\(\int \frac {1}{x^2 (a x^2+b x^3+c x^4)^{3/2}} \, dx\) [81]

Optimal result

Integrand size = 24, antiderivative size = 343 \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 \left (b^2-4 a c\right ) x^3}+\frac {b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 \left (b^2-4 a c\right ) x^2}-\frac {15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}} \] Output:

2*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/x^3/(c*x^4+b*x^3+a*x^2)^(1/2)-1/4*(-20* 
a*c+9*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^2/(-4*a*c+b^2)/x^5+1/8*b*(-68*a*c+2 
1*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^3/(-4*a*c+b^2)/x^4-1/32*(240*a^2*c^2-44 
8*a*b^2*c+105*b^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^4/(-4*a*c+b^2)/x^3+1/64*b*( 
1808*a^2*c^2-1680*a*b^2*c+315*b^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^5/(-4*a*c+b 
^2)/x^2-15/128*(16*a^2*c^2-56*a*b^2*c+21*b^4)*arctanh(1/2*x*(b*x+2*a)/a^(1 
/2)/(c*x^4+b*x^3+a*x^2)^(1/2))/a^(11/2)
 

Mathematica [A] (verified)

Time = 2.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {a} \left (-64 a^5 c-315 b^5 x^4 (b+c x)-105 a b^3 x^3 \left (b^2-18 b c x-16 c^2 x^2\right )+16 a^4 \left (b^2+6 b c x+10 c^2 x^2\right )+2 a^2 b x^2 \left (21 b^3+308 b^2 c x-1352 b c^2 x^2-904 c^3 x^3\right )-8 a^3 x \left (3 b^3+26 b^2 c x+98 b c^2 x^2-60 c^3 x^3\right )\right )-15 \left (21 b^6-140 a b^4 c+240 a^2 b^2 c^2-64 a^3 c^3\right ) x^4 \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{64 a^{11/2} \left (-b^2+4 a c\right ) x^3 \sqrt {x^2 (a+x (b+c x))}} \] Input:

Integrate[1/(x^2*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x]
 

Output:

(Sqrt[a]*(-64*a^5*c - 315*b^5*x^4*(b + c*x) - 105*a*b^3*x^3*(b^2 - 18*b*c* 
x - 16*c^2*x^2) + 16*a^4*(b^2 + 6*b*c*x + 10*c^2*x^2) + 2*a^2*b*x^2*(21*b^ 
3 + 308*b^2*c*x - 1352*b*c^2*x^2 - 904*c^3*x^3) - 8*a^3*x*(3*b^3 + 26*b^2* 
c*x + 98*b*c^2*x^2 - 60*c^3*x^3)) - 15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2 
*c^2 - 64*a^3*c^3)*x^4*Sqrt[a + x*(b + c*x)]*ArcTanh[(Sqrt[c]*x - Sqrt[a + 
 x*(b + c*x)])/Sqrt[a]])/(64*a^(11/2)*(-b^2 + 4*a*c)*x^3*Sqrt[x^2*(a + x*( 
b + c*x))])
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1971, 27, 1998, 27, 1998, 27, 1998, 27, 1998, 27, 1951, 219}

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.

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

\(\Big \downarrow \) 1971

\(\displaystyle \frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int -\frac {9 b^2+8 c x b-20 a c}{2 x^4 \sqrt {c x^4+b x^3+a x^2}}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {9 b^2+8 c x b-20 a c}{x^4 \sqrt {c x^4+b x^3+a x^2}}dx}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 1998

\(\displaystyle \frac {-\frac {\int \frac {3 \left (b \left (21 b^2-68 a c\right )+2 c \left (9 b^2-20 a c\right ) x\right )}{2 x^3 \sqrt {c x^4+b x^3+a x^2}}dx}{4 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {b \left (21 b^2-68 a c\right )+2 c \left (9 b^2-20 a c\right ) x}{x^3 \sqrt {c x^4+b x^3+a x^2}}dx}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 1998

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {105 b^4-448 a c b^2+4 c \left (21 b^2-68 a c\right ) x b+240 a^2 c^2}{2 x^2 \sqrt {c x^4+b x^3+a x^2}}dx}{3 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {105 b^4-448 a c b^2+4 c \left (21 b^2-68 a c\right ) x b+240 a^2 c^2}{x^2 \sqrt {c x^4+b x^3+a x^2}}dx}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 1998

