\(\int \frac {1}{x^3 (a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\) [586]

Optimal result

Integrand size = 26, antiderivative size = 189 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \left (a+b x^2\right ) \log (x)}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

-b/a^3/((b*x^2+a)^2)^(1/2)-1/4*b/a^2/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-1/2*(b* 
x^2+a)/a^3/x^2/((b*x^2+a)^2)^(1/2)-3*b*(b*x^2+a)*ln(x)/a^4/((b*x^2+a)^2)^( 
1/2)+3/2*b*(b*x^2+a)*ln(b*x^2+a)/a^4/((b*x^2+a)^2)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(901\) vs. \(2(189)=378\).

Time = 1.44 (sec) , antiderivative size = 901, normalized size of antiderivative = 4.77 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {-2 a^6-5 a^5 b x^2+2 a^4 b^2 x^4+4 a^3 b^3 x^6-a b^5 x^{10}+2 a^4 \sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}+3 a^3 \sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2}-5 \left (a^2\right )^{3/2} b^2 x^4 \sqrt {\left (a+b x^2\right )^2}+a \sqrt {a^2} b^3 x^6 \sqrt {\left (a+b x^2\right )^2}-\sqrt {a^2} b^4 x^8 \sqrt {\left (a+b x^2\right )^2}+6 b x^2 \left (\left (a^2\right )^{3/2} b^2 x^4+a^4 \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )+a^3 b x^2 \left (2 \sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )\right ) \text {arctanh}\left (\frac {b x^2}{\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}}\right )-6 b x^2 \left (a^5+2 a^4 b x^2-\left (a^2\right )^{3/2} b x^2 \sqrt {\left (a+b x^2\right )^2}+a^3 \left (b^2 x^4-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right )\right ) \log \left (x^2\right )+3 a^5 b x^2 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+6 a^4 b^2 x^4 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+3 a^3 b^3 x^6 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 a^3 \sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 \left (a^2\right )^{3/2} b^2 x^4 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+3 a^5 b x^2 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+6 a^4 b^2 x^4 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+3 a^3 b^3 x^6 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 a^3 \sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 \left (a^2\right )^{3/2} b^2 x^4 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )}{2 a^4 \sqrt {a^2} x^2 \left (a^2+a b x^2-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right )^2} \] Input:

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

Output:

-1/2*(-2*a^6 - 5*a^5*b*x^2 + 2*a^4*b^2*x^4 + 4*a^3*b^3*x^6 - a*b^5*x^10 + 
2*a^4*Sqrt[a^2]*Sqrt[(a + b*x^2)^2] + 3*a^3*Sqrt[a^2]*b*x^2*Sqrt[(a + b*x^ 
2)^2] - 5*(a^2)^(3/2)*b^2*x^4*Sqrt[(a + b*x^2)^2] + a*Sqrt[a^2]*b^3*x^6*Sq 
rt[(a + b*x^2)^2] - Sqrt[a^2]*b^4*x^8*Sqrt[(a + b*x^2)^2] + 6*b*x^2*((a^2) 
^(3/2)*b^2*x^4 + a^4*(Sqrt[a^2] - Sqrt[(a + b*x^2)^2]) + a^3*b*x^2*(2*Sqrt 
[a^2] - Sqrt[(a + b*x^2)^2]))*ArcTanh[(b*x^2)/(Sqrt[a^2] - Sqrt[(a + b*x^2 
)^2])] - 6*b*x^2*(a^5 + 2*a^4*b*x^2 - (a^2)^(3/2)*b*x^2*Sqrt[(a + b*x^2)^2 
] + a^3*(b^2*x^4 - Sqrt[a^2]*Sqrt[(a + b*x^2)^2]))*Log[x^2] + 3*a^5*b*x^2* 
Log[Sqrt[a^2] - b*x^2 - Sqrt[(a + b*x^2)^2]] + 6*a^4*b^2*x^4*Log[Sqrt[a^2] 
 - b*x^2 - Sqrt[(a + b*x^2)^2]] + 3*a^3*b^3*x^6*Log[Sqrt[a^2] - b*x^2 - Sq 
rt[(a + b*x^2)^2]] - 3*a^3*Sqrt[a^2]*b*x^2*Sqrt[(a + b*x^2)^2]*Log[Sqrt[a^ 
2] - b*x^2 - Sqrt[(a + b*x^2)^2]] - 3*(a^2)^(3/2)*b^2*x^4*Sqrt[(a + b*x^2) 
^2]*Log[Sqrt[a^2] - b*x^2 - Sqrt[(a + b*x^2)^2]] + 3*a^5*b*x^2*Log[Sqrt[a^ 
2] + b*x^2 - Sqrt[(a + b*x^2)^2]] + 6*a^4*b^2*x^4*Log[Sqrt[a^2] + b*x^2 - 
Sqrt[(a + b*x^2)^2]] + 3*a^3*b^3*x^6*Log[Sqrt[a^2] + b*x^2 - Sqrt[(a + b*x 
^2)^2]] - 3*a^3*Sqrt[a^2]*b*x^2*Sqrt[(a + b*x^2)^2]*Log[Sqrt[a^2] + b*x^2 
- Sqrt[(a + b*x^2)^2]] - 3*(a^2)^(3/2)*b^2*x^4*Sqrt[(a + b*x^2)^2]*Log[Sqr 
t[a^2] + b*x^2 - Sqrt[(a + b*x^2)^2]])/(a^4*Sqrt[a^2]*x^2*(a^2 + a*b*x^2 - 
 Sqrt[a^2]*Sqrt[(a + b*x^2)^2])^2)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1384, 27, 243, 54, 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.

