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

Optimal. Leaf size=419 \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 d x^2 \left (b^2-4 a c\right )}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2} d^2}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2} d}-\frac{e^2 \left (2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right )}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{e \left (-2 a c+b^2+b c x^2\right )}{a d^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{-2 a c+b^2+b c x^2}{a d x^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

-((e*(b^2 - 2*a*c + b*c*x^2))/(a*(b^2 - 4*a*c)*d^2*Sqrt[a + b*x^2 + c*x^4])) + (b^2 - 2*a*c + b*c*x^2)/(a*(b^2
- 4*a*c)*d*x^2*Sqrt[a + b*x^2 + c*x^4]) - (e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2))/((b^2 - 4*a*c
)*d^2*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - ((3*b^2 - 8*a*c)*Sqrt[a + b*x^2 + c*x^4])/(2*a^2*(b^2
- 4*a*c)*d*x^2) + (3*b*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(4*a^(5/2)*d) + (e*ArcTanh
[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(2*a^(3/2)*d^2) + (e^4*ArcTanh[(b*d - 2*a*e + (2*c*d - b*
e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(2*d^2*(c*d^2 - b*d*e + a*e^2)^(3/2))

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Rubi [A]  time = 0.564316, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {1251, 960, 740, 806, 724, 206, 12} \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 d x^2 \left (b^2-4 a c\right )}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2} d^2}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2} d}-\frac{e^2 \left (2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right )}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{e \left (-2 a c+b^2+b c x^2\right )}{a d^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{-2 a c+b^2+b c x^2}{a d x^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*(b^2 - 2*a*c + b*c*x^2))/(a*(b^2 - 4*a*c)*d^2*Sqrt[a + b*x^2 + c*x^4])) + (b^2 - 2*a*c + b*c*x^2)/(a*(b^2
- 4*a*c)*d*x^2*Sqrt[a + b*x^2 + c*x^4]) - (e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2))/((b^2 - 4*a*c
)*d^2*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - ((3*b^2 - 8*a*c)*Sqrt[a + b*x^2 + c*x^4])/(2*a^2*(b^2
- 4*a*c)*d*x^2) + (3*b*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(4*a^(5/2)*d) + (e*ArcTanh
[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(2*a^(3/2)*d^2) + (e^4*ArcTanh[(b*d - 2*a*e + (2*c*d - b*
e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(2*d^2*(c*d^2 - b*d*e + a*e^2)^(3/2))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 960

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] &&
ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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])

Rule 12

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

