\(\int \frac {1}{x^3 (a+b x^2)^2 \sqrt {c+d x^2}} \, dx\) [1021]

Optimal result

Integrand size = 24, antiderivative size = 185 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {b (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 c (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{3/2}} \] Output:

-1/2*b*(-a*d+2*b*c)*(d*x^2+c)^(1/2)/a^2/c/(-a*d+b*c)/(b*x^2+a)-1/2*(d*x^2+ 
c)^(1/2)/a/c/x^2/(b*x^2+a)+1/2*(a*d+4*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2) 
)/a^3/c^(3/2)-1/2*b^(3/2)*(-5*a*d+4*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/( 
-a*d+b*c)^(1/2))/a^3/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\frac {a \sqrt {c+d x^2} \left (-a^2 d+2 b^2 c x^2+a b \left (c-d x^2\right )\right )}{c (-b c+a d) x^2 \left (a+b x^2\right )}-\frac {b^{3/2} (4 b c-5 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{2 a^3} \] Input:

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

Output:

((a*Sqrt[c + d*x^2]*(-(a^2*d) + 2*b^2*c*x^2 + a*b*(c - d*x^2)))/(c*(-(b*c) 
 + a*d)*x^2*(a + b*x^2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c 
 + d*x^2])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2) + ((4*b*c + a*d)*ArcT 
anh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(3/2))/(2*a^3)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {354, 114, 27, 168, 174, 73, 221}

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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 b^{3/2} c (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d) (a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a \sqrt {c}}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x^2} (2 b c-a d)}{a \left (a+b x^2\right ) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x^2}}{a c x^2 \left (a+b x^2\right )}\right )\)

Input:

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

Output:

(-(Sqrt[c + d*x^2]/(a*c*x^2*(a + b*x^2))) - ((2*b*(2*b*c - a*d)*Sqrt[c + d 
*x^2])/(a*(b*c - a*d)*(a + b*x^2)) + ((-2*(b*c - a*d)*(4*b*c + a*d)*ArcTan 
h[Sqrt[c + d*x^2]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(3/2)*c*(4*b*c - 5*a*d)*Arc 
Tanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(a*( 
b*c - a*d)))/(2*a*c))/2
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (b \,x^{2}+a \right ) b^{2} \left (b c -\frac {5 a d}{4}\right ) x^{2} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\sqrt {\left (a d -b c \right ) b}\, \left (c \,x^{2} \left (b \,x^{2}+a \right ) \left (a d +4 b c \right ) \left (a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{\sqrt {c}}\right )+a \left (2 b^{2} c \,x^{2}+b \left (-x^{2} d +c \right ) a -d \,a^{2}\right ) \sqrt {x^{2} d +c}\, c^{\frac {3}{2}}\right )}{4}\right )}{\sqrt {\left (a d -b c \right ) b}\, c^{\frac {5}{2}} x^{2} a^{3} \left (a d -b c \right ) \left (b \,x^{2}+a \right )}\)	\(191\)
risch	\(-\frac {\sqrt {x^{2} d +c}}{2 c \,a^{2} x^{2}}+\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right ) d}{2 a^{2} c^{\frac {3}{2}}}+\frac {2 b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{a^{3} \sqrt {c}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}+\frac {b^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{4 a^{2} \sqrt {-a b}\, \left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {b d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 a^{2} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{4 a^{2} \sqrt {-a b}\, \left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {b d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 a^{2} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\)	\(899\)
default	\(\frac {-\frac {\sqrt {x^{2} d +c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{2 c^{\frac {3}{2}}}}{a^{2}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}+\frac {2 b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{a^{3} \sqrt {c}}+\frac {b \left (\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{2} \sqrt {-a b}}-\frac {b \left (\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{2} \sqrt {-a b}}\)	\(903\)

Input:

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

Output:

-2/((a*d-b*c)*b)^(1/2)*((b*x^2+a)*b^2*(b*c-5/4*a*d)*x^2*c^(5/2)*arctan((d* 
x^2+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))-1/4*((a*d-b*c)*b)^(1/2)*(c*x^2*(b*x^2+ 
a)*(a*d+4*b*c)*(a*d-b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))+a*(2*b^2*c*x^2+b 
*(-d*x^2+c)*a-d*a^2)*(d*x^2+c)^(1/2)*c^(3/2)))/c^(5/2)/x^2/a^3/(a*d-b*c)/( 
b*x^2+a)
 

