\(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^2} \, dx\) [95]

Optimal result

Integrand size = 21, antiderivative size = 202 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x}{6 a^2 c (b c-a d)^3 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {d^2 (6 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2}} \]

[Out]

1/6*b*(3*a*d+2*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^2+a)^(3/2)-1/2*d*x/c/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)+1/2*d^2*
(-a*d+6*b*c)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(3/2)/(-a*d+b*c)^(7/2)+1/6*b*(-3*a^2*d^2-16
*a*b*c*d+4*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12, 385, 214} \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {b x \left (-3 a^2 d^2-16 a b c d+4 b^2 c^2\right )}{6 a^2 c \sqrt {a+b x^2} (b c-a d)^3}+\frac {d^2 (6 b c-a d) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2}}-\frac {d x}{2 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)}+\frac {b x (3 a d+2 b c)}{6 a c \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

[In]

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

[Out]

(b*(2*b*c + 3*a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (b*(4*b^2*c^2 - 16*a*b*c*d - 3*a^2*d^2)*x)/(6*
a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)) + (d^2*(6*b*c - a
*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 425

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

Rule 541

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-4 b d x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = \frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}-\frac {\int \frac {-4 b^2 c^2+12 a b c d-3 a^2 d^2-2 b d (2 b c+3 a d) x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx}{6 a c (b c-a d)^2} \\ & = \frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x}{6 a^2 c (b c-a d)^3 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {\int \frac {3 a^2 d^2 (6 b c-a d)}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{6 a^2 c (b c-a d)^3} \\ & = \frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x}{6 a^2 c (b c-a d)^3 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {\left (d^2 (6 b c-a d)\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)^3} \\ & = \frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x}{6 a^2 c (b c-a d)^3 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {\left (d^2 (6 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 c (b c-a d)^3} \\ & = \frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x}{6 a^2 c (b c-a d)^3 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {d^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {x \left (3 a^4 d^3+6 a^3 b d^3 x^2-4 b^4 c^2 x^2 \left (c+d x^2\right )+3 a^2 b^2 d \left (6 c^2+6 c d x^2+d^2 x^4\right )+2 a b^3 c \left (-3 c^2+5 c d x^2+8 d^2 x^4\right )\right )}{6 a^2 c (-b c+a d)^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {d^2 (6 b c-a d) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{2 c^{3/2} (-b c+a d)^{7/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.63 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {-\left (a d -6 b c \right ) \left (d \,x^{2}+c \right ) d^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+x \sqrt {\left (a d -b c \right ) c}\, \left (\left (-\frac {4}{3} b^{4} x^{2}-2 a \,b^{3}\right ) c^{3}+6 b^{2} d \left (-\frac {2}{9} b^{2} x^{4}+\frac {5}{9} a b \,x^{2}+a^{2}\right ) c^{2}+6 x^{2} b^{2} d^{2} a \left (\frac {8 b \,x^{2}}{9}+a \right ) c +a^{2} d^{3} \left (b \,x^{2}+a \right )^{2}\right )}{2 \sqrt {\left (a d -b c \right ) c}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} c \left (d \,x^{2}+c \right ) \left (a d -b c \right )^{3} a^{2}}\)	\(208\)
default	\(\text {Expression too large to display}\)	\(3449\)

[In]

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

[Out]

1/2/((a*d-b*c)*c)^(1/2)/(b*x^2+a)^(3/2)*(-(a*d-6*b*c)*(d*x^2+c)*d^2*(b*x^2+a)^(3/2)*a^2*arctan(c*(b*x^2+a)^(1/
2)/x/((a*d-b*c)*c)^(1/2))+x*((a*d-b*c)*c)^(1/2)*((-4/3*b^4*x^2-2*a*b^3)*c^3+6*b^2*d*(-2/9*b^2*x^4+5/9*a*b*x^2+
a^2)*c^2+6*x^2*b^2*d^2*a*(8/9*b*x^2+a)*c+a^2*d^3*(b*x^2+a)^2))/c/(d*x^2+c)/(a*d-b*c)^3/a^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (178) = 356\).

