\(\int \frac {(a+b x^2)^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx\) [1182]

Optimal result

Integrand size = 26, antiderivative size = 136 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}-\frac {\sqrt {a} (3 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 c^{3/2}}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}} \] Output:

-1/2*a*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c/x^2-1/2*a^(1/2)*(-a*d+3*b*c)*arct 
anh(c^(1/2)*(b*x^2+a)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/c^(3/2)+b^(3/2)*arcta 
nh(d^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/(d*x^2+c)^(1/2))/d^(1/2)
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{c x^2}+\frac {\sqrt {a} (-3 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {354, 109, 27, 175, 66, 104, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-((a*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^2)) + ((-2*Sqrt[a]*(3*b*c - a* 
d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[c] + 
 (4*b^(3/2)*c*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])] 
)/Sqrt[d])/(2*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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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 = 0.68 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.43

method	result	size
risch	\(-\frac {a \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{2 c \,x^{2}}-\frac {\left (-\frac {b^{2} c \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{\sqrt {b d}}-\frac {a \left (a d -3 b c \right ) \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{2 \sqrt {a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{2 c \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(194\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 \sqrt {b d}}-\frac {a \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 c \,x^{2}}+\frac {a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d}{4 c \sqrt {a c}}-\frac {3 a b \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{4 \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(251\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (\ln \left (\frac {a d \,x^{2}+x^{2} b c +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}+2 a c}{x^{2}}\right ) a^{2} d \,x^{2} \sqrt {b d}-3 \ln \left (\frac {a d \,x^{2}+x^{2} b c +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}+2 a c}{x^{2}}\right ) a b c \,x^{2} \sqrt {b d}+2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c \,x^{2} \sqrt {a c}-2 a \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{4 c \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\)	\(263\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (104) = 208\).

Time = 0.47 (sec) , antiderivative size = 958, normalized size of antiderivative = 7.04 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/8*(2*b*c*x^2*sqrt(b/d)*log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^ 
2 + 8*(b^2*c*d + a*b*d^2)*x^2 + 4*(2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 
 + a)*sqrt(d*x^2 + c)*sqrt(b/d)) - (3*b*c - a*d)*x^2*sqrt(a/c)*log(((b^2*c 
^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2 + 4* 
(2*a*c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a/c)) 
/x^4) - 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a)/(c*x^2), -1/8*(4*b*c*x^2*sqrt 
(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)* 
sqrt(-b/d)/(b^2*d*x^4 + a*b*c + (b^2*c + a*b*d)*x^2)) + (3*b*c - a*d)*x^2* 
sqrt(a/c)*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^ 
2 + a^2*c*d)*x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt( 
d*x^2 + c)*sqrt(a/c))/x^4) + 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a)/(c*x^2), 
 1/4*(b*c*x^2*sqrt(b/d)*log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 
+ 8*(b^2*c*d + a*b*d^2)*x^2 + 4*(2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 + 
 a)*sqrt(d*x^2 + c)*sqrt(b/d)) + (3*b*c - a*d)*x^2*sqrt(-a/c)*arctan(1/2*( 
(b*c + a*d)*x^2 + 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a/c)/(a*b*d 
*x^4 + a^2*c + (a*b*c + a^2*d)*x^2)) - 2*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a 
)/(c*x^2), -1/4*(2*b*c*x^2*sqrt(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*s 
qrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b/d)/(b^2*d*x^4 + a*b*c + (b^2*c + a* 
b*d)*x^2)) - (3*b*c - a*d)*x^2*sqrt(-a/c)*arctan(1/2*((b*c + a*d)*x^2 + 2* 
a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a/c)/(a*b*d*x^4 + a^2*c + (a...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (104) = 208\).

Time = 0.16 (sec) , antiderivative size = 491, normalized size of antiderivative = 3.61 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=-\frac {b^{3} {\left (\frac {\log \left ({\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d}} + \frac {{\left (3 \, \sqrt {b d} a b c - \sqrt {b d} a^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b^{2} c} + \frac {2 \, {\left (\sqrt {b d} a b^{3} c^{2} - 2 \, \sqrt {b d} a^{2} b^{2} c d + \sqrt {b d} a^{3} b d^{2} - \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b c - \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )} b c}\right )}}{2 \, {\left | b \right |}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a c d -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a \,d^{2} x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b c d \,x^{2}+\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a \,d^{2} x^{2}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b c d \,x^{2}+2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b \,c^{2} x^{2}}{2 c^{2} d \,x^{2}} \] Input:

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

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d - sqrt(c)*sqrt(a)*log(sqrt(a)* 
sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*a*d**2*x**2 + 3*sqrt(c)*s 
qrt(a)*log(sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*b*c*d* 
x**2 + sqrt(c)*sqrt(a)*log(x)*a*d**2*x**2 - 3*sqrt(c)*sqrt(a)*log(x)*b*c*d 
*x**2 + 2*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d - sqrt(d)*sqrt 
(c + d*x**2)*b)*b*c**2*x**2)/(2*c**2*d*x**2)