\(\int \frac {1}{x^2 (a+b x^2)^2 (c+d x^2)^{5/2}} \, dx\) [782]

Optimal result

Integrand size = 24, antiderivative size = 279 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+20 a b c d-8 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 x \sqrt {c+d x^2}}-\frac {\left (9 b^3 c^3-18 a b^2 c^2 d+40 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x}-\frac {b^3 (3 b c-8 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}} \]

[Out]

1/6*d*(2*a*d+3*b*c)/a/c/(-a*d+b*c)^2/x/(d*x^2+c)^(3/2)+1/2*b/a/(-a*d+b*c)/x/(b*x^2+a)/(d*x^2+c)^(3/2)-1/2*b^3*
(-8*a*d+3*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(5/2)/(-a*d+b*c)^(7/2)+1/6*d*(-8*a^2*d^2+2
0*a*b*c*d+3*b^2*c^2)/a/c^2/(-a*d+b*c)^3/x/(d*x^2+c)^(1/2)-1/6*(-16*a^3*d^3+40*a^2*b*c*d^2-18*a*b^2*c^2*d+9*b^3
*c^3)*(d*x^2+c)^(1/2)/a^2/c^3/(-a*d+b*c)^3/x

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {483, 593, 597, 12, 385, 211} \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {b^3 (3 b c-8 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}}+\frac {d \left (-8 a^2 d^2+20 a b c d+3 b^2 c^2\right )}{6 a c^2 x \sqrt {c+d x^2} (b c-a d)^3}-\frac {\sqrt {c+d x^2} \left (-16 a^3 d^3+40 a^2 b c d^2-18 a b^2 c^2 d+9 b^3 c^3\right )}{6 a^2 c^3 x (b c-a d)^3}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c x \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[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 12

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

Rule 211

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

Rule 385

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 483

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 593

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 597

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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] (verified)

Time = 1.09 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {-9 b^4 c^3 x^2 \left (c+d x^2\right )^2-6 a b^3 c^2 \left (c-3 d x^2\right ) \left (c+d x^2\right )^2+2 a^4 d^3 \left (3 c^2+12 c d x^2+8 d^2 x^4\right )+2 a^2 b^2 c d \left (9 c^3+9 c^2 d x^2-21 c d^2 x^4-20 d^3 x^6\right )-2 a^3 b d^2 \left (9 c^3+27 c^2 d x^2+8 c d^2 x^4-8 d^3 x^6\right )}{6 a^2 c^3 (b c-a d)^3 x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {b^3 (3 b c-8 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2} (b c-a d)^{7/2}} \]

[In]

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

[Out]

(-9*b^4*c^3*x^2*(c + d*x^2)^2 - 6*a*b^3*c^2*(c - 3*d*x^2)*(c + d*x^2)^2 + 2*a^4*d^3*(3*c^2 + 12*c*d*x^2 + 8*d^
2*x^4) + 2*a^2*b^2*c*d*(9*c^3 + 9*c^2*d*x^2 - 21*c*d^2*x^4 - 20*d^3*x^6) - 2*a^3*b*d^2*(9*c^3 + 27*c^2*d*x^2 +
8*c*d^2*x^4 - 8*d^3*x^6))/(6*a^2*c^3*(b*c - a*d)^3*x*(a + b*x^2)*(c + d*x^2)^(3/2)) + (b^3*(3*b*c - 8*a*d)*Ar
cTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(5/2)*(b*c - a*d)^(7/2))

Maple [A] (verified)

Time = 3.35 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.63

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{2}+c}}{a^{2} x}+\frac {b^{3} c^{3} \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (8 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 \left (a d -b c \right )^{3} a^{2}}+\frac {d^{4} x^{3}}{3 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 d^{3} \left (a d -2 b c \right ) x}{\left (a d -b c \right )^{3} \sqrt {d \,x^{2}+c}}}{c^{3}}\)	\(176\)
risch	\(\text {Expression too large to display}\)	\(1622\)
default	\(\text {Expression too large to display}\)	\(3544\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (251) = 502\).

Time = 1.08 (sec) , antiderivative size = 1662, normalized size of antiderivative = 5.96 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[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)]

Sympy [F]

\[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (251) = 502\).

Time = 1.01 (sec) , antiderivative size = 938, normalized size of antiderivative = 3.36 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {{\left (\frac {{\left (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}\right )} x^{2}}{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}} + \frac {6 \, {\left (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}\right )}}{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}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (3 \, 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^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{4} c^{3} \sqrt {d} - 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{3} c^{2} d^{\frac {3}{2}} + 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b^{2} c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{3} b d^{\frac {7}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{4} c^{4} \sqrt {d} + 22 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{3} c^{3} d^{\frac {3}{2}} - 36 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b^{2} c^{2} d^{\frac {5}{2}} + 28 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} b c d^{\frac {7}{2}} - 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{4} d^{\frac {9}{2}} + 3 \, b^{4} c^{5} \sqrt {d} - 6 \, a b^{3} c^{4} d^{\frac {3}{2}} + 6 \, a^{2} b^{2} c^{3} d^{\frac {5}{2}} - 2 \, a^{3} b c^{2} d^{\frac {7}{2}}}{{\left (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}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )}} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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