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\int \frac {b \left (315 b^4-1680 a c b^2+1808 a^2 c^2\right )+2 c \left (105 b^4-448 a c b^2+240 a^2 c^2\right ) x}{2 x \sqrt {c x^4+b x^3+a x^2}}dx}{2 a}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\int \frac {b \left (315 b^4-1680 a c b^2+1808 a^2 c^2\right )+2 c \left (105 b^4-448 a c b^2+240 a^2 c^2\right ) x}{x \sqrt {c x^4+b x^3+a x^2}}dx}{4 a}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 1998

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {-\frac {\int \frac {15 \left (b^2-4 a c\right ) \left (21 b^4-56 a c b^2+16 a^2 c^2\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{a}-\frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {-\frac {15 \left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \int \frac {1}{\sqrt {c x^4+b x^3+a x^2}}dx}{2 a}-\frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 1951

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \int \frac {1}{4 a-\frac {x^2 (2 a+b x)^2}{c x^4+b x^3+a x^2}}d\frac {x (2 a+b x)}{\sqrt {c x^4+b x^3+a x^2}}}{a}-\frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}-\frac {\frac {15 \left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}}{6 a}-\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}\right )}{8 a}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a x^5}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

Input:

Int[1/(x^2*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x]
 

Output:

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*x^3*Sqrt[a*x^2 + b*x^3 + c*x^4] 
) + (-1/4*((9*b^2 - 20*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^5) - (3*(-1/ 
3*(b*(21*b^2 - 68*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^4) - (-1/2*((105* 
b^4 - 448*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^3) - (- 
((b*(315*b^4 - 1680*a*b^2*c + 1808*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/( 
a*x^2)) + (15*(b^2 - 4*a*c)*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*ArcTanh[(x* 
(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(2*a^(3/2)))/(4*a)) 
/(6*a)))/(8*a))/(a*(b^2 - 4*a*c))
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1951

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] : 
> Simp[-2/(n - 2)   Subst[Int[1/(4*a - x^2), x], x, x*((2*a + b*x^(n - 2))/ 
Sqrt[a*x^2 + b*x^n + c*x^r])], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r, 2* 
n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]
 

rule 1971

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[(-x^(m - q + 1))*(b^2 - 2*a*c + b*c*x^(n - q))*((a*x^q 
 + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))), x] + 
Simp[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))   Int[x^(m - q)*(b^2*(m + p*q + (n 
 - q)*(p + 1) + 1) - 2*a*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + b*c*(m + p*q 
 + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1 
), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !Int 
egerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, 
 q] && LtQ[m + p*q + 1, n - q]
 

