\(\int \frac {(c+d x^2)^{3/2}}{x^3 (a+b x^2)} \, dx\) [933]

Optimal result

Integrand size = 24, antiderivative size = 114 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x^2}}{2 a x^2}+\frac {\sqrt {c} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}-\frac {(b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 \sqrt {b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {-\frac {a c \sqrt {c+d x^2}}{x^2}+\frac {2 (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b}}+\sqrt {c} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {354, 109, 27, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-((c*Sqrt[c + d*x^2])/(a*x^2)) - ((-2*Sqrt[c]*(2*b*c - 3*a*d)*ArcTanh[Sqr 
t[c + d*x^2]/Sqrt[c]])/a + (4*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + 
d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(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 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 = 0.78 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {\frac {c \left (-\frac {\sqrt {x^{2} d +c}\, a}{x^{2}}-\frac {\left (3 a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{2}+\frac {\left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a^{2}}\)	\(100\)
risch	\(-\frac {c \sqrt {x^{2} d +c}}{2 a \,x^{2}}+\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a b \sqrt {-\frac {a d -b c}{b}}}-\frac {\sqrt {c}\, \left (3 a d -2 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{a}}{2 a}\)	\(412\)
default	\(\text {Expression too large to display}\)	\(1377\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 738, normalized size of antiderivative = 6.47 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=\left [-\frac {{\left (b c - a d\right )} x^{2} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} x^{2}}, -\frac {2 \, {\left (2 \, b c - 3 \, a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} \sqrt {-c}}{c}\right ) + {\left (b c - a d\right )} x^{2} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} x^{2}}, -\frac {2 \, {\left (b c - a d\right )} x^{2} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} x^{2}}, -\frac {{\left (b c - a d\right )} x^{2} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} \sqrt {-c}}{c}\right ) + \sqrt {d x^{2} + c} a c}{2 \, a^{2} x^{2}}\right ] \] Input:

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

Output:

[-1/4*((b*c - a*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*b*c - 3*a*d)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) 
+ 2*c)/x^2) + 2*sqrt(d*x^2 + c)*a*c)/(a^2*x^2), -1/4*(2*(2*b*c - 3*a*d)*sq 
rt(-c)*x^2*arctan(sqrt(d*x^2 + c)*sqrt(-c)/c) + (b*c - a*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*sqrt(d*x^2 + c)*a*c)/(a^2 
*x^2), -1/4*(2*(b*c - a*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*b*c - 3*a*d)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + 
 c)*sqrt(c) + 2*c)/x^2) + 2*sqrt(d*x^2 + c)*a*c)/(a^2*x^2), -1/2*((b*c - a 
*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*b* 
c - 3*a*d)*sqrt(-c)*x^2*arctan(sqrt(d*x^2 + c)*sqrt(-c)/c) + sqrt(d*x^2 + 
c)*a*c)/(a^2*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.05 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c}} - \frac {\sqrt {d x^{2} + c} c}{2 \, a x^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 560, normalized size of antiderivative = 4.91 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c\,\sqrt {d\,x^2+c}}{2\,a\,x^2}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {29\,b^2\,c^{3/2}\,d^6\,\sqrt {d\,x^2+c}}{4\,\left (\frac {29\,b^2\,c^2\,d^6}{4}-3\,a\,b\,c\,d^7-\frac {23\,b^3\,c^3\,d^5}{4\,a}+\frac {3\,b^4\,c^4\,d^4}{2\,a^2}\right )}+\frac {23\,b^3\,c^{5/2}\,d^5\,\sqrt {d\,x^2+c}}{4\,\left (\frac {23\,b^3\,c^3\,d^5}{4}-\frac {29\,a\,b^2\,c^2\,d^6}{4}-\frac {3\,b^4\,c^4\,d^4}{2\,a}+3\,a^2\,b\,c\,d^7\right )}+\frac {3\,b^4\,c^{7/2}\,d^4\,\sqrt {d\,x^2+c}}{2\,\left (-3\,a^3\,b\,c\,d^7+\frac {29\,a^2\,b^2\,c^2\,d^6}{4}-\frac {23\,a\,b^3\,c^3\,d^5}{4}+\frac {3\,b^4\,c^4\,d^4}{2}\right )}-\frac {3\,a\,b\,\sqrt {c}\,d^7\,\sqrt {d\,x^2+c}}{\frac {29\,b^2\,c^2\,d^6}{4}-3\,a\,b\,c\,d^7-\frac {23\,b^3\,c^3\,d^5}{4\,a}+\frac {3\,b^4\,c^4\,d^4}{2\,a^2}}\right )\,\left (3\,a\,d-2\,b\,c\right )}{2\,a^2}-\frac {\mathrm {atanh}\left (\frac {3\,b^2\,c^2\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}}{2\,\left (-2\,a^3\,b\,c\,d^7+\frac {11\,a^2\,b^2\,c^2\,d^6}{2}-5\,a\,b^3\,c^3\,d^5+\frac {3\,b^4\,c^4\,d^4}{2}\right )}+\frac {2\,b\,c\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}}{5\,b^3\,c^3\,d^5-\frac {11\,a\,b^2\,c^2\,d^6}{2}-\frac {3\,b^4\,c^4\,d^4}{2\,a}+2\,a^2\,b\,c\,d^7}\right )\,\sqrt {-b\,{\left (a\,d-b\,c\right )}^3}}{a^2\,b} \] Input:

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

Output:

- (c*(c + d*x^2)^(1/2))/(2*a*x^2) - (c^(1/2)*atanh((29*b^2*c^(3/2)*d^6*(c 
+ d*x^2)^(1/2))/(4*((29*b^2*c^2*d^6)/4 - 3*a*b*c*d^7 - (23*b^3*c^3*d^5)/(4 
*a) + (3*b^4*c^4*d^4)/(2*a^2))) + (23*b^3*c^(5/2)*d^5*(c + d*x^2)^(1/2))/( 
4*((23*b^3*c^3*d^5)/4 - (29*a*b^2*c^2*d^6)/4 - (3*b^4*c^4*d^4)/(2*a) + 3*a 
^2*b*c*d^7)) + (3*b^4*c^(7/2)*d^4*(c + d*x^2)^(1/2))/(2*((3*b^4*c^4*d^4)/2 
 - (23*a*b^3*c^3*d^5)/4 + (29*a^2*b^2*c^2*d^6)/4 - 3*a^3*b*c*d^7)) - (3*a* 
b*c^(1/2)*d^7*(c + d*x^2)^(1/2))/((29*b^2*c^2*d^6)/4 - 3*a*b*c*d^7 - (23*b 
^3*c^3*d^5)/(4*a) + (3*b^4*c^4*d^4)/(2*a^2)))*(3*a*d - 2*b*c))/(2*a^2) - ( 
atanh((3*b^2*c^2*d^4*(c + d*x^2)^(1/2)*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c* 
d^2 - 3*a*b^3*c^2*d)^(1/2))/(2*((3*b^4*c^4*d^4)/2 - 5*a*b^3*c^3*d^5 + (11* 
a^2*b^2*c^2*d^6)/2 - 2*a^3*b*c*d^7)) + (2*b*c*d^5*(c + d*x^2)^(1/2)*(b^4*c 
^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)^(1/2))/(5*b^3*c^3*d^5 - 
(11*a*b^2*c^2*d^6)/2 - (3*b^4*c^4*d^4)/(2*a) + 2*a^2*b*c*d^7))*(-b*(a*d - 
b*c)^3)^(1/2))/(a^2*b)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1461, normalized size of antiderivative = 12.82 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^(3/2)/x^3/(b*x^2+a),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*d*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)*sqr 
t(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)))*b*c*x**2 - 2*sqrt(b)*sqrt(2*sq 
rt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(c + d*x**2)*b + sq 
rt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)) 
)*a**2*d**2*x**2 + 4*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)*s 
qrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)))*a*b*c*d*x**2 - 2*sqrt(b)*sqrt(2*sq 
rt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(c + d*x**2)*b + sq 
rt(d)*b*x)/(sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)) 
)*b**2*c**2*x**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)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt( 
a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)* 
a*d*x**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)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b* 
c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x**2 
 - sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a...