3.782 \(\int \frac{1}{x^2 (a+b x^2)^2 (c+d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=279 \[ -\frac{\sqrt{c+d x^2} \left (40 a^2 b c d^2-16 a^3 d^3-18 a b^2 c^2 d+9 b^3 c^3\right )}{6 a^2 c^3 x (b c-a d)^3}+\frac{d \left (-8 a^2 d^2+20 a b c d+3 b^2 c^2\right )}{6 a c^2 x \sqrt{c+d x^2} (b c-a d)^3}-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}}+\frac{b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)^(
3/2)) + (d*(3*b^2*c^2 + 20*a*b*c*d - 8*a^2*d^2))/(6*a*c^2*(b*c - a*d)^3*x*Sqrt[c + d*x^2]) - ((9*b^3*c^3 - 18*
a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^2*c^3*(b*c - a*d)^3*x) - (b^3*(3*b*c - 8*a*d)
*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.443202, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {472, 579, 583, 12, 377, 205} \[ -\frac{\sqrt{c+d x^2} \left (40 a^2 b c d^2-16 a^3 d^3-18 a b^2 c^2 d+9 b^3 c^3\right )}{6 a^2 c^3 x (b c-a d)^3}+\frac{d \left (-8 a^2 d^2+20 a b c d+3 b^2 c^2\right )}{6 a c^2 x \sqrt{c+d x^2} (b c-a d)^3}-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}}+\frac{b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)^(
3/2)) + (d*(3*b^2*c^2 + 20*a*b*c*d - 8*a^2*d^2))/(6*a*c^2*(b*c - a*d)^3*x*Sqrt[c + d*x^2]) - ((9*b^3*c^3 - 18*
a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^2*c^3*(b*c - a*d)^3*x) - (b^3*(3*b*c - 8*a*d)
*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(7/2))

Rule 472

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

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 12

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

Rule 377

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{-3 b c+2 a d-6 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{2 a (b c-a d)}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{-9 b^2 c^2+12 a b c d-8 a^2 d^2-4 b d (3 b c+2 a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{6 a c (b c-a d)^2}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 x \sqrt{c+d x^2}}-\frac{\int \frac{-9 b^3 c^3+18 a b^2 c^2 d-40 a^2 b c d^2+16 a^3 d^3-2 b d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{6 a c^2 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 x \sqrt{c+d x^2}}-\frac{\left (9 b^3 c^3-18 a b^2 c^2 d+40 a^2 b c d^2-16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x}+\frac{\int -\frac{3 b^3 c^3 (3 b c-8 a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{6 a^2 c^3 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 x \sqrt{c+d x^2}}-\frac{\left (9 b^3 c^3-18 a b^2 c^2 d+40 a^2 b c d^2-16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x}-\frac{\left (b^3 (3 b c-8 a d)\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 a^2 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 x \sqrt{c+d x^2}}-\frac{\left (9 b^3 c^3-18 a b^2 c^2 d+40 a^2 b c d^2-16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x}-\frac{\left (b^3 (3 b c-8 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{2 a^2 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 x \sqrt{c+d x^2}}-\frac{\left (9 b^3 c^3-18 a b^2 c^2 d+40 a^2 b c d^2-16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x}-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 5.53379, size = 188, normalized size = 0.67 \[ \sqrt{c+d x^2} \left (\frac{b^4 x}{2 a^2 \left (a+b x^2\right ) (a d-b c)^3}-\frac{1}{a^2 c^3 x}+\frac{d^3 x (5 a d-11 b c)}{3 c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3 x}{3 c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right )-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

Sqrt[c + d*x^2]*(-(1/(a^2*c^3*x)) + (b^4*x)/(2*a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (d^3*x)/(3*c^2*(b*c - a*d)^
2*(c + d*x^2)^2) + (d^3*(-11*b*c + 5*a*d)*x)/(3*c^3*(b*c - a*d)^3*(c + d*x^2))) - (b^3*(3*b*c - 8*a*d)*ArcTan[
(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(7/2))

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Maple [B]  time = 0.018, size = 2513, normalized size = 9. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