rule 1998

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ 
.)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp[A*x^(m - q + 1)*((a*x^q + b 
*x^n + c*x^(2*n - q))^(p + 1)/(a*(m + p*q + 1))), x] + Simp[1/(a*(m + p*q + 
 1))   Int[x^(m + n - q)*Simp[a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(p 
 + 1) + 1) - A*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q), x]*(a*x^q + b 
*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - 
q] && EqQ[j, 2*n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] 
 && RationalQ[m, p, q] && ((GeQ[p, -1] && LtQ[p, 0]) || EqQ[m + p*q + (n - 
q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {15 \left (\frac {2 a^{\frac {11}{2}} c}{15}+\frac {7 b^{3} x^{3} \left (-16 c^{2} x^{2}-18 b c x +b^{2}\right ) a^{\frac {3}{2}}}{32}-\frac {7 \left (-\frac {904}{21} c^{3} x^{3}-\frac {1352}{21} b \,c^{2} x^{2}+\frac {44}{3} b^{2} c x +b^{3}\right ) b \,x^{2} a^{\frac {5}{2}}}{80}+\frac {\left (-20 c^{3} x^{3}+\frac {98}{3} b \,c^{2} x^{2}+\frac {26}{3} b^{2} c x +b^{3}\right ) x \,a^{\frac {7}{2}}}{20}+\left (-\frac {1}{5} b c x -\frac {1}{30} b^{2}-\frac {1}{3} c^{2} x^{2}\right ) a^{\frac {9}{2}}+x^{4} \left (\frac {21 b^{5} \left (c x +b \right ) \sqrt {a}}{32}+\sqrt {c \,x^{2}+b x +a}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right ) \left (a^{2} c^{2}-\frac {7}{2} a \,b^{2} c +\frac {21}{16} b^{4}\right ) \left (a c -\frac {b^{2}}{4}\right )\right )\right )}{8 a^{\frac {11}{2}} \sqrt {c \,x^{2}+b x +a}\, x^{4} \left (a c -\frac {b^{2}}{4}\right )}\)	\(258\)
risch	\(-\frac {\left (c \,x^{2}+b x +a \right ) \left (292 a b c \,x^{3}-187 b^{3} x^{3}-56 a^{2} c \,x^{2}+82 a \,b^{2} x^{2}-40 b \,a^{2} x +16 a^{3}\right )}{64 a^{5} x^{3} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}+\frac {\left (c \left (112 a^{2} c^{2}-456 a \,b^{2} c +187 b^{4}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {374 b^{5} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {656 a \,b^{3} c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+15 a \left (16 a^{2} c^{2}-56 a \,b^{2} c +21 b^{4}\right ) \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {288 a^{2} b \,c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right ) x \sqrt {c \,x^{2}+b x +a}}{128 a^{5} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\)	\(417\)
default	\(-\frac {\left (c \,x^{2}+b x +a \right ) \left (3616 a^{\frac {7}{2}} b \,c^{3} x^{5}-3360 a^{\frac {5}{2}} b^{3} c^{2} x^{5}+630 a^{\frac {3}{2}} b^{5} c \,x^{5}+960 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{3} x^{4}-3600 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c^{2} x^{4}+2100 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} c \,x^{4}-315 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{6} x^{4}-960 a^{\frac {9}{2}} c^{3} x^{4}+5408 a^{\frac {7}{2}} b^{2} c^{2} x^{4}-3780 a^{\frac {5}{2}} b^{4} c \,x^{4}+630 a^{\frac {3}{2}} b^{6} x^{4}+1568 a^{\frac {9}{2}} b \,c^{2} x^{3}-1232 a^{\frac {7}{2}} b^{3} c \,x^{3}+210 a^{\frac {5}{2}} b^{5} x^{3}-320 a^{\frac {11}{2}} c^{2} x^{2}+416 a^{\frac {9}{2}} b^{2} c \,x^{2}-84 a^{\frac {7}{2}} b^{4} x^{2}-192 a^{\frac {11}{2}} b c x +48 a^{\frac {9}{2}} b^{3} x +128 a^{\frac {13}{2}} c -32 a^{\frac {11}{2}} b^{2}\right )}{128 x \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {13}{2}} \left (4 a c -b^{2}\right )}\)	\(446\)

Input:

int(1/x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-15/8/a^(11/2)/(c*x^2+b*x+a)^(1/2)*(2/15*a^(11/2)*c+7/32*b^3*x^3*(-16*c^2* 
x^2-18*b*c*x+b^2)*a^(3/2)-7/80*(-904/21*c^3*x^3-1352/21*b*c^2*x^2+44/3*b^2 
*c*x+b^3)*b*x^2*a^(5/2)+1/20*(-20*c^3*x^3+98/3*b*c^2*x^2+26/3*b^2*c*x+b^3) 
*x*a^(7/2)+(-1/5*b*c*x-1/30*b^2-1/3*c^2*x^2)*a^(9/2)+x^4*(21/32*b^5*(c*x+b 
)*a^(1/2)+(c*x^2+b*x+a)^(1/2)*(-ln(2)+ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^ 
(1/2))/x/a^(1/2)))*(a^2*c^2-7/2*a*b^2*c+21/16*b^4)*(a*c-1/4*b^2)))/x^4/(a* 
c-1/4*b^2)
 

