\(\int \frac {(c+d x^2)^{3/2}}{x^4 (a+b x^2)^2} \, dx\) [1004]

Optimal result

Integrand size = 24, antiderivative size = 166 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {(5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {(15 b c-11 a d) \sqrt {c+d x^2}}{6 a^3 x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x^3 \left (a+b x^2\right )}+\frac {(5 b c-2 a d) \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2}} \] Output:

-1/6*(-3*a*d+5*b*c)*(d*x^2+c)^(1/2)/a^2/b/x^3+1/6*(-11*a*d+15*b*c)*(d*x^2+ 
c)^(1/2)/a^3/x+1/2*(-a*d+b*c)*(d*x^2+c)^(1/2)/a/b/x^3/(b*x^2+a)+1/2*(-2*a* 
d+5*b*c)*(-a*d+b*c)^(1/2)*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2 
))/a^(7/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1668\) vs. \(2(166)=332\).

Time = 7.45 (sec) , antiderivative size = 1668, normalized size of antiderivative = 10.05 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((Sqrt[a]*(2048*c^(13/2) + 7168*c^(11/2)*d*x^2 + 9728*c^(9/2)*d^2*x^4 + 64 
00*c^(7/2)*d^3*x^6 + 2072*c^(5/2)*d^4*x^8 + 292*c^(3/2)*d^5*x^10 + 12*Sqrt 
[c]*d^6*x^12 - 2048*c^6*Sqrt[c + d*x^2] - 6144*c^5*d*x^2*Sqrt[c + d*x^2] - 
 6912*c^4*d^2*x^4*Sqrt[c + d*x^2] - 3584*c^3*d^3*x^6*Sqrt[c + d*x^2] - 840 
*c^2*d^4*x^8*Sqrt[c + d*x^2] - 72*c*d^5*x^10*Sqrt[c + d*x^2] - d^6*x^12*Sq 
rt[c + d*x^2])*(-15*b^2*c*x^4 + 2*a^2*(c + 4*d*x^2) + a*b*(-10*c*x^2 + 11* 
d*x^4)))/(x^3*(a + b*x^2)*(2048*c^6 + 6144*c^5*d*x^2 + 6912*c^4*d^2*x^4 + 
3584*c^3*d^3*x^6 + 840*c^2*d^4*x^8 + 72*c*d^5*x^10 + d^6*x^12 - 2048*c^(11 
/2)*Sqrt[c + d*x^2] - 5120*c^(9/2)*d*x^2*Sqrt[c + d*x^2] - 4608*c^(7/2)*d^ 
2*x^4*Sqrt[c + d*x^2] - 1792*c^(5/2)*d^3*x^6*Sqrt[c + d*x^2] - 280*c^(3/2) 
*d^4*x^8*Sqrt[c + d*x^2] - 12*Sqrt[c]*d^5*x^10*Sqrt[c + d*x^2])) + (21*a*b 
*c*d*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqr 
t[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*S 
qrt[b*c - a*d]] + (15*b^(3/2)*c^(3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b*c - 
 a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + 
d*x^2]))])/Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] + (21*a*b 
*c*d*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqr 
t[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*S 
qrt[b*c - a*d]] + (6*a*Sqrt[b]*Sqrt[c]*d*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b* 
c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqr...
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {370, 25, 445, 27, 445, 27, 291, 218}

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.

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

\(\Big \downarrow \) 370

\(\displaystyle \frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}-\frac {\int -\frac {2 d (2 b c-a d) x^2+c (5 b c-3 a d)}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 d (2 b c-a d) x^2+c (5 b c-3 a d)}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {-\frac {\int \frac {b c \left (2 d (5 b c-3 a d) x^2+c (15 b c-11 a d)\right )}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {c+d x^2} (5 b c-3 a d)}{3 a x^3}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {b \int \frac {2 d (5 b c-3 a d) x^2+c (15 b c-11 a d)}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{3 a}-\frac {\sqrt {c+d x^2} (5 b c-3 a d)}{3 a x^3}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {-\frac {b \left (-\frac {\int \frac {3 c (5 b c-2 a d) (b c-a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {c+d x^2} (15 b c-11 a d)}{a x}\right )}{3 a}-\frac {\sqrt {c+d x^2} (5 b c-3 a d)}{3 a x^3}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {b \left (-\frac {3 (5 b c-2 a d) (b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {c+d x^2} (15 b c-11 a d)}{a x}\right )}{3 a}-\frac {\sqrt {c+d x^2} (5 b c-3 a d)}{3 a x^3}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {-\frac {b \left (-\frac {3 (5 b c-2 a d) (b c-a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{a}-\frac {\sqrt {c+d x^2} (15 b c-11 a d)}{a x}\right )}{3 a}-\frac {\sqrt {c+d x^2} (5 b c-3 a d)}{3 a x^3}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {b \left (-\frac {3 (5 b c-2 a d) \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2}}-\frac {\sqrt {c+d x^2} (15 b c-11 a d)}{a x}\right )}{3 a}-\frac {\sqrt {c+d x^2} (5 b c-3 a d)}{3 a x^3}}{2 a b}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )}\)

Input:

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

Output:

((b*c - a*d)*Sqrt[c + d*x^2])/(2*a*b*x^3*(a + b*x^2)) + (-1/3*((5*b*c - 3* 
a*d)*Sqrt[c + d*x^2])/(a*x^3) - (b*(-(((15*b*c - 11*a*d)*Sqrt[c + d*x^2])/ 
(a*x)) - (3*(5*b*c - 2*a*d)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sq 
rt[a]*Sqrt[c + d*x^2])])/a^(3/2)))/(3*a))/(2*a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 218

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

rule 291

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

rule 370

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

rule 445

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {x^{2} d +c}\, \left (4 a d \,x^{2}-6 x^{2} b c +a c \right )}{3 x^{3}}-\frac {b \left (a d -b c \right ) \sqrt {x^{2} d +c}\, x}{2 \left (b \,x^{2}+a \right )}+\frac {\left (2 a^{2} d^{2}-7 a b c d +5 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{2 \sqrt {\left (a d -b c \right ) a}}}{a^{3}}\)	\(132\)
risch	\(-\frac {\sqrt {x^{2} d +c}\, \left (4 a d \,x^{2}-6 x^{2} b c +a c \right )}{3 a^{3} x^{3}}-\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 )}{4 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 )}{4 b}+\frac {\left (a^{2} d^{2}-6 a b c d +5 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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{2} d^{2}-6 a b c d +5 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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}}{a^{3}}\)	\(937\)
default	\(\text {Expression too large to display}\)	\(3573\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.67 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{2} c - 2 \, a b d\right )} x^{5} + {\left (5 \, a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (15 \, b^{2} c - 11 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \, {\left (5 \, a b c - 4 \, a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{2} c - 2 \, a b d\right )} x^{5} + {\left (5 \, a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (15 \, b^{2} c - 11 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \, {\left (5 \, a b c - 4 \, a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \] Input:

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

Output:

[-1/24*(3*((5*b^2*c - 2*a*b*d)*x^5 + (5*a*b*c - 2*a^2*d)*x^3)*sqrt(-(b*c - 
 a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c 
^2 - 4*a^2*c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)* 
sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*((15*b^2*c - 11*a*b 
*d)*x^4 - 2*a^2*c + 2*(5*a*b*c - 4*a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^5 
 + a^4*x^3), 1/12*(3*((5*b^2*c - 2*a*b*d)*x^5 + (5*a*b*c - 2*a^2*d)*x^3)*s 
qrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sq 
rt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 - a*c*d)*x)) + 2*((15*b^2* 
c - 11*a*b*d)*x^4 - 2*a^2*c + 2*(5*a*b*c - 4*a^2*d)*x^2)*sqrt(d*x^2 + c))/ 
(a^3*b*x^5 + a^4*x^3)]
 

Sympy [F]

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

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

Output:

Integral((c + d*x**2)**(3/2)/(x**4*(a + b*x**2)**2), x)
 

Maxima [F]

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

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

Output:

integrate((d*x^2 + c)^(3/2)/((b*x^2 + a)^2*x^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (142) = 284\).

Time = 0.42 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.66 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (5 \, b^{2} c^{2} \sqrt {d} - 7 \, a b c d^{\frac {3}{2}} + 2 \, a^{2} d^{\frac {5}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {5}{2}} - b^{2} c^{3} \sqrt {d} + a b c^{2} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a^{3}} - \frac {4 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a c d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{3} \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c^{2} d^{\frac {3}{2}} + 3 \, b c^{4} \sqrt {d} - 2 \, a c^{3} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3}} \] Input:

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

Output:

-1/2*(5*b^2*c^2*sqrt(d) - 7*a*b*c*d^(3/2) + 2*a^2*d^(5/2))*arctan(1/2*((sq 
rt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sq 
rt(a*b*c*d - a^2*d^2)*a^3) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^2*sqrt 
(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c*d^(3/2) + 2*(sqrt(d)*x - sqr 
t(d*x^2 + c))^2*a^2*d^(5/2) - b^2*c^3*sqrt(d) + a*b*c^2*d^(3/2))/(((sqrt(d 
)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sq 
rt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*a^3) - 4/3*(3*(sqrt(d)*x - sqrt( 
d*x^2 + c))^4*b*c^2*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*c*d^(3/2 
) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^3*sqrt(d) + 3*(sqrt(d)*x - sqrt( 
d*x^2 + c))^2*a*c^2*d^(3/2) + 3*b*c^4*sqrt(d) - 2*a*c^3*d^(3/2))/(((sqrt(d 
)*x - sqrt(d*x^2 + c))^2 - c)^3*a^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 1460, normalized size of antiderivative = 8.80 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(24*sqrt(a)*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**3*d**2*x 
**3 - 90*sqrt(a)*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**2*b 
*c*d*x**3 + 24*sqrt(a)*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**2*b*d**2*x**5 + 75*sqrt(a)*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*b**2*c**2*x**3 - 90*sqrt(a)*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*b**2*c*d*x**5 + 75*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*s 
qrt(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**3*c**2*x**5 + 24*sqrt(a)*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**3*d**2*x**3 - 90*sqrt(a)*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**2*b*c*d*x**3 + 24*sqrt(a)*sqrt(a*d - b*c)*log(s 
qrt(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**2*b*d**2*x**5 + 75*sqrt(a)*sqrt(a*d - b*c)*l 
og(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt...