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

Optimal. Leaf size=343 \[ \frac{b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^5 x^2 \left (b^2-4 a c\right )}-\frac{\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a^4 x^3 \left (b^2-4 a c\right )}-\frac{15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}}+\frac{b \left (21 b^2-68 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{8 a^3 x^4 \left (b^2-4 a c\right )}-\frac{\left (9 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^2 x^5 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]

[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))

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Rubi [A]  time = 0.620551, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1924, 1951, 12, 1904, 206} \[ \frac{b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^5 x^2 \left (b^2-4 a c\right )}-\frac{\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a^4 x^3 \left (b^2-4 a c\right )}-\frac{15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}}+\frac{b \left (21 b^2-68 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{8 a^3 x^4 \left (b^2-4 a c\right )}-\frac{\left (9 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^2 x^5 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

[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 1924

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 1951

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]

Rule 12

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

Rule 1904

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 206

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

Rubi steps

\begin{align*} \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{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 ) \operatorname{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]  time = 0.286878, size = 272, normalized size = 0.79 \[ \frac{15 x^4 \left (240 a^2 b^2 c^2-64 a^3 c^3-140 a b^4 c+21 b^6\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} \left (-16 a^4 \left (b^2+6 b c x+10 c^2 x^2\right )+8 a^3 x \left (26 b^2 c x+3 b^3+98 b c^2 x^2-60 c^3 x^3\right )+2 a^2 b x^2 \left (-308 b^2 c x-21 b^3+1352 b c^2 x^2+904 c^3 x^3\right )+64 a^5 c+105 a b^3 x^3 \left (b^2-18 b c x-16 c^2 x^2\right )+315 b^5 x^4 (b+c x)\right )}{128 a^{11/2} x^3 \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*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) + 8*a^3*x*(3*b^3 + 26*b^2*c*x + 98*b*c^2*x^2 - 60*c^3*x^3) + 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)) + 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[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(128*a^(11/2)*(-b^2 + 4*a*c)*x^3*Sqrt[x^
2*(a + x*(b + c*x))])

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Maple [A]  time = 0.008, size = 446, normalized size = 1.3 \begin{align*} -{\frac{c{x}^{2}+bx+a}{128\,x \left ( 4\,ac-{b}^{2} \right ) } \left ( 3616\,{a}^{7/2}{x}^{5}b{c}^{3}-3360\,{a}^{5/2}{x}^{5}{b}^{3}{c}^{2}+630\,{a}^{3/2}{x}^{5}{b}^{5}c-960\,{a}^{9/2}{x}^{4}{c}^{3}+5408\,{a}^{7/2}{x}^{4}{b}^{2}{c}^{2}-3780\,{a}^{5/2}{x}^{4}{b}^{4}c+630\,{a}^{3/2}{x}^{4}{b}^{6}+960\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{4}{c}^{3}-3600\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{3}{b}^{2}{c}^{2}+2100\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{2}{b}^{4}c-315\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{4}a{b}^{6}+1568\,{a}^{9/2}{x}^{3}b{c}^{2}-1232\,{a}^{7/2}{x}^{3}{b}^{3}c+210\,{a}^{5/2}{x}^{3}{b}^{5}-320\,{a}^{11/2}{x}^{2}{c}^{2}+416\,{a}^{9/2}{x}^{2}{b}^{2}c-84\,{a}^{7/2}{x}^{2}{b}^{4}-192\,{a}^{11/2}xbc+48\,{a}^{9/2}x{b}^{3}+128\,{a}^{13/2}c-32\,{a}^{11/2}{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128/x*(c*x^2+b*x+a)*(3616*a^(7/2)*x^5*b*c^3-3360*a^(5/2)*x^5*b^3*c^2+630*a^(3/2)*x^5*b^5*c-960*a^(9/2)*x^4*
c^3+5408*a^(7/2)*x^4*b^2*c^2-3780*a^(5/2)*x^4*b^4*c+630*a^(3/2)*x^4*b^6+960*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a
)^(1/2))/x)*(c*x^2+b*x+a)^(1/2)*x^4*a^4*c^3-3600*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*(c*x^2+b*x+a)^(
1/2)*x^4*a^3*b^2*c^2+2100*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*(c*x^2+b*x+a)^(1/2)*x^4*a^2*b^4*c-315*
ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*(c*x^2+b*x+a)^(1/2)*x^4*a*b^6+1568*a^(9/2)*x^3*b*c^2-1232*a^(7/2
)*x^3*b^3*c+210*a^(5/2)*x^3*b^5-320*a^(11/2)*x^2*c^2+416*a^(9/2)*x^2*b^2*c-84*a^(7/2)*x^2*b^4-192*a^(11/2)*x*b
*c+48*a^(9/2)*x*b^3+128*a^(13/2)*c-32*a^(11/2)*b^2)/(c*x^4+b*x^3+a*x^2)^(3/2)/a^(13/2)/(4*a*c-b^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 4.08513, size = 1901, normalized size = 5.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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