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{d x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{e}{d^2 x \left (a+b x+c x^2\right )^{3/2}}+\frac{e^2}{d^2 (d+e x) \left (a+b x+c x^2\right )^{3/2}}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 d}-\frac{e \operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac{e \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) d^2 \sqrt{a+b x^2+c x^4}}+\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) d x^2 \sqrt{a+b x^2+c x^4}}-\frac{e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x^2\right )}{\left (b^2-4 a c\right ) d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{a \left (b^2-4 a c\right ) d}+\frac{e \operatorname{Subst}\left (\int \frac{-\frac{b^2}{2}+2 a c}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{a \left (b^2-4 a c\right ) d^2}-\frac{e^2 \operatorname{Subst}\left (\int -\frac{\left (b^2-4 a c\right ) e^2}{2 (d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{\left (b^2-4 a c\right ) d^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) d^2 \sqrt{a+b x^2+c x^4}}+\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) d x^2 \sqrt{a+b x^2+c x^4}}-\frac{e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x^2\right )}{\left (b^2-4 a c\right ) d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) d x^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 a^2 d}-\frac{e \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 a d^2}+\frac{e^4 \operatorname{Subst}\left (\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) d^2 \sqrt{a+b x^2+c x^4}}+\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) d x^2 \sqrt{a+b x^2+c x^4}}-\frac{e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x^2\right )}{\left (b^2-4 a c\right ) d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) d x^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{2 a^2 d}+\frac{e \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{a d^2}-\frac{e^4 \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x^2}{\sqrt{a+b x^2+c x^4}}\right )}{d^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) d^2 \sqrt{a+b x^2+c x^4}}+\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) d x^2 \sqrt{a+b x^2+c x^4}}-\frac{e^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x^2\right )}{\left (b^2-4 a c\right ) d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) d x^2}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2} d}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2} d^2}+\frac{e^4 \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x^2}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x^2+c x^4}}\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.63858, size = 350, normalized size = 0.84 \[ \frac{\frac{2 d \left (a^2 \left (b^2 e^2+4 b c e \left (d-e x^2\right )-4 c^2 \left (d^2+d e x^2+e^2 x^4\right )\right )-4 a^3 c e^2+a \left (b^2 c \left (d^2+12 d e x^2+e^2 x^4\right )+b^3 e \left (e x^2-d\right )-10 b c^2 d x^2 \left (d-e x^2\right )-8 c^3 d^2 x^4\right )+3 b^2 d x^2 \left (b+c x^2\right ) (c d-b e)\right )}{a^2 x^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (e (b d-a e)-c d^2\right )}+\frac{(2 a e+3 b d) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{a^{5/2}}+\frac{2 e^4 \tanh ^{-1}\left (\frac{-2 a e+b \left (d-e x^2\right )+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*d*(-4*a^3*c*e^2 + 3*b^2*d*(c*d - b*e)*x^2*(b + c*x^2) + a^2*(b^2*e^2 + 4*b*c*e*(d - e*x^2) - 4*c^2*(d^2 +
