3.784 \(\int \frac{1}{x^4 (a+b x^2)^2 (c+d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=362 \[ \frac{\sqrt{c+d x^2} \left (-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4-20 a b^3 c^3 d+15 b^4 c^4\right )}{6 a^3 c^4 x (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left (32 a^2 b c d^2-16 a^3 d^3-6 a b^2 c^2 d+5 b^3 c^3\right )}{6 a^2 c^3 x^3 (b c-a d)^3}+\frac{d \left (-4 a^2 d^2+8 a b c d+b^2 c^2\right )}{2 a c^2 x^3 \sqrt{c+d x^2} (b c-a d)^3}+\frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^
2)^(3/2)) + (d*(b^2*c^2 + 8*a*b*c*d - 4*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*x^3*Sqrt[c + d*x^2]) - ((5*b^3*c^3 -
6*a*b^2*c^2*d + 32*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^3) + ((15*b^4*c^4 - 2
0*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 64*a^3*b*c*d^3 - 32*a^4*d^4)*Sqrt[c + d*x^2])/(6*a^3*c^4*(b*c - a*d)^3*x)
+ (5*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.596045, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{\sqrt{c+d x^2} \left (-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4-20 a b^3 c^3 d+15 b^4 c^4\right )}{6 a^3 c^4 x (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left (32 a^2 b c d^2-16 a^3 d^3-6 a b^2 c^2 d+5 b^3 c^3\right )}{6 a^2 c^3 x^3 (b c-a d)^3}+\frac{d \left (-4 a^2 d^2+8 a b c d+b^2 c^2\right )}{2 a c^2 x^3 \sqrt{c+d x^2} (b c-a d)^3}+\frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(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^3*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^
2)^(3/2)) + (d*(b^2*c^2 + 8*a*b*c*d - 4*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*x^3*Sqrt[c + d*x^2]) - ((5*b^3*c^3 -
6*a*b^2*c^2*d + 32*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^3) + ((15*b^4*c^4 - 2
0*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 64*a^3*b*c*d^3 - 32*a^4*d^4)*Sqrt[c + d*x^2])/(6*a^3*c^4*(b*c - a*d)^3*x)
+ (5*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(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^4 \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^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{-5 b c+2 a d-8 b d x^2}{x^4 \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^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{-3 \left (5 b^2 c^2-4 a b c d+4 a^2 d^2\right )-6 b d (3 b c+2 a d) x^2}{x^4 \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^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt{c+d x^2}}-\frac{\int \frac{-3 \left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right )-12 b d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right ) x^2}{x^4 \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^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt{c+d x^2}}-\frac{\left (5 b^3 c^3-6 a b^2 c^2 d+32 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^3}+\frac{\int \frac{-3 \left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right )-6 b d \left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{18 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^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt{c+d x^2}}-\frac{\left (5 b^3 c^3-6 a b^2 c^2 d+32 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^3}+\frac{\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt{c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}-\frac{\int -\frac{45 b^4 c^4 (b c-2 a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{18 a^3 c^4 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt{c+d x^2}}-\frac{\left (5 b^3 c^3-6 a b^2 c^2 d+32 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^3}+\frac{\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt{c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac{\left (5 b^4 (b c-2 a d)\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 a^3 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt{c+d x^2}}-\frac{\left (5 b^3 c^3-6 a b^2 c^2 d+32 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^3}+\frac{\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt{c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac{\left (5 b^4 (b c-2 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^3 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt{c+d x^2}}-\frac{\left (5 b^3 c^3-6 a b^2 c^2 d+32 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^3}+\frac{\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt{c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 5.69681, size = 210, normalized size = 0.58 \[ \frac{\sqrt{c+d x^2} \left (-\frac{3 b^5 x^4}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{4 x^2 (4 a d+3 b c)}{a^3 c^4}-\frac{2}{a^2 c^3}+\frac{4 d^4 x^4 (7 b c-4 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{2 d^4 x^4}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right )}{6 x^3}+\frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

(Sqrt[c + d*x^2]*(-2/(a^2*c^3) + (4*(3*b*c + 4*a*d)*x^2)/(a^3*c^4) - (3*b^5*x^4)/(a^3*(-(b*c) + a*d)^3*(a + b*
x^2)) + (2*d^4*x^4)/(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^4*(7*b*c - 4*a*d)*x^4)/(c^4*(b*c - a*d)^3*(c + d*
x^2))))/(6*x^3) + (5*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c
- a*d)^(7/2))

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Maple [B]  time = 0.018, size = 2623, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

