\(\int \frac {(c+d x^2)^{3/2}}{x^3 (a+b x^2)^2} \, dx\) [1003]

Optimal result

Integrand size = 24, antiderivative size = 170 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {(2 b c-a d) \sqrt {c+d x^2}}{2 a^2 \left (a+b x^2\right )}-\frac {c \sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\frac {a \sqrt {c+d x^2} \left (-a c-2 b c x^2+a d x^2\right )}{x^2 \left (a+b x^2\right )}+\frac {\left (4 b^2 c^2-5 a b c d+a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}}+\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {354, 109, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-((c*Sqrt[c + d*x^2])/(a*x^2*(a + b*x^2))) - ((2*(2*b*c - a*d)*Sqrt[c + d 
*x^2])/(a*(a + b*x^2)) + ((-2*Sqrt[c]*(4*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[ 
Sqrt[c + d*x^2]/Sqrt[c]])/a + (2*(b*c - a*d)^(3/2)*(4*b*c - a*d)*ArcTanh[( 
Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(a*(b*c - a*d)))/( 
2*a))/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 109

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

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 = 0.83 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {c}\, \left (b \,x^{2}+a \right ) \left (b c -\frac {a d}{4}\right ) \left (-a d +b c \right ) x^{2} \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\left (3 \left (b \,x^{2}+a \right ) c \left (a d -\frac {4 b c}{3}\right ) x^{2} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{\sqrt {c}}\right )+\sqrt {c}\, a \left (2 x^{2} b c +a \left (-x^{2} d +c \right )\right ) \sqrt {x^{2} d +c}\right ) \sqrt {\left (a d -b c \right ) b}}{2}}{a^{3} x^{2} \sqrt {c}\, \left (b \,x^{2}+a \right ) \sqrt {\left (a d -b c \right ) b}}\)	\(170\)
risch	\(-\frac {c \sqrt {x^{2} d +c}}{2 a^{2} x^{2}}-\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \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 )}{2 b \sqrt {-a b}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \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 )}{2 b \sqrt {-a b}}+\frac {\sqrt {c}\, \left (3 a d -4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{a}-\frac {2 c \left (a d -b c \right ) \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 \sqrt {-\frac {a d -b c}{b}}}-\frac {2 c \left (a d -b c \right ) \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 \sqrt {-\frac {a d -b c}{b}}}}{2 a^{2}}\)	\(939\)
default	\(\text {Expression too large to display}\)	\(3521\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1040, normalized size of antiderivative = 6.12 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/8*(((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt((b*c - a*d)/b) 
*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*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a* 
d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*((4*b^2*c - 3*a*b*d)*x^4 + (4*a*b* 
c - 3*a^2*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x 
^2) + 4*(a^2*c + (2*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^4 + a^4* 
x^2), -1/8*(4*((4*b^2*c - 3*a*b*d)*x^4 + (4*a*b*c - 3*a^2*d)*x^2)*sqrt(-c) 
*arctan(sqrt(d*x^2 + c)*sqrt(-c)/c) + ((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - 
a^2*d)*x^2)*sqrt((b*c - a*d)/b)*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*(b^2*d*x^2 + 2*b^2*c - a*b*d) 
*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(a^ 
2*c + (2*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^4 + a^4*x^2), -1/4* 
(((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt(-(b*c - a*d)/b)*arct 
an(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/b)/(b*c^ 
2 - a*c*d + (b*c*d - a*d^2)*x^2)) + ((4*b^2*c - 3*a*b*d)*x^4 + (4*a*b*c - 
3*a^2*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) 
+ 2*(a^2*c + (2*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^4 + a^4*x^2) 
, -1/4*(((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt(-(b*c - a*d)/ 
b)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/b 
)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + 2*((4*b^2*c - 3*a*b*d)*x^4 +...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.27 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b c d - 2 \, \sqrt {d x^{2} + c} b c^{2} d - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a d^{2} + 2 \, \sqrt {d x^{2} + c} a c d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )} a^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 2.13 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.59 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b^2\,c^2\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{\frac {a^2\,b\,c\,d^7}{4}-\frac {5\,a\,b^2\,c^2\,d^6}{4}+b^3\,c^3\,d^5}-\frac {b\,c\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{4\,\left (\frac {a\,b\,c\,d^7}{4}-\frac {5\,b^2\,c^2\,d^6}{4}+\frac {b^3\,c^3\,d^5}{a}\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-4\,b\,c\right )}{2\,a^3\,b}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {3\,b\,\sqrt {c}\,d^7\,\sqrt {d\,x^2+c}}{4\,\left (\frac {3\,b\,c\,d^7}{4}-\frac {7\,b^2\,c^2\,d^6}{4\,a}+\frac {b^3\,c^3\,d^5}{a^2}\right )}-\frac {7\,b^2\,c^{3/2}\,d^6\,\sqrt {d\,x^2+c}}{4\,\left (\frac {3\,a\,b\,c\,d^7}{4}-\frac {7\,b^2\,c^2\,d^6}{4}+\frac {b^3\,c^3\,d^5}{a}\right )}+\frac {b^3\,c^{5/2}\,d^5\,\sqrt {d\,x^2+c}}{\frac {3\,a^2\,b\,c\,d^7}{4}-\frac {7\,a\,b^2\,c^2\,d^6}{4}+b^3\,c^3\,d^5}\right )\,\left (3\,a\,d-4\,b\,c\right )}{2\,a^3}-\frac {\frac {\left (a\,c\,d^2-b\,c^2\,d\right )\,\sqrt {d\,x^2+c}}{a^2}-\frac {d\,{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-2\,b\,c\right )}{2\,a^2}}{\left (d\,x^2+c\right )\,\left (a\,d-2\,b\,c\right )+b\,{\left (d\,x^2+c\right )}^2+b\,c^2-a\,c\,d} \] Input:

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

Output:

(atanh((b^2*c^2*d^5*(c + d*x^2)^(1/2)*(b^2*c - a*b*d)^(1/2))/(b^3*c^3*d^5 
- (5*a*b^2*c^2*d^6)/4 + (a^2*b*c*d^7)/4) - (b*c*d^6*(c + d*x^2)^(1/2)*(b^2 
*c - a*b*d)^(1/2))/(4*((a*b*c*d^7)/4 - (5*b^2*c^2*d^6)/4 + (b^3*c^3*d^5)/a 
)))*(-b*(a*d - b*c))^(1/2)*(a*d - 4*b*c))/(2*a^3*b) - (c^(1/2)*atanh((3*b* 
c^(1/2)*d^7*(c + d*x^2)^(1/2))/(4*((3*b*c*d^7)/4 - (7*b^2*c^2*d^6)/(4*a) + 
 (b^3*c^3*d^5)/a^2)) - (7*b^2*c^(3/2)*d^6*(c + d*x^2)^(1/2))/(4*((3*a*b*c* 
d^7)/4 - (7*b^2*c^2*d^6)/4 + (b^3*c^3*d^5)/a)) + (b^3*c^(5/2)*d^5*(c + d*x 
^2)^(1/2))/(b^3*c^3*d^5 - (7*a*b^2*c^2*d^6)/4 + (3*a^2*b*c*d^7)/4))*(3*a*d 
 - 4*b*c))/(2*a^3) - (((a*c*d^2 - b*c^2*d)*(c + d*x^2)^(1/2))/a^2 - (d*(c 
+ d*x^2)^(3/2)*(a*d - 2*b*c))/(2*a^2))/((c + d*x^2)*(a*d - 2*b*c) + b*(c + 
 d*x^2)^2 + b*c^2 - a*c*d)
 

Reduce [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 2988, normalized size of antiderivative = 17.58 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*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)*sq 
rt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a**2*d*x**2 - 8*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)*sqrt(2*sq 
rt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*c*x**2 + 2*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)*sqrt(2*sqrt(d)*sq 
rt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*d*x**4 - 8*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)*sqrt(2*sqrt(d)*sqrt(a)*sq 
rt(a*d - b*c) + 2*a*d - b*c)))*b**2*c*x**4 - 2*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*d** 
2*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)*sq 
rt(a*d - b*c) + 2*a*d - b*c)))*a**2*b*c*d*x**2 - 2*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*d**2*x**4 - 8*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d...