d*e*x^2 + e^2*x^4)) + a*(-8*c^3*d^2*x^4 - 10*b*c^2*d*x^2*(d - e*x^2) + b^3*e*(-d + e*x^2) + b^2*c*(d^2 + 12*d*
e*x^2 + e^2*x^4))))/(a^2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*x^2*Sqrt[a + b*x^2 + c*x^4]) + ((3*b*d + 2*a
*e)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/a^(5/2) + (2*e^4*ArcTanh[(-2*a*e + 2*c*d*x^2 +
b*(d - e*x^2))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))
/(4*d^2)

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Maple [B]  time = 0.02, size = 863, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d^2*e/a/(c*x^4+b*x^2+a)^(1/2)+1/d^2*e*b/a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*c*x^2+1/2/d^2*e*b^2/a/(4*a*c-
b^2)/(c*x^4+b*x^2+a)^(1/2)+1/2/d^2*e/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-1/2/d/a/x^2/(
c*x^4+b*x^2+a)^(1/2)-3/4/d*b/a^2/(c*x^4+b*x^2+a)^(1/2)+3/2/d*b^2/a^2/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*c*x^2+3
/4/d*b^3/a^2/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)+3/4/d*b/a^(5/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/
x^2)-4/d*c^2/a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*x^2-2/d*c/a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*b-2*e^2/d^2*c/(
e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)/(-4*a*c+b^2)/(x^2+1/2*b/c-1/2*(-4*a*c+b^2)^(1/2)/c)*(c*(x^2-1/2*(-b+(-4*a*c+b^
2)^(1/2))/c)^2+(-4*a*c+b^2)^(1/2)*(x^2-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))^(1/2)+2*e^3/d^2*c/(e*(-4*a*c+b^2)^(1/2)
-b*e+2*c*d)/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*
e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d
^2)/e^2)^(1/2))/(x^2+d/e))+2*e^2/d^2*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(-4*a*c+b^2)/(x^2+1/2*(-4*a*c+b^2)^(1/
2)/c+1/2*b/c)*(c*(x^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2-(-4*a*c+b^2)^(1/2)*(x^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(
1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 55.3532, size = 13005, normalized size = 31.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*((a^3*b^2*c - 4*a^4*c^2)*e^4*x^6 + (a^3*b^3 - 4*a^4*b*c)*e^4*x^4 + (a^4*b^2 - 4*a^5*c)*e^4*x^2)*sqrt(c
*d^2 - b*d*e + a*e^2)*log(-((8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4
*a*c)*d^2 + 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(c*d^2 - b*d*e
+ a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e))/(e^2*x^4 + 2*d*e*x^2 + d^2)) + ((3*(b^3*c^3 - 4*a*b*c^4)*d^5 - 2*
(3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^4*e + (3*b^5*c - 10*a*b^3*c^2 - 8*a^2*b*c^3)*d^3*e^2 - 4*(a*b^4*c - 5
*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^3 - (a^2*b^3*c - 4*a^3*b*c^2)*d*e^4 + 2*(a^3*b^2*c - 4*a^4*c^2)*e^5)*x^6 + (3*
(b^4*c^2 - 4*a*b^2*c^3)*d^5 - 2*(3*b^5*c - 13*a*b^3*c^2 + 4*a^2*b*c^3)*d^4*e + (3*b^6 - 10*a*b^4*c - 8*a^2*b^2
*c^2)*d^3*e^2 - 4*(a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^3 - (a^2*b^4 - 4*a^3*b^2*c)*d*e^4 + 2*(a^3*b^3 - 4
*a^4*b*c)*e^5)*x^4 + (3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5 - 2*(3*a*b^4*c - 13*a^2*b^2*c^2 + 4*a^3*c^3)*d^4*e + (3*