Fricas [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 1413, normalized size of antiderivative = 7.64 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^4 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^2 
)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 
 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2 
*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b 
*x^2 + a^2)) + 2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 
 - 3*a^2*b*c*d - a^3*d^2)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqr 
t(c) + 2*c)/x^2) - 4*(a^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b*c*d)*x^2) 
*sqrt(d*x^2 + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^4 + (a^4*b*c^3 - a^5*c^2* 
d)*x^2), -1/8*(4*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 
 - 3*a^2*b*c*d - a^3*d^2)*x^2)*sqrt(-c)*arctan(sqrt(d*x^2 + c)*sqrt(-c)/c) 
 - ((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^4 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^2)*s 
qrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2* 
(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c* 
d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^ 
2 + a^2)) + 4*(a^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b*c*d)*x^2)*sqrt(d 
*x^2 + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^4 + (a^4*b*c^3 - a^5*c^2*d)*x^2) 
, 1/4*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^4 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^ 
2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c) 
*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b 
*d^2)*x^4 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^2)*sqrt(c)*log(-(d*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(1/((b*x^2 + a)^2*sqrt(d*x^2 + c)*x^3), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x^{2} + c} b^{2} c^{2} d - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x^{2} + c} a b c d^{2} - \sqrt {d x^{2} + c} a^{2} d^{3}}{2 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c} c} \] Input:

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

Output:

1/2*(4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/( 
(a^3*b*c - a^4*d)*sqrt(-b^2*c + a*b*d)) - 1/2*(2*(d*x^2 + c)^(3/2)*b^2*c*d 
 - 2*sqrt(d*x^2 + c)*b^2*c^2*d - (d*x^2 + c)^(3/2)*a*b*d^2 + 2*sqrt(d*x^2 
+ c)*a*b*c*d^2 - sqrt(d*x^2 + c)*a^2*d^3)/((a^2*b*c^2 - a^3*c*d)*((d*x^2 + 
 c)^2*b - 2*(d*x^2 + c)*b*c + b*c^2 + (d*x^2 + c)*a*d - a*c*d)) - 1/2*(4*b 
*c + a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*c)
 

Mupad [B] (verification not implemented)

Time = 3.17 (sec) , antiderivative size = 3837, normalized size of antiderivative = 20.74 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

(((c + d*x^2)^(1/2)*(a^2*d^3 + 2*b^2*c^2*d - 2*a*b*c*d^2))/(2*a^2*(b*c^2 - 
 a*c*d)) + (b*(c + d*x^2)^(3/2)*(a*d^2 - 2*b*c*d))/(2*a^2*(b*c^2 - a*c*d)) 
)/((c + d*x^2)*(a*d - 2*b*c) + b*(c + d*x^2)^2 + b*c^2 - a*c*d) + (atan((( 
(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^2)^(1/2)*(a^4*b^3*d^ 
6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d 
^4))/(2*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) + ((-b^3*(a*d - b*c)^ 
3)^(1/2)*(5*a*d - 4*b*c)*((2*a^9*b^2*c*d^6 + 4*a^6*b^5*c^4*d^3 - 8*a^7*b^4 
*c^3*d^4 + 2*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) 
- ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x^2)^(1/2)*(5*a*d - 4*b*c)*(32*a^6*b^ 
5*c^5*d^2 - 80*a^7*b^4*c^4*d^3 + 64*a^8*b^3*c^3*d^4 - 16*a^9*b^2*c^2*d^5)) 
/(8*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3 
*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))/(4*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c 
^2*d - 3*a^5*b*c*d^2)))*1i)/(4*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 
3*a^5*b*c*d^2)) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^2 
)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 
 + 26*a^2*b^5*c^2*d^4))/(2*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) - 
((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*((2*a^9*b^2*c*d^6 + 4*a^6*b^5* 
c^4*d^3 - 8*a^7*b^4*c^3*d^4 + 2*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^ 
2 - 2*a^7*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x^2)^(1/2)*(5*a*d 
- 4*b*c)*(32*a^6*b^5*c^5*d^2 - 80*a^7*b^4*c^4*d^3 + 64*a^8*b^3*c^3*d^4 ...
 

Reduce [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 3434, normalized size of antiderivative = 18.56 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(10*sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d 
 - b*c)*sqrt(a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*s 
qrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a**2*c*d*x**2 - 8*s 
qrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b* 
c)*sqrt(a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2 
*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*c**2*x**2 + 10*sqrt( 
d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*s 
qrt(a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*c*d*x**4 - 8*sqrt(d)*sqr 
t(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*sqrt(a* 
d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*s 
qrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*b**2*c**2*x**4 - 10*sqrt(b)*sqrt(2 
*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(c + d*x**2)*b + 
 sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b* 
c)))*a**3*c*d**2*x**2 + 18*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
+ 2*a*d - b*c)*atan((sqrt(c + d*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a**2*b*c**2*d*x**2 - 10*sqrt 
(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(c + d 
*x**2)*b + sqrt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 
2*a*d - b*c)))*a**2*b*c*d**2*x**4 - 8*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sq...