1/12/a^2*d/(a*d-b*c)*b/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*
x+5/4/a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*
d-b*c)/b)^(1/2)-3/4*b^3/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/
b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2)
)-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))-8/3/a^2*d/c^3*x/(d*x^2+c)^(1/2)+1/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)
/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-3/4*b^3/a^2/(-a*b)^(1/2)
/(a*d-b*c)^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/4*b^2/a^2/
(-a*b)^(1/2)/(a*d-b*c)/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+3/
4*b^3/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)
/b)^(1/2)+1/4/a^2/(a*d-b*c)*b/(x+1/b*(-a*b)^(1/2))/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^
(1/2))-(a*d-b*c)/b)^(3/2)-1/a^2/c/x/(d*x^2+c)^(3/2)+3/4*b^3/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*
ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))-5/12/a^2*d*(-a*b)^(1/2)/(a*d-b
*c)^2*b/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+5/12/a^2*d*(-a*b)
^(1/2)/(a*d-b*c)^2*b/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-5/4/
a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)
/b)^(1/2)+5/4/a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x
+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a
*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))+1/12*b/a^2*d/(a*d-b*c)/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*
(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+1/6*b/a^2*d/(a*d-b*c)/c^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)
/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+3/4*b^2/a^2/(a*d-b*c)^2/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-5/12/a*d^2*b/(a*d-b*c)^2/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a
*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-5/6/a*d^2*b/(a*d-b*c)^2/c^2/((x-1/b*(-a*b)^(1/2))^2*d+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+5/4/a*d^2*b^2/(a*d-b*c)^3/c/((x-1/b*(-a*b)^(1/2))^2
*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-5/4/a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^2/(-(a*d-
b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)
^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))+3/4*b^2/a^2/(a*d
-b*c)^2/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-5/12/a*d^2*
b/(a*d-b*c)^2/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-5/6/a*d
^2*b/(a*d-b*c)^2/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+5/
4/a*d^2*b^2/(a*d-b*c)^3/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)
*x+1/6/a^2*d/(a*d-b*c)*b/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1
/2)*x+1/4/a^2/(a*d-b*c)*b/(x-1/b*(-a*b)^(1/2))/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2
))-(a*d-b*c)/b)^(3/2)-4/3/a^2*d/c^2*x/(d*x^2+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^2), x)

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Fricas [B]  time = 10.4207, size = 3333, normalized size = 11.95 \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/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/24*(3*((3*b^5*c^4*d^2 - 8*a*b^4*c^3*d^3)*x^7 + (6*b^5*c^5*d - 13*a*b^4*c^4*d^2 - 8*a^2*b^3*c^3*d^3)*x^5 +
(3*b^5*c^6 - 2*a*b^4*c^5*d - 16*a^2*b^3*c^4*d^2)*x^3 + (3*a*b^4*c^6 - 8*a^2*b^3*c^5*d)*x)*sqrt(-a*b*c + a^2*d)
*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*((b*c - 2*a*d)*x^3 -
a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(6*a^2*b^4*c^6 - 24*a^3*b^3*c^5
*d + 36*a^4*b^2*c^4*d^2 - 24*a^5*b*c^3*d^3 + 6*a^6*c^2*d^4 + (9*a*b^5*c^4*d^2 - 27*a^2*b^4*c^3*d^3 + 58*a^3*b^
3*c^2*d^4 - 56*a^4*b^2*c*d^5 + 16*a^5*b*d^6)*x^6 + 2*(9*a*b^5*c^5*d - 24*a^2*b^4*c^4*d^2 + 36*a^3*b^3*c^3*d^3
- 13*a^4*b^2*c^2*d^4 - 16*a^5*b*c*d^5 + 8*a^6*d^6)*x^4 + 3*(3*a*b^5*c^6 - 5*a^2*b^4*c^5*d - 4*a^3*b^3*c^4*d^2
+ 24*a^4*b^2*c^3*d^3 - 26*a^5*b*c^2*d^4 + 8*a^6*c*d^5)*x^2)*sqrt(d*x^2 + c))/((a^3*b^5*c^7*d^2 - 4*a^4*b^4*c^6
*d^3 + 6*a^5*b^3*c^5*d^4 - 4*a^6*b^2*c^4*d^5 + a^7*b*c^3*d^6)*x^7 + (2*a^3*b^5*c^8*d - 7*a^4*b^4*c^7*d^2 + 8*a
^5*b^3*c^6*d^3 - 2*a^6*b^2*c^5*d^4 - 2*a^7*b*c^4*d^5 + a^8*c^3*d^6)*x^5 + (a^3*b^5*c^9 - 2*a^4*b^4*c^8*d - 2*a
^5*b^3*c^7*d^2 + 8*a^6*b^2*c^6*d^3 - 7*a^7*b*c^5*d^4 + 2*a^8*c^4*d^5)*x^3 + (a^4*b^4*c^9 - 4*a^5*b^3*c^8*d + 6
*a^6*b^2*c^7*d^2 - 4*a^7*b*c^6*d^3 + a^8*c^5*d^4)*x), -1/12*(3*((3*b^5*c^4*d^2 - 8*a*b^4*c^3*d^3)*x^7 + (6*b^5
*c^5*d - 13*a*b^4*c^4*d^2 - 8*a^2*b^3*c^3*d^3)*x^5 + (3*b^5*c^6 - 2*a*b^4*c^5*d - 16*a^2*b^3*c^4*d^2)*x^3 + (3
*a*b^4*c^6 - 8*a^2*b^3*c^5*d)*x)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*
sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*(6*a^2*b^4*c^6 - 24*a^3*b^3*c^5*d + 36*
a^4*b^2*c^4*d^2 - 24*a^5*b*c^3*d^3 + 6*a^6*c^2*d^4 + (9*a*b^5*c^4*d^2 - 27*a^2*b^4*c^3*d^3 + 58*a^3*b^3*c^2*d^
4 - 56*a^4*b^2*c*d^5 + 16*a^5*b*d^6)*x^6 + 2*(9*a*b^5*c^5*d - 24*a^2*b^4*c^4*d^2 + 36*a^3*b^3*c^3*d^3 - 13*a^4
*b^2*c^2*d^4 - 16*a^5*b*c*d^5 + 8*a^6*d^6)*x^4 + 3*(3*a*b^5*c^6 - 5*a^2*b^4*c^5*d - 4*a^3*b^3*c^4*d^2 + 24*a^4
*b^2*c^3*d^3 - 26*a^5*b*c^2*d^4 + 8*a^6*c*d^5)*x^2)*sqrt(d*x^2 + c))/((a^3*b^5*c^7*d^2 - 4*a^4*b^4*c^6*d^3 + 6
*a^5*b^3*c^5*d^4 - 4*a^6*b^2*c^4*d^5 + a^7*b*c^3*d^6)*x^7 + (2*a^3*b^5*c^8*d - 7*a^4*b^4*c^7*d^2 + 8*a^5*b^3*c
^6*d^3 - 2*a^6*b^2*c^5*d^4 - 2*a^7*b*c^4*d^5 + a^8*c^3*d^6)*x^5 + (a^3*b^5*c^9 - 2*a^4*b^4*c^8*d - 2*a^5*b^3*c
^7*d^2 + 8*a^6*b^2*c^6*d^3 - 7*a^7*b*c^5*d^4 + 2*a^8*c^4*d^5)*x^3 + (a^4*b^4*c^9 - 4*a^5*b^3*c^8*d + 6*a^6*b^2
*c^7*d^2 - 4*a^7*b*c^6*d^3 + a^8*c^5*d^4)*x)]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 9.19209, size = 1266, normalized size = 4.54 \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/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