a*b^5 - 10*a^2*b^3*c - 8*a^3*b*c^2)*d^3*e^2 - 4*(a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*d^2*e^3 - (a^3*b^3 - 4*a^4
*b*c)*d*e^4 + 2*(a^4*b^2 - 4*a^5*c)*e^5)*x^2)*sqrt(a)*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 4*sqrt(c*x^4 + b*x
^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*((a^2*b^2*c^2 - 4*a^3*c^3)*d^5 - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d
^4*e + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^3*e^2 - 2*(a^3*b^3 - 4*a^4*b*c)*d^2*e^3 + (a^4*b^2 - 4*a^5*c)*d*e
^4 + ((3*a*b^2*c^3 - 8*a^2*c^4)*d^5 - 6*(a*b^3*c^2 - 3*a^2*b*c^3)*d^4*e + 3*(a*b^4*c - 2*a^2*b^2*c^2 - 4*a^3*c
^3)*d^3*e^2 - 2*(2*a^2*b^3*c - 7*a^3*b*c^2)*d^2*e^3 + (a^3*b^2*c - 4*a^4*c^2)*d*e^4)*x^4 + ((3*a*b^3*c^2 - 10*
a^2*b*c^3)*d^5 - 2*(3*a*b^4*c - 11*a^2*b^2*c^2 + 2*a^3*c^3)*d^4*e + (3*a*b^5 - 8*a^2*b^3*c - 10*a^3*b*c^2)*d^3
*e^2 - 4*(a^2*b^4 - 4*a^3*b^2*c + a^4*c^2)*d^2*e^3 + (a^3*b^3 - 4*a^4*b*c)*d*e^4)*x^2)*sqrt(c*x^4 + b*x^2 + a)
)/(((a^3*b^2*c^3 - 4*a^4*c^4)*d^6 - 2*(a^3*b^3*c^2 - 4*a^4*b*c^3)*d^5*e + (a^3*b^4*c - 2*a^4*b^2*c^2 - 8*a^5*c
^3)*d^4*e^2 - 2*(a^4*b^3*c - 4*a^5*b*c^2)*d^3*e^3 + (a^5*b^2*c - 4*a^6*c^2)*d^2*e^4)*x^6 + ((a^3*b^3*c^2 - 4*a
^4*b*c^3)*d^6 - 2*(a^3*b^4*c - 4*a^4*b^2*c^2)*d^5*e + (a^3*b^5 - 2*a^4*b^3*c - 8*a^5*b*c^2)*d^4*e^2 - 2*(a^4*b
^4 - 4*a^5*b^2*c)*d^3*e^3 + (a^5*b^3 - 4*a^6*b*c)*d^2*e^4)*x^4 + ((a^4*b^2*c^2 - 4*a^5*c^3)*d^6 - 2*(a^4*b^3*c
- 4*a^5*b*c^2)*d^5*e + (a^4*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*d^4*e^2 - 2*(a^5*b^3 - 4*a^6*b*c)*d^3*e^3 + (a^6*b
^2 - 4*a^7*c)*d^2*e^4)*x^2), 1/8*(4*((a^3*b^2*c - 4*a^4*c^2)*e^4*x^6 + (a^3*b^3 - 4*a^4*b*c)*e^4*x^4 + (a^4*b^
2 - 4*a^5*c)*e^4*x^2)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c*d^2 + b*d*e - a
*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e)/((c^2*d^2 - b*c*d*e + a*c*e^2)*x^4 + a*c*d^2 - a*b*d*e + a^2*e^2 + (b*
c*d^2 - b^2*d*e + a*b*e^2)*x^2)) + ((3*(b^3*c^3 - 4*a*b*c^4)*d^5 - 2*(3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^
4*e + (3*b^5*c - 10*a*b^3*c^2 - 8*a^2*b*c^3)*d^3*e^2 - 4*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^3 - (a^2*
b^3*c - 4*a^3*b*c^2)*d*e^4 + 2*(a^3*b^2*c - 4*a^4*c^2)*e^5)*x^6 + (3*(b^4*c^2 - 4*a*b^2*c^3)*d^5 - 2*(3*b^5*c
- 13*a*b^3*c^2 + 4*a^2*b*c^3)*d^4*e + (3*b^6 - 10*a*b^4*c - 8*a^2*b^2*c^2)*d^3*e^2 - 4*(a*b^5 - 5*a^2*b^3*c +
4*a^3*b*c^2)*d^2*e^3 - (a^2*b^4 - 4*a^3*b^2*c)*d*e^4 + 2*(a^3*b^3 - 4*a^4*b*c)*e^5)*x^4 + (3*(a*b^3*c^2 - 4*a^
2*b*c^3)*d^5 - 2*(3*a*b^4*c - 13*a^2*b^2*c^2 + 4*a^3*c^3)*d^4*e + (3*a*b^5 - 10*a^2*b^3*c - 8*a^3*b*c^2)*d^3*e
^2 - 4*(a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*d^2*e^3 - (a^3*b^3 - 4*a^4*b*c)*d*e^4 + 2*(a^4*b^2 - 4*a^5*c)*e^5)*
x^2)*sqrt(a)*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^
4) - 4*((a^2*b^2*c^2 - 4*a^3*c^3)*d^5 - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^4*e + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2
)*d^3*e^2 - 2*(a^3*b^3 - 4*a^4*b*c)*d^2*e^3 + (a^4*b^2 - 4*a^5*c)*d*e^4 + ((3*a*b^2*c^3 - 8*a^2*c^4)*d^5 - 6*(
a*b^3*c^2 - 3*a^2*b*c^3)*d^4*e + 3*(a*b^4*c - 2*a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^2 - 2*(2*a^2*b^3*c - 7*a^3*b*c^