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

\(\Big \downarrow \) 1384

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 54

\(\displaystyle \frac {\left (a+b x^2\right ) \int \left (\frac {3 b^2}{a^4 \left (b x^2+a\right )}+\frac {2 b^2}{a^3 \left (b x^2+a\right )^2}+\frac {b^2}{a^2 \left (b x^2+a\right )^3}-\frac {3 b}{a^4 x^2}+\frac {1}{a^3 x^4}\right )dx^2}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (a+b x^2\right ) \left (-\frac {3 b \log \left (x^2\right )}{a^4}+\frac {3 b \log \left (a+b x^2\right )}{a^4}-\frac {2 b}{a^3 \left (a+b x^2\right )}-\frac {1}{a^3 x^2}-\frac {b}{2 a^2 \left (a+b x^2\right )^2}\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\)

Input:

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

Output:

((a + b*x^2)*(-(1/(a^3*x^2)) - b/(2*a^2*(a + b*x^2)^2) - (2*b)/(a^3*(a + b 
*x^2)) - (3*b*Log[x^2])/a^4 + (3*b*Log[a + b*x^2])/a^4))/(2*Sqrt[a^2 + 2*a 
*b*x^2 + b^2*x^4])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 1384

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac 
Part[p]))   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n 
- 1)] && NeQ[u, x^(2*n - 1)] &&  !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
 

rule 2009

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.48

method	result	size
pseudoelliptic	\(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (-3 b \,x^{2} \left (b \,x^{2}+a \right )^{2} \ln \left (b \,x^{2}+a \right )+3 b \,x^{2} \left (b \,x^{2}+a \right )^{2} \ln \left (x^{2}\right )+a \left (3 b^{2} x^{4}+\frac {9}{2} a b \,x^{2}+a^{2}\right )\right )}{2 \left (b \,x^{2}+a \right )^{2} x^{2} a^{4}}\)	\(90\)
risch	\(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {3 b^{2} x^{4}}{2 a^{3}}-\frac {9 b \,x^{2}}{4 a^{2}}-\frac {1}{2 a}\right )}{\left (b \,x^{2}+a \right )^{3} x^{2}}-\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a^{4}}+\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \ln \left (-b \,x^{2}-a \right )}{2 \left (b \,x^{2}+a \right ) a^{4}}\)	\(117\)
default	\(\frac {\left (6 \ln \left (b \,x^{2}+a \right ) x^{6} b^{3}-12 \ln \left (x \right ) x^{6} b^{3}+12 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2}-24 \ln \left (x \right ) x^{4} a \,b^{2}-6 b^{2} x^{4} a +6 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b -12 \ln \left (x \right ) x^{2} a^{2} b -9 a^{2} b \,x^{2}-2 a^{3}\right ) \left (b \,x^{2}+a \right )}{4 x^{2} a^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\)	\(133\)

Input:

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

Output:

-1/2*csgn(b*x^2+a)*(-3*b*x^2*(b*x^2+a)^2*ln(b*x^2+a)+3*b*x^2*(b*x^2+a)^2*l 
n(x^2)+a*(3*b^2*x^4+9/2*a*b*x^2+a^2))/(b*x^2+a)^2/x^2/a^4
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \] Input:

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

Output:

-1/4*(6*a*b^2*x^4 + 9*a^2*b*x^2 + 2*a^3 - 6*(b^3*x^6 + 2*a*b^2*x^4 + a^2*b 
*x^2)*log(b*x^2 + a) + 12*(b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*log(x))/(a^4 
*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)
 

Sympy [F]

\[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral(1/(x**3*((a + b*x**2)**2)**(3/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {6 \, b^{2} x^{4} + 9 \, a b x^{2} + 2 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} + \frac {3 \, b \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac {3 \, b \log \left (x\right )}{a^{4}} \] Input:

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

Output:

-1/4*(6*b^2*x^4 + 9*a*b*x^2 + 2*a^2)/(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2) 
 + 3/2*b*log(b*x^2 + a)/a^4 - 3*b*log(x)/a^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {9 \, b^{3} x^{4} + 22 \, a b^{2} x^{2} + 14 \, a^{2} b}{4 \, {\left (b x^{2} + a\right )}^{2} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b x^{2} - a}{2 \, a^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \] Input:

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

Output:

-3/2*b*log(x^2)/(a^4*sgn(b*x^2 + a)) + 3/2*b*log(abs(b*x^2 + a))/(a^4*sgn( 
b*x^2 + a)) - 1/4*(9*b^3*x^4 + 22*a*b^2*x^2 + 14*a^2*b)/((b*x^2 + a)^2*a^4 
*sgn(b*x^2 + a)) + 1/2*(3*b*x^2 - a)/(a^4*x^2*sgn(b*x^2 + a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {6 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b \,x^{2}+12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} x^{4}+6 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} x^{6}-12 \,\mathrm {log}\left (x \right ) a^{2} b \,x^{2}-24 \,\mathrm {log}\left (x \right ) a \,b^{2} x^{4}-12 \,\mathrm {log}\left (x \right ) b^{3} x^{6}-2 a^{3}-6 a^{2} b \,x^{2}+3 b^{3} x^{6}}{4 a^{4} x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

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

Output:

(6*log(a + b*x**2)*a**2*b*x**2 + 12*log(a + b*x**2)*a*b**2*x**4 + 6*log(a 
+ b*x**2)*b**3*x**6 - 12*log(x)*a**2*b*x**2 - 24*log(x)*a*b**2*x**4 - 12*l 
og(x)*b**3*x**6 - 2*a**3 - 6*a**2*b*x**2 + 3*b**3*x**6)/(4*a**4*x**2*(a**2 
 + 2*a*b*x**2 + b**2*x**4))