Time = 0.95 (sec) , antiderivative size = 1440, normalized size of antiderivative = 7.13 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/24*(3*(6*a^4*b*c^2*d^2 - a^5*c*d^3 + (6*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (6*a^2*b^3*c^2*d^2 + 11*a^3*b^2*
c*d^3 - 2*a^4*b*d^4)*x^4 + (12*a^3*b^2*c^2*d^2 + 4*a^4*b*c*d^3 - a^5*d^4)*x^2)*sqrt(b*c^2 - a*c*d)*log(((8*b^2
*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt
(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((4*b^5*c^4*d - 20*a*b^4*c^3*d^2 + 13*a^2*b^
3*c^2*d^3 + 3*a^3*b^2*c*d^4)*x^5 + 2*(2*b^5*c^5 - 7*a*b^4*c^4*d - 4*a^2*b^3*c^3*d^2 + 6*a^3*b^2*c^2*d^3 + 3*a^
4*b*c*d^4)*x^3 + 3*(2*a*b^4*c^5 - 8*a^2*b^3*c^4*d + 6*a^3*b^2*c^3*d^2 - a^4*b*c^2*d^3 + a^5*c*d^4)*x)*sqrt(b*x
^2 + a))/(a^4*b^4*c^7 - 4*a^5*b^3*c^6*d + 6*a^6*b^2*c^5*d^2 - 4*a^7*b*c^4*d^3 + a^8*c^3*d^4 + (a^2*b^6*c^6*d -
4*a^3*b^5*c^5*d^2 + 6*a^4*b^4*c^4*d^3 - 4*a^5*b^3*c^3*d^4 + a^6*b^2*c^2*d^5)*x^6 + (a^2*b^6*c^7 - 2*a^3*b^5*c
^6*d - 2*a^4*b^4*c^5*d^2 + 8*a^5*b^3*c^4*d^3 - 7*a^6*b^2*c^3*d^4 + 2*a^7*b*c^2*d^5)*x^4 + (2*a^3*b^5*c^7 - 7*a
^4*b^4*c^6*d + 8*a^5*b^3*c^5*d^2 - 2*a^6*b^2*c^4*d^3 - 2*a^7*b*c^3*d^4 + a^8*c^2*d^5)*x^2), -1/12*(3*(6*a^4*b*
c^2*d^2 - a^5*c*d^3 + (6*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (6*a^2*b^3*c^2*d^2 + 11*a^3*b^2*c*d^3 - 2*a^4*b*d^
4)*x^4 + (12*a^3*b^2*c^2*d^2 + 4*a^4*b*c*d^3 - a^5*d^4)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c
*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*((4*b^5*c
^4*d - 20*a*b^4*c^3*d^2 + 13*a^2*b^3*c^2*d^3 + 3*a^3*b^2*c*d^4)*x^5 + 2*(2*b^5*c^5 - 7*a*b^4*c^4*d - 4*a^2*b^3
*c^3*d^2 + 6*a^3*b^2*c^2*d^3 + 3*a^4*b*c*d^4)*x^3 + 3*(2*a*b^4*c^5 - 8*a^2*b^3*c^4*d + 6*a^3*b^2*c^3*d^2 - a^4
*b*c^2*d^3 + a^5*c*d^4)*x)*sqrt(b*x^2 + a))/(a^4*b^4*c^7 - 4*a^5*b^3*c^6*d + 6*a^6*b^2*c^5*d^2 - 4*a^7*b*c^4*d
^3 + a^8*c^3*d^4 + (a^2*b^6*c^6*d - 4*a^3*b^5*c^5*d^2 + 6*a^4*b^4*c^4*d^3 - 4*a^5*b^3*c^3*d^4 + a^6*b^2*c^2*d^
5)*x^6 + (a^2*b^6*c^7 - 2*a^3*b^5*c^6*d - 2*a^4*b^4*c^5*d^2 + 8*a^5*b^3*c^4*d^3 - 7*a^6*b^2*c^3*d^4 + 2*a^7*b*
c^2*d^5)*x^4 + (2*a^3*b^5*c^7 - 7*a^4*b^4*c^6*d + 8*a^5*b^3*c^5*d^2 - 2*a^6*b^2*c^4*d^3 - 2*a^7*b*c^3*d^4 + a^
8*c^2*d^5)*x^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (178) = 356\).

Time = 0.89 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.07 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {{\left (\frac {2 \, {\left (b^{8} c^{4} - 7 \, a b^{7} c^{3} d + 15 \, a^{2} b^{6} c^{2} d^{2} - 13 \, a^{3} b^{5} c d^{3} + 4 \, a^{4} b^{4} d^{4}\right )} x^{2}}{a^{2} b^{7} c^{6} - 6 \, a^{3} b^{6} c^{5} d + 15 \, a^{4} b^{5} c^{4} d^{2} - 20 \, a^{5} b^{4} c^{3} d^{3} + 15 \, a^{6} b^{3} c^{2} d^{4} - 6 \, a^{7} b^{2} c d^{5} + a^{8} b d^{6}} + \frac {3 \, {\left (a b^{7} c^{4} - 6 \, a^{2} b^{6} c^{3} d + 12 \, a^{3} b^{5} c^{2} d^{2} - 10 \, a^{4} b^{4} c d^{3} + 3 \, a^{5} b^{3} d^{4}\right )}}{a^{2} b^{7} c^{6} - 6 \, a^{3} b^{6} c^{5} d + 15 \, a^{4} b^{5} c^{4} d^{2} - 20 \, a^{5} b^{4} c^{3} d^{3} + 15 \, a^{6} b^{3} c^{2} d^{4} - 6 \, a^{7} b^{2} c d^{5} + a^{8} b d^{6}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {{\left (6 \, b^{\frac {3}{2}} c d^{2} - a \sqrt {b} d^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c d^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d^{3} + a^{2} \sqrt {b} d^{3}}{{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}} \]

[In]

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

[Out]

1/3*(2*(b^8*c^4 - 7*a*b^7*c^3*d + 15*a^2*b^6*c^2*d^2 - 13*a^3*b^5*c*d^3 + 4*a^4*b^4*d^4)*x^2/(a^2*b^7*c^6 - 6*
a^3*b^6*c^5*d + 15*a^4*b^5*c^4*d^2 - 20*a^5*b^4*c^3*d^3 + 15*a^6*b^3*c^2*d^4 - 6*a^7*b^2*c*d^5 + a^8*b*d^6) +
3*(a*b^7*c^4 - 6*a^2*b^6*c^3*d + 12*a^3*b^5*c^2*d^2 - 10*a^4*b^4*c*d^3 + 3*a^5*b^3*d^4)/(a^2*b^7*c^6 - 6*a^3*b
^6*c^5*d + 15*a^4*b^5*c^4*d^2 - 20*a^5*b^4*c^3*d^3 + 15*a^6*b^3*c^2*d^4 - 6*a^7*b^2*c*d^5 + a^8*b*d^6))*x/(b*x
^2 + a)^(3/2) + 1/2*(6*b^(3/2)*c*d^2 - a*sqrt(b)*d^3)*arctan(-1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c -
a*d)/sqrt(-b^2*c^2 + a*b*c*d))/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*sqrt(-b^2*c^2 + a*b*c
*d)) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c*d^2 - (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*d^3 + a^2*
sqrt(b)*d^3)/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(
sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d))

Mupad [F(-1)]

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

[In]

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

[Out]

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