\(\int \frac {1}{x^2 (a x^2+b x^3+c x^4)^{3/2}} \, dx\) [64]

Optimal result

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

[Out]

-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)+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+21*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^2-448
*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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1938, 1965, 12, 1918, 212} \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 x^4 \left (b^2-4 a c\right )}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^5 \left (b^2-4 a c\right )}-\frac {15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}}+\frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 x^2 \left (b^2-4 a c\right )}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 x^3 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]

[In]

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

[Out]

(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]) - ((9*b^2 - 20*a*c)*Sqrt[a*x^2 + b
*x^3 + c*x^4])/(4*a^2*(b^2 - 4*a*c)*x^5) + (b*(21*b^2 - 68*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(8*a^3*(b^2 - 4*a
*c)*x^4) - ((105*b^4 - 448*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(32*a^4*(b^2 - 4*a*c)*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])/(64*a^5*(b^2 - 4*a*c)*x^2) - (15*(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])])/(128*a^(11/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1918

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-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 1938

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] + Dist[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] &&  !IntegerQ[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 1965

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> 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] + Dist[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] && Eq
Q[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]

Rubi steps \begin{align*} \text {integral}& = \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 {2 \int \frac {-\frac {9 b^2}{2}+10 a c-4 b c x}{x^4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{a \left (b^2-4 a c\right )} \\ & = \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 {\int \frac {-\frac {3}{4} b \left (21 b^2-68 a c\right )-\frac {3}{2} c \left (9 b^2-20 a c\right ) x}{x^3 \sqrt {a x^2+b x^3+c x^4}} \, dx}{2 a^2 \left (b^2-4 a c\right )} \\ & = \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 {\int \frac {-\frac {3}{8} \left (105 b^4-448 a b^2 c+240 a^2 c^2\right )-\frac {3}{2} b c \left (21 b^2-68 a c\right ) x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{6 a^3 \left (b^2-4 a c\right )} \\ & = \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 {\int \frac {-\frac {3}{16} b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right )-\frac {3}{8} c \left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{12 a^4 \left (b^2-4 a c\right )} \\ & = \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 {\int -\frac {45 \left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )}{32 \sqrt {a x^2+b x^3+c x^4}} \, dx}{12 a^5 \left (b^2-4 a c\right )} \\ & = \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 {\left (15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 a^5} \\ & = \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 {\left (15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{64 a^5} \\ & = \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 ) \tanh ^{-1}\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {a} \left (-64 a^5 c-315 b^5 x^4 (b+c x)-105 a b^3 x^3 \left (b^2-18 b c x-16 c^2 x^2\right )+16 a^4 \left (b^2+6 b c x+10 c^2 x^2\right )+2 a^2 b x^2 \left (21 b^3+308 b^2 c x-1352 b c^2 x^2-904 c^3 x^3\right )-8 a^3 x \left (3 b^3+26 b^2 c x+98 b c^2 x^2-60 c^3 x^3\right )\right )-15 \left (21 b^6-140 a b^4 c+240 a^2 b^2 c^2-64 a^3 c^3\right ) x^4 \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{64 a^{11/2} \left (-b^2+4 a c\right ) x^3 \sqrt {x^2 (a+x (b+c x))}} \]

[In]

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

[Out]

(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))])

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {15 \left (\frac {2 a^{\frac {11}{2}} c}{15}+\frac {7 b^{3} x^{3} \left (-16 c^{2} x^{2}-18 b c x +b^{2}\right ) a^{\frac {3}{2}}}{32}-\frac {7 \left (-\frac {904}{21} c^{3} x^{3}-\frac {1352}{21} b \,c^{2} x^{2}+\frac {44}{3} b^{2} c x +b^{3}\right ) x^{2} b \,a^{\frac {5}{2}}}{80}+\frac {x \left (-20 c^{3} x^{3}+\frac {98}{3} b \,c^{2} x^{2}+\frac {26}{3} b^{2} c x +b^{3}\right ) a^{\frac {7}{2}}}{20}+\left (-\frac {1}{5} b c x -\frac {1}{3} c^{2} x^{2}-\frac {1}{30} b^{2}\right ) a^{\frac {9}{2}}+\left (\frac {21 b^{5} \left (c x +b \right ) \sqrt {a}}{32}+\left (-\ln \left (2\right )+\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right ) \left (a^{2} c^{2}-\frac {7}{2} a \,b^{2} c +\frac {21}{16} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (a c -\frac {b^{2}}{4}\right )\right ) x^{4}\right )}{8 a^{\frac {11}{2}} \sqrt {c \,x^{2}+b x +a}\, x^{4} \left (a c -\frac {b^{2}}{4}\right )}\)	\(258\)
risch	\(-\frac {\left (c \,x^{2}+b x +a \right ) \left (292 a b c \,x^{3}-187 b^{3} x^{3}-56 a^{2} c \,x^{2}+82 a \,b^{2} x^{2}-40 a^{2} b x +16 a^{3}\right )}{64 a^{5} x^{3} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}+\frac {\left (\frac {374 b^{5} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {288 a^{2} b \,c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {656 a \,b^{3} c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (112 a^{2} c^{3}-456 b^{2} a \,c^{2}+187 b^{4} c \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+15 a \left (16 a^{2} c^{2}-56 a \,b^{2} c +21 b^{4}\right ) \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )\right ) x \sqrt {c \,x^{2}+b x +a}}{128 a^{5} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\)	\(419\)
default	\(-\frac {\left (c \,x^{2}+b x +a \right ) \left (3616 a^{\frac {7}{2}} b \,c^{3} x^{5}-3360 a^{\frac {5}{2}} b^{3} c^{2} x^{5}+630 a^{\frac {3}{2}} b^{5} c \,x^{5}-960 a^{\frac {9}{2}} c^{3} x^{4}+5408 a^{\frac {7}{2}} b^{2} c^{2} x^{4}-3780 a^{\frac {5}{2}} b^{4} c \,x^{4}+630 a^{\frac {3}{2}} b^{6} x^{4}+960 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{3} x^{4}-3600 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c^{2} x^{4}+2100 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} c \,x^{4}-315 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{6} x^{4}+1568 a^{\frac {9}{2}} b \,c^{2} x^{3}-1232 a^{\frac {7}{2}} b^{3} c \,x^{3}+210 a^{\frac {5}{2}} b^{5} x^{3}-320 a^{\frac {11}{2}} c^{2} x^{2}+416 a^{\frac {9}{2}} b^{2} c \,x^{2}-84 a^{\frac {7}{2}} b^{4} x^{2}-192 a^{\frac {11}{2}} b c x +48 a^{\frac {9}{2}} b^{3} x +128 a^{\frac {13}{2}} c -32 a^{\frac {11}{2}} b^{2}\right )}{128 x \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {13}{2}} \left (4 a c -b^{2}\right )}\)	\(446\)

