\(\int \frac {1}{x^2 (b x^2+c x^4)^3} \, dx\) [92]

Optimal result

Integrand size = 17, antiderivative size = 111 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=-\frac {1}{7 b^3 x^7}+\frac {3 c}{5 b^4 x^5}-\frac {2 c^2}{b^5 x^3}+\frac {10 c^3}{b^6 x}+\frac {c^4 x}{4 b^5 \left (b+c x^2\right )^2}+\frac {19 c^4 x}{8 b^6 \left (b+c x^2\right )}+\frac {99 c^{7/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{13/2}} \] Output:

-1/7/b^3/x^7+3/5*c/b^4/x^5-2*c^2/b^5/x^3+10*c^3/b^6/x+1/4*c^4*x/b^5/(c*x^2 
+b)^2+19/8*c^4*x/b^6/(c*x^2+b)+99/8*c^(7/2)*arctan(c^(1/2)*x/b^(1/2))/b^(1 
3/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=\frac {-40 b^5+88 b^4 c x^2-264 b^3 c^2 x^4+1848 b^2 c^3 x^6+5775 b c^4 x^8+3465 c^5 x^{10}}{280 b^6 x^7 \left (b+c x^2\right )^2}+\frac {99 c^{7/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{13/2}} \] Input:

Integrate[1/(x^2*(b*x^2 + c*x^4)^3),x]
 

Output:

(-40*b^5 + 88*b^4*c*x^2 - 264*b^3*c^2*x^4 + 1848*b^2*c^3*x^6 + 5775*b*c^4* 
x^8 + 3465*c^5*x^10)/(280*b^6*x^7*(b + c*x^2)^2) + (99*c^(7/2)*ArcTan[(Sqr 
t[c]*x)/Sqrt[b]])/(8*b^(13/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {9, 253, 253, 264, 264, 264, 264, 218}

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.

\(\displaystyle \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int \frac {1}{x^8 \left (b+c x^2\right )^3}dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \int \frac {1}{x^8 \left (c x^2+b\right )^2}dx}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \int \frac {1}{x^8 \left (c x^2+b\right )}dx}{2 b}+\frac {1}{2 b x^7 \left (b+c x^2\right )}\right )}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (-\frac {c \int \frac {1}{x^6 \left (c x^2+b\right )}dx}{b}-\frac {1}{7 b x^7}\right )}{2 b}+\frac {1}{2 b x^7 \left (b+c x^2\right )}\right )}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (-\frac {c \left (-\frac {c \int \frac {1}{x^4 \left (c x^2+b\right )}dx}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {1}{7 b x^7}\right )}{2 b}+\frac {1}{2 b x^7 \left (b+c x^2\right )}\right )}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (-\frac {c \left (-\frac {c \left (-\frac {c \int \frac {1}{x^2 \left (c x^2+b\right )}dx}{b}-\frac {1}{3 b x^3}\right )}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {1}{7 b x^7}\right )}{2 b}+\frac {1}{2 b x^7 \left (b+c x^2\right )}\right )}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (-\frac {c \left (-\frac {c \left (-\frac {c \left (-\frac {c \int \frac {1}{c x^2+b}dx}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{3 b x^3}\right )}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {1}{7 b x^7}\right )}{2 b}+\frac {1}{2 b x^7 \left (b+c x^2\right )}\right )}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {11 \left (\frac {9 \left (-\frac {c \left (-\frac {c \left (-\frac {c \left (-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x}\right )}{b}-\frac {1}{3 b x^3}\right )}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {1}{7 b x^7}\right )}{2 b}+\frac {1}{2 b x^7 \left (b+c x^2\right )}\right )}{4 b}+\frac {1}{4 b x^7 \left (b+c x^2\right )^2}\)

Input:

Int[1/(x^2*(b*x^2 + c*x^4)^3),x]
 

Output:

1/(4*b*x^7*(b + c*x^2)^2) + (11*(1/(2*b*x^7*(b + c*x^2)) + (9*(-1/7*1/(b*x 
^7) - (c*(-1/5*1/(b*x^5) - (c*(-1/3*1/(b*x^3) - (c*(-(1/(b*x)) - (Sqrt[c]* 
ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)))/b))/b))/b))/(2*b)))/(4*b)
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {1}{7 b^{3} x^{7}}+\frac {10 c^{3}}{b^{6} x}-\frac {2 c^{2}}{b^{5} x^{3}}+\frac {3 c}{5 b^{4} x^{5}}+\frac {c^{4} \left (\frac {\frac {19}{8} c \,x^{3}+\frac {21}{8} b x}{\left (c \,x^{2}+b \right )^{2}}+\frac {99 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{b^{6}}\)	\(86\)
risch	\(\frac {\frac {99 c^{5} x^{10}}{8 b^{6}}+\frac {165 c^{4} x^{8}}{8 b^{5}}+\frac {33 c^{3} x^{6}}{5 b^{4}}-\frac {33 c^{2} x^{4}}{35 b^{3}}+\frac {11 c \,x^{2}}{35 b^{2}}-\frac {1}{7 b}}{x^{7} \left (c \,x^{2}+b \right )^{2}}+\frac {99 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{13} \textit {\_Z}^{2}+c^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{13}+2 c^{7}\right ) x -b^{7} c^{3} \textit {\_R} \right )\right )}{16}\)	\(120\)

Input:

int(1/x^2/(c*x^4+b*x^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/7/b^3/x^7+10*c^3/b^6/x-2*c^2/b^5/x^3+3/5*c/b^4/x^5+1/b^6*c^4*((19/8*c*x 
^3+21/8*b*x)/(c*x^2+b)^2+99/8/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.58 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=\left [\frac {6930 \, c^{5} x^{10} + 11550 \, b c^{4} x^{8} + 3696 \, b^{2} c^{3} x^{6} - 528 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} - 80 \, b^{5} + 3465 \, {\left (c^{5} x^{11} + 2 \, b c^{4} x^{9} + b^{2} c^{3} x^{7}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right )}{560 \, {\left (b^{6} c^{2} x^{11} + 2 \, b^{7} c x^{9} + b^{8} x^{7}\right )}}, \frac {3465 \, c^{5} x^{10} + 5775 \, b c^{4} x^{8} + 1848 \, b^{2} c^{3} x^{6} - 264 \, b^{3} c^{2} x^{4} + 88 \, b^{4} c x^{2} - 40 \, b^{5} + 3465 \, {\left (c^{5} x^{11} + 2 \, b c^{4} x^{9} + b^{2} c^{3} x^{7}\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right )}{280 \, {\left (b^{6} c^{2} x^{11} + 2 \, b^{7} c x^{9} + b^{8} x^{7}\right )}}\right ] \] Input:

integrate(1/x^2/(c*x^4+b*x^2)^3,x, algorithm="fricas")
 

Output:

[1/560*(6930*c^5*x^10 + 11550*b*c^4*x^8 + 3696*b^2*c^3*x^6 - 528*b^3*c^2*x 
^4 + 176*b^4*c*x^2 - 80*b^5 + 3465*(c^5*x^11 + 2*b*c^4*x^9 + b^2*c^3*x^7)* 
sqrt(-c/b)*log((c*x^2 + 2*b*x*sqrt(-c/b) - b)/(c*x^2 + b)))/(b^6*c^2*x^11 
+ 2*b^7*c*x^9 + b^8*x^7), 1/280*(3465*c^5*x^10 + 5775*b*c^4*x^8 + 1848*b^2 
*c^3*x^6 - 264*b^3*c^2*x^4 + 88*b^4*c*x^2 - 40*b^5 + 3465*(c^5*x^11 + 2*b* 
c^4*x^9 + b^2*c^3*x^7)*sqrt(c/b)*arctan(x*sqrt(c/b)))/(b^6*c^2*x^11 + 2*b^ 
7*c*x^9 + b^8*x^7)]
 

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=- \frac {99 \sqrt {- \frac {c^{7}}{b^{13}}} \log {\left (- \frac {b^{7} \sqrt {- \frac {c^{7}}{b^{13}}}}{c^{4}} + x \right )}}{16} + \frac {99 \sqrt {- \frac {c^{7}}{b^{13}}} \log {\left (\frac {b^{7} \sqrt {- \frac {c^{7}}{b^{13}}}}{c^{4}} + x \right )}}{16} + \frac {- 40 b^{5} + 88 b^{4} c x^{2} - 264 b^{3} c^{2} x^{4} + 1848 b^{2} c^{3} x^{6} + 5775 b c^{4} x^{8} + 3465 c^{5} x^{10}}{280 b^{8} x^{7} + 560 b^{7} c x^{9} + 280 b^{6} c^{2} x^{11}} \] Input:

integrate(1/x**2/(c*x**4+b*x**2)**3,x)
 

Output:

-99*sqrt(-c**7/b**13)*log(-b**7*sqrt(-c**7/b**13)/c**4 + x)/16 + 99*sqrt(- 
c**7/b**13)*log(b**7*sqrt(-c**7/b**13)/c**4 + x)/16 + (-40*b**5 + 88*b**4* 
c*x**2 - 264*b**3*c**2*x**4 + 1848*b**2*c**3*x**6 + 5775*b*c**4*x**8 + 346 
5*c**5*x**10)/(280*b**8*x**7 + 560*b**7*c*x**9 + 280*b**6*c**2*x**11)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=\frac {3465 \, c^{5} x^{10} + 5775 \, b c^{4} x^{8} + 1848 \, b^{2} c^{3} x^{6} - 264 \, b^{3} c^{2} x^{4} + 88 \, b^{4} c x^{2} - 40 \, b^{5}}{280 \, {\left (b^{6} c^{2} x^{11} + 2 \, b^{7} c x^{9} + b^{8} x^{7}\right )}} + \frac {99 \, c^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{6}} \] Input:

integrate(1/x^2/(c*x^4+b*x^2)^3,x, algorithm="maxima")
 

Output:

1/280*(3465*c^5*x^10 + 5775*b*c^4*x^8 + 1848*b^2*c^3*x^6 - 264*b^3*c^2*x^4 
 + 88*b^4*c*x^2 - 40*b^5)/(b^6*c^2*x^11 + 2*b^7*c*x^9 + b^8*x^7) + 99/8*c^ 
4*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^6)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=\frac {99 \, c^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{6}} + \frac {19 \, c^{5} x^{3} + 21 \, b c^{4} x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{6}} + \frac {350 \, c^{3} x^{6} - 70 \, b c^{2} x^{4} + 21 \, b^{2} c x^{2} - 5 \, b^{3}}{35 \, b^{6} x^{7}} \] Input:

integrate(1/x^2/(c*x^4+b*x^2)^3,x, algorithm="giac")
 

Output:

99/8*c^4*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^6) + 1/8*(19*c^5*x^3 + 21*b*c^ 
4*x)/((c*x^2 + b)^2*b^6) + 1/35*(350*c^3*x^6 - 70*b*c^2*x^4 + 21*b^2*c*x^2 
 - 5*b^3)/(b^6*x^7)
 

Mupad [B] (verification not implemented)

Time = 17.87 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {11\,c\,x^2}{35\,b^2}-\frac {1}{7\,b}-\frac {33\,c^2\,x^4}{35\,b^3}+\frac {33\,c^3\,x^6}{5\,b^4}+\frac {165\,c^4\,x^8}{8\,b^5}+\frac {99\,c^5\,x^{10}}{8\,b^6}}{b^2\,x^7+2\,b\,c\,x^9+c^2\,x^{11}}+\frac {99\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,b^{13/2}} \] Input:

int(1/(x^2*(b*x^2 + c*x^4)^3),x)
 

Output:

((11*c*x^2)/(35*b^2) - 1/(7*b) - (33*c^2*x^4)/(35*b^3) + (33*c^3*x^6)/(5*b 
^4) + (165*c^4*x^8)/(8*b^5) + (99*c^5*x^10)/(8*b^6))/(b^2*x^7 + c^2*x^11 + 
 2*b*c*x^9) + (99*c^(7/2)*atan((c^(1/2)*x)/b^(1/2)))/(8*b^(13/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^3} \, dx=\frac {3465 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {b}}\right ) b^{2} c^{3} x^{7}+6930 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {b}}\right ) b \,c^{4} x^{9}+3465 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {b}}\right ) c^{5} x^{11}-40 b^{6}+88 b^{5} c \,x^{2}-264 b^{4} c^{2} x^{4}+1848 b^{3} c^{3} x^{6}+5775 b^{2} c^{4} x^{8}+3465 b \,c^{5} x^{10}}{280 b^{7} x^{7} \left (c^{2} x^{4}+2 b c \,x^{2}+b^{2}\right )} \] Input:

int(1/x^2/(c*x^4+b*x^2)^3,x)
 

Output:

(3465*sqrt(c)*sqrt(b)*atan((c*x)/(sqrt(c)*sqrt(b)))*b**2*c**3*x**7 + 6930* 
sqrt(c)*sqrt(b)*atan((c*x)/(sqrt(c)*sqrt(b)))*b*c**4*x**9 + 3465*sqrt(c)*s 
qrt(b)*atan((c*x)/(sqrt(c)*sqrt(b)))*c**5*x**11 - 40*b**6 + 88*b**5*c*x**2 
 - 264*b**4*c**2*x**4 + 1848*b**3*c**3*x**6 + 5775*b**2*c**4*x**8 + 3465*b 
*c**5*x**10)/(280*b**7*x**7*(b**2 + 2*b*c*x**2 + c**2*x**4))