2)*d^2*e^3 + (a^3*b^2*c - 4*a^4*c^2)*d*e^4)*x^4 + ((3*a*b^3*c^2 - 10*a^2*b*c^3)*d^5 - 2*(3*a*b^4*c - 11*a^2*b^
2*c^2 + 2*a^3*c^3)*d^4*e + (3*a*b^5 - 8*a^2*b^3*c - 10*a^3*b*c^2)*d^3*e^2 - 4*(a^2*b^4 - 4*a^3*b^2*c + a^4*c^2
)*d^2*e^3 + (a^3*b^3 - 4*a^4*b*c)*d*e^4)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(((a^3*b^2*c^3 - 4*a^4*c^4)*d^6 - 2*(a^
3*b^3*c^2 - 4*a^4*b*c^3)*d^5*e + (a^3*b^4*c - 2*a^4*b^2*c^2 - 8*a^5*c^3)*d^4*e^2 - 2*(a^4*b^3*c - 4*a^5*b*c^2)
*d^3*e^3 + (a^5*b^2*c - 4*a^6*c^2)*d^2*e^4)*x^6 + ((a^3*b^3*c^2 - 4*a^4*b*c^3)*d^6 - 2*(a^3*b^4*c - 4*a^4*b^2*
c^2)*d^5*e + (a^3*b^5 - 2*a^4*b^3*c - 8*a^5*b*c^2)*d^4*e^2 - 2*(a^4*b^4 - 4*a^5*b^2*c)*d^3*e^3 + (a^5*b^3 - 4*
a^6*b*c)*d^2*e^4)*x^4 + ((a^4*b^2*c^2 - 4*a^5*c^3)*d^6 - 2*(a^4*b^3*c - 4*a^5*b*c^2)*d^5*e + (a^4*b^4 - 2*a^5*
b^2*c - 8*a^6*c^2)*d^4*e^2 - 2*(a^5*b^3 - 4*a^6*b*c)*d^3*e^3 + (a^6*b^2 - 4*a^7*c)*d^2*e^4)*x^2), -1/4*(((3*(b
^3*c^3 - 4*a*b*c^4)*d^5 - 2*(3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^4*e + (3*b^5*c - 10*a*b^3*c^2 - 8*a^2*b*c
^3)*d^3*e^2 - 4*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^3 - (a^2*b^3*c - 4*a^3*b*c^2)*d*e^4 + 2*(a^3*b^2*c
- 4*a^4*c^2)*e^5)*x^6 + (3*(b^4*c^2 - 4*a*b^2*c^3)*d^5 - 2*(3*b^5*c - 13*a*b^3*c^2 + 4*a^2*b*c^3)*d^4*e + (3*
b^6 - 10*a*b^4*c - 8*a^2*b^2*c^2)*d^3*e^2 - 4*(a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^3 - (a^2*b^4 - 4*a^3*b
^2*c)*d*e^4 + 2*(a^3*b^3 - 4*a^4*b*c)*e^5)*x^4 + (3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5 - 2*(3*a*b^4*c - 13*a^2*b^2*
c^2 + 4*a^3*c^3)*d^4*e + (3*a*b^5 - 10*a^2*b^3*c - 8*a^3*b*c^2)*d^3*e^2 - 4*(a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2
)*d^2*e^3 - (a^3*b^3 - 4*a^4*b*c)*d*e^4 + 2*(a^4*b^2 - 4*a^5*c)*e^5)*x^2)*sqrt(-a)*arctan(1/2*sqrt(c*x^4 + b*x
^2 + a)*(b*x^2 + 2*a)*sqrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) - ((a^3*b^2*c - 4*a^4*c^2)*e^4*x^6 + (a^3*b^3 - 4*a^
4*b*c)*e^4*x^4 + (a^4*b^2 - 4*a^5*c)*e^4*x^2)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-((8*c^2*d^2 - 8*b*c*d*e + (b^2
+ 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e
)*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(c*d^2 - b*d*e + a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e))/(e^2*x^4 + 2*
d*e*x^2 + d^2)) + 2*((a^2*b^2*c^2 - 4*a^3*c^3)*d^5 - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^4*e + (a^2*b^4 - 2*a^3*b^2*
c - 8*a^4*c^2)*d^3*e^2 - 2*(a^3*b^3 - 4*a^4*b*c)*d^2*e^3 + (a^4*b^2 - 4*a^5*c)*d*e^4 + ((3*a*b^2*c^3 - 8*a^2*c
^4)*d^5 - 6*(a*b^3*c^2 - 3*a^2*b*c^3)*d^4*e + 3*(a*b^4*c - 2*a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^2 - 2*(2*a^2*b^3*c
- 7*a^3*b*c^2)*d^2*e^3 + (a^3*b^2*c - 4*a^4*c^2)*d*e^4)*x^4 + ((3*a*b^3*c^2 - 10*a^2*b*c^3)*d^5 - 2*(3*a*b^4*
c - 11*a^2*b^2*c^2 + 2*a^3*c^3)*d^4*e + (3*a*b^5 - 8*a^2*b^3*c - 10*a^3*b*c^2)*d^3*e^2 - 4*(a^2*b^4 - 4*a^3*b^
2*c + a^4*c^2)*d^2*e^3 + (a^3*b^3 - 4*a^4*b*c)*d*e^4)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(((a^3*b^2*c^3 - 4*a^4*c^4
)*d^6 - 2*(a^3*b^3*c^2 - 4*a^4*b*c^3)*d^5*e + (a^3*b^4*c - 2*a^4*b^2*c^2 - 8*a^5*c^3)*d^4*e^2 - 2*(a^4*b^3*c -
4*a^5*b*c^2)*d^3*e^3 + (a^5*b^2*c - 4*a^6*c^2)*d^2*e^4)*x^6 + ((a^3*b^3*c^2 - 4*a^4*b*c^3)*d^6 - 2*(a^3*b^4*c
