.30 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {{\left ({\left ({\left (c x + 1\right )} {\left ({\left (c x + 1\right )} {\left (\frac {8192 \, {\left (c x + 1\right )}}{c} - \frac {69003}{c}\right )} + \frac {220248}{c}\right )} - \frac {317520}{c}\right )} {\left (c x + 1\right )} + \frac {176400}{c}\right )} \sqrt {c x + 1}}{80640 \, {\left (c x - 1\right )}^{\frac {9}{2}}} + \frac {4725 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{16} + 85050 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{14} + 685860 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{12} + 3273480 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{10} + 9968112 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{8} + 16533216 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{6} + 16152768 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 8832384 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} + 2097152}{20160 \, {\left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} + 2\right )}^{9} c} \] Input:

integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^5,x, algo 
rithm="giac")
 

Output:

1/80640*(((c*x + 1)*((c*x + 1)*(8192*(c*x + 1)/c - 69003/c) + 220248/c) - 
317520/c)*(c*x + 1) + 176400/c)*sqrt(c*x + 1)/(c*x - 1)^(9/2) + 1/20160*(4 
725*(sqrt(c*x + 1) - sqrt(c*x - 1))^16 + 85050*(sqrt(c*x + 1) - sqrt(c*x - 
 1))^14 + 685860*(sqrt(c*x + 1) - sqrt(c*x - 1))^12 + 3273480*(sqrt(c*x + 
1) - sqrt(c*x - 1))^10 + 9968112*(sqrt(c*x + 1) - sqrt(c*x - 1))^8 + 16533 
216*(sqrt(c*x + 1) - sqrt(c*x - 1))^6 + 16152768*(sqrt(c*x + 1) - sqrt(c*x 
 - 1))^4 + 8832384*(sqrt(c*x + 1) - sqrt(c*x - 1))^2 + 2097152)/(((sqrt(c* 
x + 1) - sqrt(c*x - 1))^2 + 2)^9*c)
 

Mupad [B] (verification not implemented)

Time = 22.81 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {315\,x\,\sqrt {c\,x-1}-420\,c^2\,x^3\,\sqrt {c\,x-1}+504\,c^4\,x^5\,\sqrt {c\,x-1}-288\,c^6\,x^7\,\sqrt {c\,x-1}+64\,c^8\,x^9\,\sqrt {c\,x-1}}{{\left (c\,x-1\right )}^5\,\sqrt {c\,x+1}\,\left (\left (\left (c\,x-1\right )\,\left (\left (c\,x-1\right )\,\left (315\,c\,x+2205\right )+7560\right )+10080\right )\,\left (c\,x-1\right )+5040\right )} \] Input:

int(-(4*c^2*x^2 + 1)/((c^2*x^2 - 1)^5*(c*x - 1)^(1/2)*(c*x + 1)^(1/2)),x)
 

Output:

(315*x*(c*x - 1)^(1/2) - 420*c^2*x^3*(c*x - 1)^(1/2) + 504*c^4*x^5*(c*x - 
1)^(1/2) - 288*c^6*x^7*(c*x - 1)^(1/2) + 64*c^8*x^9*(c*x - 1)^(1/2))/((c*x 
 - 1)^5*(c*x + 1)^(1/2)*(((c*x - 1)*((c*x - 1)*(315*c*x + 2205) + 7560) + 
10080)*(c*x - 1) + 5040))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.41 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^5} \, dx=\frac {-64 \sqrt {c x -1}\, c^{9} x^{9}-64 \sqrt {c x -1}\, c^{8} x^{8}+256 \sqrt {c x -1}\, c^{7} x^{7}+256 \sqrt {c x -1}\, c^{6} x^{6}-384 \sqrt {c x -1}\, c^{5} x^{5}-384 \sqrt {c x -1}\, c^{4} x^{4}+256 \sqrt {c x -1}\, c^{3} x^{3}+256 \sqrt {c x -1}\, c^{2} x^{2}-64 \sqrt {c x -1}\, c x -64 \sqrt {c x -1}+64 \sqrt {c x +1}\, c^{9} x^{9}-288 \sqrt {c x +1}\, c^{7} x^{7}+504 \sqrt {c x +1}\, c^{5} x^{5}-420 \sqrt {c x +1}\, c^{3} x^{3}+315 \sqrt {c x +1}\, c x}{315 \sqrt {c x -1}\, c \left (c^{9} x^{9}+c^{8} x^{8}-4 c^{7} x^{7}-4 c^{6} x^{6}+6 c^{5} x^{5}+6 c^{4} x^{4}-4 c^{3} x^{3}-4 c^{2} x^{2}+c x +1\right )} \] Input:

int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^5,x)
 

Output:

( - 64*sqrt(c*x - 1)*c**9*x**9 - 64*sqrt(c*x - 1)*c**8*x**8 + 256*sqrt(c*x 
 - 1)*c**7*x**7 + 256*sqrt(c*x - 1)*c**6*x**6 - 384*sqrt(c*x - 1)*c**5*x** 
5 - 384*sqrt(c*x - 1)*c**4*x**4 + 256*sqrt(c*x - 1)*c**3*x**3 + 256*sqrt(c 
*x - 1)*c**2*x**2 - 64*sqrt(c*x - 1)*c*x - 64*sqrt(c*x - 1) + 64*sqrt(c*x 
+ 1)*c**9*x**9 - 288*sqrt(c*x + 1)*c**7*x**7 + 504*sqrt(c*x + 1)*c**5*x**5 
 - 420*sqrt(c*x + 1)*c**3*x**3 + 315*sqrt(c*x + 1)*c*x)/(315*sqrt(c*x - 1) 
*c*(c**9*x**9 + c**8*x**8 - 4*c**7*x**7 - 4*c**6*x**6 + 6*c**5*x**5 + 6*c* 
*4*x**4 - 4*c**3*x**3 - 4*c**2*x**2 + c*x + 1))