Fricas [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.52 \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/256*(15*((21*b^6*c - 140*a*b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*x^7 
+ (21*b^7 - 140*a*b^5*c + 240*a^2*b^3*c^2 - 64*a^3*b*c^3)*x^6 + (21*a*b^6 
- 140*a^2*b^4*c + 240*a^3*b^2*c^2 - 64*a^4*c^3)*x^5)*sqrt(a)*log(-(8*a*b*x 
^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2* 
a)*sqrt(a))/x^3) - 4*(16*a^5*b^2 - 64*a^6*c - (315*a*b^5*c - 1680*a^2*b^3* 
c^2 + 1808*a^3*b*c^3)*x^5 - (315*a*b^6 - 1890*a^2*b^4*c + 2704*a^3*b^2*c^2 
 - 480*a^4*c^3)*x^4 - 7*(15*a^2*b^5 - 88*a^3*b^3*c + 112*a^4*b*c^2)*x^3 + 
2*(21*a^3*b^4 - 104*a^4*b^2*c + 80*a^5*c^2)*x^2 - 24*(a^4*b^3 - 4*a^5*b*c) 
*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/((a^6*b^2*c - 4*a^7*c^2)*x^7 + (a^6*b^3 - 
 4*a^7*b*c)*x^6 + (a^7*b^2 - 4*a^8*c)*x^5), 1/128*(15*((21*b^6*c - 140*a*b 
^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*x^7 + (21*b^7 - 140*a*b^5*c + 240*a 
^2*b^3*c^2 - 64*a^3*b*c^3)*x^6 + (21*a*b^6 - 140*a^2*b^4*c + 240*a^3*b^2*c 
^2 - 64*a^4*c^3)*x^5)*sqrt(-a)*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x 
 + 2*a)*sqrt(-a)/(a*c*x^3 + a*b*x^2 + a^2*x)) - 2*(16*a^5*b^2 - 64*a^6*c - 
 (315*a*b^5*c - 1680*a^2*b^3*c^2 + 1808*a^3*b*c^3)*x^5 - (315*a*b^6 - 1890 
*a^2*b^4*c + 2704*a^3*b^2*c^2 - 480*a^4*c^3)*x^4 - 7*(15*a^2*b^5 - 88*a^3* 
b^3*c + 112*a^4*b*c^2)*x^3 + 2*(21*a^3*b^4 - 104*a^4*b^2*c + 80*a^5*c^2)*x 
^2 - 24*(a^4*b^3 - 4*a^5*b*c)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/((a^6*b^2*c 
- 4*a^7*c^2)*x^7 + (a^6*b^3 - 4*a^7*b*c)*x^6 + (a^7*b^2 - 4*a^8*c)*x^5)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/x**2/(c*x**4+b*x**3+a*x**2)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:

integrate(1/x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^4 + b*x^3 + a*x^2)^(3/2)*x^2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \] Input:

int(1/(x^2*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x)
 

Output:

int(1/(x^2*(a*x^2 + b*x^3 + c*x^4)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 1055, normalized size of antiderivative = 3.08 \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x)
 

Output:

( - 128*sqrt(a + b*x + c*x**2)*a**6*c + 32*sqrt(a + b*x + c*x**2)*a**5*b** 
2 + 192*sqrt(a + b*x + c*x**2)*a**5*b*c*x + 320*sqrt(a + b*x + c*x**2)*a** 
5*c**2*x**2 - 48*sqrt(a + b*x + c*x**2)*a**4*b**3*x - 416*sqrt(a + b*x + c 
*x**2)*a**4*b**2*c*x**2 - 1568*sqrt(a + b*x + c*x**2)*a**4*b*c**2*x**3 + 9 
60*sqrt(a + b*x + c*x**2)*a**4*c**3*x**4 + 84*sqrt(a + b*x + c*x**2)*a**3* 
b**4*x**2 + 1232*sqrt(a + b*x + c*x**2)*a**3*b**3*c*x**3 - 5408*sqrt(a + b 
*x + c*x**2)*a**3*b**2*c**2*x**4 - 3616*sqrt(a + b*x + c*x**2)*a**3*b*c**3 
*x**5 - 210*sqrt(a + b*x + c*x**2)*a**2*b**5*x**3 + 3780*sqrt(a + b*x + c* 
x**2)*a**2*b**4*c*x**4 + 3360*sqrt(a + b*x + c*x**2)*a**2*b**3*c**2*x**5 - 
 630*sqrt(a + b*x + c*x**2)*a*b**6*x**4 - 630*sqrt(a + b*x + c*x**2)*a*b** 
5*c*x**5 + 960*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a 
**4*c**3*x**4 - 3600*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - 
b*x)*a**3*b**2*c**2*x**4 + 960*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2 
) - 2*a - b*x)*a**3*b*c**3*x**5 + 960*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + 
 c*x**2) - 2*a - b*x)*a**3*c**4*x**6 + 2100*sqrt(a)*log(2*sqrt(a)*sqrt(a + 
 b*x + c*x**2) - 2*a - b*x)*a**2*b**4*c*x**4 - 3600*sqrt(a)*log(2*sqrt(a)* 
sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**3*c**2*x**5 - 3600*sqrt(a)*log 
(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**2*c**3*x**6 - 315*s 
qrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**6*x**4 + 210 
0*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**5*c*x*...