- 4*a^4*b^2*c^2)*d^5*e + (a^3*b^5 - 2*a^4*b^3*c - 8*a^5*b*c^2)*d^4*e^2 - 2*(a^4*b^4 - 4*a^5*b^2*c)*d^3*e^3 +
(a^5*b^3 - 4*a^6*b*c)*d^2*e^4)*x^4 + ((a^4*b^2*c^2 - 4*a^5*c^3)*d^6 - 2*(a^4*b^3*c - 4*a^5*b*c^2)*d^5*e + (a^4
*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*d^4*e^2 - 2*(a^5*b^3 - 4*a^6*b*c)*d^3*e^3 + (a^6*b^2 - 4*a^7*c)*d^2*e^4)*x^2),
1/4*(2*((a^3*b^2*c - 4*a^4*c^2)*e^4*x^6 + (a^3*b^3 - 4*a^4*b*c)*e^4*x^4 + (a^4*b^2 - 4*a^5*c)*e^4*x^2)*sqrt(-
c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c*d^2 + b*d*e - a*e^2)*((2*c*d - b*e)*x^2 + b
*d - 2*a*e)/((c^2*d^2 - b*c*d*e + a*c*e^2)*x^4 + a*c*d^2 - a*b*d*e + a^2*e^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x
^2)) - ((3*(b^3*c^3 - 4*a*b*c^4)*d^5 - 2*(3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^4*e + (3*b^5*c - 10*a*b^3*c^
2 - 8*a^2*b*c^3)*d^3*e^2 - 4*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^3 - (a^2*b^3*c - 4*a^3*b*c^2)*d*e^4 +
2*(a^3*b^2*c - 4*a^4*c^2)*e^5)*x^6 + (3*(b^4*c^2 - 4*a*b^2*c^3)*d^5 - 2*(3*b^5*c - 13*a*b^3*c^2 + 4*a^2*b*c^3
)*d^4*e + (3*b^6 - 10*a*b^4*c - 8*a^2*b^2*c^2)*d^3*e^2 - 4*(a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^3 - (a^2*
b^4 - 4*a^3*b^2*c)*d*e^4 + 2*(a^3*b^3 - 4*a^4*b*c)*e^5)*x^4 + (3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5 - 2*(3*a*b^4*c
- 13*a^2*b^2*c^2 + 4*a^3*c^3)*d^4*e + (3*a*b^5 - 10*a^2*b^3*c - 8*a^3*b*c^2)*d^3*e^2 - 4*(a^2*b^4 - 5*a^3*b^2*
c + 4*a^4*c^2)*d^2*e^3 - (a^3*b^3 - 4*a^4*b*c)*d*e^4 + 2*(a^4*b^2 - 4*a^5*c)*e^5)*x^2)*sqrt(-a)*arctan(1/2*sqr
t(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) - 2*((a^2*b^2*c^2 - 4*a^3*c^3)*d^5 - 2*
(a^2*b^3*c - 4*a^3*b*c^2)*d^4*e + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^3*e^2 - 2*(a^3*b^3 - 4*a^4*b*c)*d^2*e^
3 + (a^4*b^2 - 4*a^5*c)*d*e^4 + ((3*a*b^2*c^3 - 8*a^2*c^4)*d^5 - 6*(a*b^3*c^2 - 3*a^2*b*c^3)*d^4*e + 3*(a*b^4*
c - 2*a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^2 - 2*(2*a^2*b^3*c - 7*a^3*b*c^2)*d^2*e^3 + (a^3*b^2*c - 4*a^4*c^2)*d*e^4
)*x^4 + ((3*a*b^3*c^2 - 10*a^2*b*c^3)*d^5 - 2*(3*a*b^4*c - 11*a^2*b^2*c^2 + 2*a^3*c^3)*d^4*e + (3*a*b^5 - 8*a^
2*b^3*c - 10*a^3*b*c^2)*d^3*e^2 - 4*(a^2*b^4 - 4*a^3*b^2*c + a^4*c^2)*d^2*e^3 + (a^3*b^3 - 4*a^4*b*c)*d*e^4)*x
^2)*sqrt(c*x^4 + b*x^2 + a))/(((a^3*b^2*c^3 - 4*a^4*c^4)*d^6 - 2*(a^3*b^3*c^2 - 4*a^4*b*c^3)*d^5*e + (a^3*b^4*
c - 2*a^4*b^2*c^2 - 8*a^5*c^3)*d^4*e^2 - 2*(a^4*b^3*c - 4*a^5*b*c^2)*d^3*e^3 + (a^5*b^2*c - 4*a^6*c^2)*d^2*e^4
)*x^6 + ((a^3*b^3*c^2 - 4*a^4*b*c^3)*d^6 - 2*(a^3*b^4*c - 4*a^4*b^2*c^2)*d^5*e + (a^3*b^5 - 2*a^4*b^3*c - 8*a^
5*b*c^2)*d^4*e^2 - 2*(a^4*b^4 - 4*a^5*b^2*c)*d^3*e^3 + (a^5*b^3 - 4*a^6*b*c)*d^2*e^4)*x^4 + ((a^4*b^2*c^2 - 4*
a^5*c^3)*d^6 - 2*(a^4*b^3*c - 4*a^5*b*c^2)*d^5*e + (a^4*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*d^4*e^2 - 2*(a^5*b^3 -
4*a^6*b*c)*d^3*e^3 + (a^6*b^2 - 4*a^7*c)*d^2*e^4)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**3*(d + e*x**2)*(a + b*x**2 + c*x**4)**(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^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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