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

Optimal. Leaf size=277 \[ \frac{\sqrt{c+d x^2} \left (-8 a^2 b c d^2+16 a^3 d^3-14 a b^2 c^2 d+15 b^3 c^3\right )}{6 a^3 c^3 x (b c-a d)^2}-\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-4 a b c d+5 b^2 c^2\right )}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x^3 \sqrt{c+d x^2} (b c-a d)^2} \]

[Out]

(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x^3*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*Sqrt[c + d*x^
2]) - ((5*b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*Sqrt[c + d*x^2])/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((15*b^3*c^3 - 14*
a*b^2*c^2*d - 8*a^2*b*c*d^2 + 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^3*c^3*(b*c - a*d)^2*x) + (b^3*(5*b*c - 8*a*d)*
ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.396469, antiderivative size = 277, 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} \[ \frac{\sqrt{c+d x^2} \left (-8 a^2 b c d^2+16 a^3 d^3-14 a b^2 c^2 d+15 b^3 c^3\right )}{6 a^3 c^3 x (b c-a d)^2}-\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-4 a b c d+5 b^2 c^2\right )}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x^3 \sqrt{c+d x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x^3*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*Sqrt[c + d*x^
2]) - ((5*b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*Sqrt[c + d*x^2])/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((15*b^3*c^3 - 14*
a*b^2*c^2*d - 8*a^2*b*c*d^2 + 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^3*c^3*(b*c - a*d)^2*x) + (b^3*(5*b*c - 8*a*d)*
ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(5/2))

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 )^{3/2}} \, dx &=\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\int \frac{-5 b c+2 a d-6 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{2 a (b c-a d)}\\ &=\frac{d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt{c+d x^2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\int \frac{-5 b^2 c^2+4 a b c d-8 a^2 d^2-4 b d (b c+2 a d) x^2}{x^4 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 a c (b c-a d)^2}\\ &=\frac{d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt{c+d x^2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt{c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{\int \frac{-15 b^3 c^3+14 a b^2 c^2 d+8 a^2 b c d^2-16 a^3 d^3-2 b d \left (5 b^2 c^2-4 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^2 c^2 (b c-a d)^2}\\ &=\frac{d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt{c+d x^2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt{c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}-\frac{\int -\frac{3 b^3 c^3 (5 b c-8 a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{6 a^3 c^3 (b c-a d)^2}\\ &=\frac{d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt{c+d x^2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt{c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}+\frac{\left (b^3 (5 b c-8 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)^2}\\ &=\frac{d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt{c+d x^2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt{c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}+\frac{\left (b^3 (5 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^3 (b c-a d)^2}\\ &=\frac{d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt{c+d x^2}}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt{c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt{c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}+\frac{b^3 (5 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^{7/2} (b c-a d)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 5.43444, size = 167, normalized size = 0.6 \[ \sqrt{c+d x^2} \left (\frac{\frac{b^4 x}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{2 b}{c^2 x}}{a^3}-\frac{c-5 d x^2}{3 a^2 c^3 x^3}+\frac{d^4 x}{c^3 \left (c+d x^2\right ) (b c-a d)^2}\right )+\frac{b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 1608, normalized size = 5.8 \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)^(3/2),x)

[Out]

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

________________________________________________________________________________________

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{3}{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)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 8.48678, size = 2504, normalized size = 9.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*((5*b^5*c^4*d - 8*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 3*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4
*c^5 - 8*a^2*b^3*c^4*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^3*c^5 - 6*a^4*b^2*c^4*d + 6*a^5*b*c^3*d^2 - 2*a^6*c^2*d^3 - (15*a*b^5*c^4*d - 29*a^
2*b^4*c^3*d^2 + 6*a^3*b^3*c^2*d^3 + 24*a^4*b^2*c*d^4 - 16*a^5*b*d^5)*x^6 - (15*a*b^5*c^5 - 19*a^2*b^4*c^4*d -
14*a^3*b^3*c^3*d^2 + 18*a^4*b^2*c^2*d^3 + 16*a^5*b*c*d^4 - 16*a^6*d^5)*x^4 - 2*(5*a^2*b^4*c^5 - 11*a^3*b^3*c^4
*d + 3*a^4*b^2*c^3*d^2 + 7*a^5*b*c^2*d^3 - 4*a^6*c*d^4)*x^2)*sqrt(d*x^2 + c))/((a^4*b^4*c^6*d - 3*a^5*b^3*c^5*
d^2 + 3*a^6*b^2*c^4*d^3 - a^7*b*c^3*d^4)*x^7 + (a^4*b^4*c^7 - 2*a^5*b^3*c^6*d + 2*a^7*b*c^4*d^3 - a^8*c^3*d^4)
*x^5 + (a^5*b^3*c^7 - 3*a^6*b^2*c^6*d + 3*a^7*b*c^5*d^2 - a^8*c^4*d^3)*x^3), 1/12*(3*((5*b^5*c^4*d - 8*a*b^4*c
^3*d^2)*x^7 + (5*b^5*c^5 - 3*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 8*a^2*b^3*c^4*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^3*c^5 - 6*a^4*b^2*c^4*d + 6*a^5*b*c^3*d^2 - 2*a^6*c^2*d^3 - (15*a*b^
5*c^4*d - 29*a^2*b^4*c^3*d^2 + 6*a^3*b^3*c^2*d^3 + 24*a^4*b^2*c*d^4 - 16*a^5*b*d^5)*x^6 - (15*a*b^5*c^5 - 19*a
^2*b^4*c^4*d - 14*a^3*b^3*c^3*d^2 + 18*a^4*b^2*c^2*d^3 + 16*a^5*b*c*d^4 - 16*a^6*d^5)*x^4 - 2*(5*a^2*b^4*c^5 -
11*a^3*b^3*c^4*d + 3*a^4*b^2*c^3*d^2 + 7*a^5*b*c^2*d^3 - 4*a^6*c*d^4)*x^2)*sqrt(d*x^2 + c))/((a^4*b^4*c^6*d -
3*a^5*b^3*c^5*d^2 + 3*a^6*b^2*c^4*d^3 - a^7*b*c^3*d^4)*x^7 + (a^4*b^4*c^7 - 2*a^5*b^3*c^6*d + 2*a^7*b*c^4*d^3
- a^8*c^3*d^4)*x^5 + (a^5*b^3*c^7 - 3*a^6*b^2*c^6*d + 3*a^7*b*c^5*d^2 - a^8*c^4*d^3)*x^3)]

________________________________________________________________________________________

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)**(3/2),x)

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [A]  time = 11.8805, size = 656, normalized size = 2.37 \begin{align*} \frac{d^{4} x}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt{d x^{2} + c}} - \frac{{\left (5 \, b^{4} c \sqrt{d} - 8 \, a b^{3} d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt{a b c d - a^{2} d^{2}}} - \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{4} c \sqrt{d} - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b^{3} d^{\frac{3}{2}} - b^{4} c^{2} \sqrt{d}}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} - \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b c \sqrt{d} + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c^{2} \sqrt{d} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a c d^{\frac{3}{2}} + 6 \, b c^{3} \sqrt{d} + 5 \, a c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3} c^{2}} \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)^(3/2),x, algorithm="giac")

[Out]

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