-1/3*((11*b^4*c^6*d^5 - 38*a*b^3*c^5*d^6 + 48*a^2*b^2*c^4*d^7 - 26*a^3*b*c^3*d^8 + 5*a^4*c^2*d^9)*x^2/(b^6*c^1
1*d - 6*a*b^5*c^10*d^2 + 15*a^2*b^4*c^9*d^3 - 20*a^3*b^3*c^8*d^4 + 15*a^4*b^2*c^7*d^5 - 6*a^5*b*c^6*d^6 + a^6*
c^5*d^7) + 6*(2*b^4*c^7*d^4 - 7*a*b^3*c^6*d^5 + 9*a^2*b^2*c^5*d^6 - 5*a^3*b*c^4*d^7 + a^4*c^3*d^8)/(b^6*c^11*d
- 6*a*b^5*c^10*d^2 + 15*a^2*b^4*c^9*d^3 - 20*a^3*b^3*c^8*d^4 + 15*a^4*b^2*c^7*d^5 - 6*a^5*b*c^6*d^6 + a^6*c^5
*d^7))*x/(d*x^2 + c)^(3/2) + 1/2*(3*b^4*c*sqrt(d) - 8*a*b^3*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))
^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*sqrt(a
*b*c*d - a^2*d^2)) + (3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^4*c^3*sqrt(d) - 8*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*
b^3*c^2*d^(3/2) + 6*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*b^2*c*d^(5/2) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^3*
b*d^(7/2) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^4*c^4*sqrt(d) + 22*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b^3*c^3*d
^(3/2) - 36*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*b^2*c^2*d^(5/2) + 28*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^3*b*c*d
^(7/2) - 8*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^4*d^(9/2) + 3*b^4*c^5*sqrt(d) - 6*a*b^3*c^4*d^(3/2) + 6*a^2*b^2*c
^3*d^(5/2) - 2*a^3*b*c^2*d^(7/2))/((a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*((sqrt(d)*x
- sqrt(d*x^2 + c))^6*b - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d + 3*(s
qrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2 - 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c*d - b*c^3))