[In]

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

[Out]

-15/8/a^(11/2)*(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)*x^2*b*a^(5/2)+1/20*x*(-20*c^3*x^3+98/3*b*c^2*x^2+26/3*b^2*c*x+b^3)*a^(7/2)+(-1/5*b
*c*x-1/3*c^2*x^2-1/30*b^2)*a^(9/2)+(21/32*b^5*(c*x+b)*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)*(c*x^2+b*x+a)^(1/2)*(a*c-1/4*b^2))*x^4)/(c*x^2+b*x+a)^(1/2)/x^4
/(a*c-1/4*b^2)

Fricas [A] (verification not implemented)

none

Time = 0.51 (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=\left [\frac {15 \, {\left ({\left (21 \, b^{6} c - 140 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} x^{7} + {\left (21 \, b^{7} - 140 \, a b^{5} c + 240 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} x^{6} + {\left (21 \, a b^{6} - 140 \, a^{2} b^{4} c + 240 \, a^{3} b^{2} c^{2} - 64 \, a^{4} c^{3}\right )} x^{5}\right )} \sqrt {a} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, {\left (16 \, a^{5} b^{2} - 64 \, a^{6} c - {\left (315 \, a b^{5} c - 1680 \, a^{2} b^{3} c^{2} + 1808 \, a^{3} b c^{3}\right )} x^{5} - {\left (315 \, a b^{6} - 1890 \, a^{2} b^{4} c + 2704 \, a^{3} b^{2} c^{2} - 480 \, a^{4} c^{3}\right )} x^{4} - 7 \, {\left (15 \, a^{2} b^{5} - 88 \, a^{3} b^{3} c + 112 \, a^{4} b c^{2}\right )} x^{3} + 2 \, {\left (21 \, a^{3} b^{4} - 104 \, a^{4} b^{2} c + 80 \, a^{5} c^{2}\right )} x^{2} - 24 \, {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{256 \, {\left ({\left (a^{6} b^{2} c - 4 \, a^{7} c^{2}\right )} x^{7} + {\left (a^{6} b^{3} - 4 \, a^{7} b c\right )} x^{6} + {\left (a^{7} b^{2} - 4 \, a^{8} c\right )} x^{5}\right )}}, \frac {15 \, {\left ({\left (21 \, b^{6} c - 140 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} x^{7} + {\left (21 \, b^{7} - 140 \, a b^{5} c + 240 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} x^{6} + {\left (21 \, a b^{6} - 140 \, a^{2} b^{4} c + 240 \, a^{3} b^{2} c^{2} - 64 \, a^{4} c^{3}\right )} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, {\left (16 \, a^{5} b^{2} - 64 \, a^{6} c - {\left (315 \, a b^{5} c - 1680 \, a^{2} b^{3} c^{2} + 1808 \, a^{3} b c^{3}\right )} x^{5} - {\left (315 \, a b^{6} - 1890 \, a^{2} b^{4} c + 2704 \, a^{3} b^{2} c^{2} - 480 \, a^{4} c^{3}\right )} x^{4} - 7 \, {\left (15 \, a^{2} b^{5} - 88 \, a^{3} b^{3} c + 112 \, a^{4} b c^{2}\right )} x^{3} + 2 \, {\left (21 \, a^{3} b^{4} - 104 \, a^{4} b^{2} c + 80 \, a^{5} c^{2}\right )} x^{2} - 24 \, {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{128 \, {\left ({\left (a^{6} b^{2} c - 4 \, a^{7} c^{2}\right )} x^{7} + {\left (a^{6} b^{3} - 4 \, a^{7} b c\right )} x^{6} + {\left (a^{7} b^{2} - 4 \, a^{8} c\right )} x^{5}\right )}}\right ] \]

[In]

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

[Out]

[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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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