5/12*b^2/a^2*d^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)^
(3/2)*x+5/12*b^2/a^2*d^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)^(3/2)*x-5/4*b^3/a^2*d^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*b^3/a^3*d*(-a*b)^(1/2)/(a*d-b*c)^3/(-(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^2/a^3*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^2/a^3*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-5/4*b^3/a^3/(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+5/6*b^2/a^2*d^2/(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+2/a^2*d/c^2/x/(d*x^2+c)^(3/2)+8/3/
a^2*d^2/c^3*x/(d*x^2+c)^(3/2)+1/12*b^2/a^3*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^2/a^3*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+16/3/a^2*d^2/c^4*x/(d*x^2+c)^(1/2)-5/12*b^3/a^3/(-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)+5/4*b^4/a^3/(-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^3/(
a*d-b*c)/(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)+2*b/a^3/c/x/(d*x^2+c)^(3/2)+5/12*b^3/a^3/(-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)-5/4*b^4/a^3/(-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^3/(a*d-b*c)/(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)+5/6*b^2/a^2*d^2/(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*b^3/a^2*d^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*b^3/a^3*d*(-a*b)
^(1/2)/(a*d-b*c)^3/(-(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)))-5/4*b^3/a^3/(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-1/3/a^2/c/x^3/(d*x^2+c)^(3/2)+5/4*b^3/a^3*d*(-a*b)^(1/2)/(a*d-b*c)^3/((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*b^4/a^3/(-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*b^2/a^3*d*(-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)^(3/2)-5/4*b^4/
a^3/(-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*b^2/a^3*d*(-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)^(3/2)-5/4*b^3/a^3*d*(-a*b)^(1/2)/(a*d-b*c)^3/((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)+8/3*b/a^3*d/c^2*x/(d*x^2+c)^(3/2)+16/3*b/a^3*d/c^3*x/(d*x^2+c
)^(1/2)

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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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(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^4), x)

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Fricas [B]  time = 9.49402, size = 3780, normalized size = 10.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/24*(15*((b^6*c^5*d^2 - 2*a*b^5*c^4*d^3)*x^9 + (2*b^6*c^6*d - 3*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3)*x^7 + (b
^6*c^7 - 4*a^2*b^4*c^5*d^2)*x^5 + (a*b^5*c^7 - 2*a^2*b^4*c^6*d)*x^3)*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*(2*a^3*b^4*c^7 - 8*a^4*b^3*c^6*d + 12*a^5*b^2*c^5*d^
2 - 8*a^6*b*c^4*d^3 + 2*a^7*c^3*d^4 - (15*a*b^6*c^5*d^2 - 35*a^2*b^5*c^4*d^3 + 8*a^3*b^4*c^3*d^4 + 76*a^4*b^3*
c^2*d^5 - 96*a^5*b^2*c*d^6 + 32*a^6*b*d^7)*x^8 - 2*(15*a*b^6*c^6*d - 30*a^2*b^5*c^5*d^2 - 4*a^3*b^4*c^4*d^3 +
61*a^4*b^3*c^3*d^4 - 34*a^5*b^2*c^2*d^5 - 24*a^6*b*c*d^6 + 16*a^7*d^7)*x^6 - 3*(5*a*b^6*c^7 - 5*a^2*b^5*c^6*d
- 14*a^3*b^4*c^5*d^2 + 16*a^4*b^3*c^4*d^3 + 26*a^5*b^2*c^3*d^4 - 44*a^6*b*c^2*d^5 + 16*a^7*c*d^6)*x^4 - 2*(5*a
^2*b^5*c^7 - 14*a^3*b^4*c^6*d + 6*a^4*b^3*c^5*d^2 + 16*a^5*b^2*c^4*d^3 - 19*a^6*b*c^3*d^4 + 6*a^7*c^2*d^5)*x^2
)*sqrt(d*x^2 + c))/((a^4*b^5*c^8*d^2 - 4*a^5*b^4*c^7*d^3 + 6*a^6*b^3*c^6*d^4 - 4*a^7*b^2*c^5*d^5 + a^8*b*c^4*d
^6)*x^9 + (2*a^4*b^5*c^9*d - 7*a^5*b^4*c^8*d^2 + 8*a^6*b^3*c^7*d^3 - 2*a^7*b^2*c^6*d^4 - 2*a^8*b*c^5*d^5 + a^9
*c^4*d^6)*x^7 + (a^4*b^5*c^10 - 2*a^5*b^4*c^9*d - 2*a^6*b^3*c^8*d^2 + 8*a^7*b^2*c^7*d^3 - 7*a^8*b*c^6*d^4 + 2*
a^9*c^5*d^5)*x^5 + (a^5*b^4*c^10 - 4*a^6*b^3*c^9*d + 6*a^7*b^2*c^8*d^2 - 4*a^8*b*c^7*d^3 + a^9*c^6*d^4)*x^3),
1/12*(15*((b^6*c^5*d^2 - 2*a*b^5*c^4*d^3)*x^9 + (2*b^6*c^6*d - 3*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3)*x^7 + (b^6
*c^7 - 4*a^2*b^4*c^5*d^2)*x^5 + (a*b^5*c^7 - 2*a^2*b^4*c^6*d)*x^3)*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*(2*a^
3*b^4*c^7 - 8*a^4*b^3*c^6*d + 12*a^5*b^2*c^5*d^2 - 8*a^6*b*c^4*d^3 + 2*a^7*c^3*d^4 - (15*a*b^6*c^5*d^2 - 35*a^
2*b^5*c^4*d^3 + 8*a^3*b^4*c^3*d^4 + 76*a^4*b^3*c^2*d^5 - 96*a^5*b^2*c*d^6 + 32*a^6*b*d^7)*x^8 - 2*(15*a*b^6*c^
6*d - 30*a^2*b^5*c^5*d^2 - 4*a^3*b^4*c^4*d^3 + 61*a^4*b^3*c^3*d^4 - 34*a^5*b^2*c^2*d^5 - 24*a^6*b*c*d^6 + 16*a
^7*d^7)*x^6 - 3*(5*a*b^6*c^7 - 5*a^2*b^5*c^6*d - 14*a^3*b^4*c^5*d^2 + 16*a^4*b^3*c^4*d^3 + 26*a^5*b^2*c^3*d^4
- 44*a^6*b*c^2*d^5 + 16*a^7*c*d^6)*x^4 - 2*(5*a^2*b^5*c^7 - 14*a^3*b^4*c^6*d + 6*a^4*b^3*c^5*d^2 + 16*a^5*b^2*
c^4*d^3 - 19*a^6*b*c^3*d^4 + 6*a^7*c^2*d^5)*x^2)*sqrt(d*x^2 + c))/((a^4*b^5*c^8*d^2 - 4*a^5*b^4*c^7*d^3 + 6*a^
6*b^3*c^6*d^4 - 4*a^7*b^2*c^5*d^5 + a^8*b*c^4*d^6)*x^9 + (2*a^4*b^5*c^9*d - 7*a^5*b^4*c^8*d^2 + 8*a^6*b^3*c^7*
d^3 - 2*a^7*b^2*c^6*d^4 - 2*a^8*b*c^5*d^5 + a^9*c^4*d^6)*x^7 + (a^4*b^5*c^10 - 2*a^5*b^4*c^9*d - 2*a^6*b^3*c^8
*d^2 + 8*a^7*b^2*c^7*d^3 - 7*a^8*b*c^6*d^4 + 2*a^9*c^5*d^5)*x^5 + (a^5*b^4*c^10 - 4*a^6*b^3*c^9*d + 6*a^7*b^2*
c^8*d^2 - 4*a^8*b*c^7*d^3 + a^9*c^6*d^4)*x^3)]

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 20.5077, size = 1065, normalized size = 2.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/3*(2*(7*b^4*c^7*d^6 - 25*a*b^3*c^6*d^7 + 33*a^2*b^2*c^5*d^8 - 19*a^3*b*c^4*d^9 + 4*a^4*c^3*d^10)*x^2/(b^6*c^
13*d - 6*a*b^5*c^12*d^2 + 15*a^2*b^4*c^11*d^3 - 20*a^3*b^3*c^10*d^4 + 15*a^4*b^2*c^9*d^5 - 6*a^5*b*c^8*d^6 + a
^6*c^7*d^7) + 3*(5*b^4*c^8*d^5 - 18*a*b^3*c^7*d^6 + 24*a^2*b^2*c^6*d^7 - 14*a^3*b*c^5*d^8 + 3*a^4*c^4*d^9)/(b^
6*c^13*d - 6*a*b^5*c^12*d^2 + 15*a^2*b^4*c^11*d^3 - 20*a^3*b^3*c^10*d^4 + 15*a^4*b^2*c^9*d^5 - 6*a^5*b*c^8*d^6
+ a^6*c^7*d^7))*x/(d*x^2 + c)^(3/2) - 5/2*(b^5*c*sqrt(d) - 2*a*b^4*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^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3
)*sqrt(a*b*c*d - a^2*d^2)) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^5*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^
2*a*b^4*d^(3/2) - b^5*c^2*sqrt(d))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*((sqrt(d)*x - sq
rt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)) -
4/3*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c*sqrt(d) + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d^(3/2) - 6*(sqrt(d)
*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) - 9*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c*d^(3/2) + 3*b*c^3*sqrt(d) + 4*a*
c^2*d^(3/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3*a^3*c^3)