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

Optimal. Leaf size=202 \[ \frac {b (2 b c+3 a d) x}{6 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x}{6 a^2 c (b c-a d)^3 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {d^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2}} \]

[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]
time = 0.15, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12, 385, 214} \begin {gather*} \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) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx &=-\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]
time = 1.30, size = 219, normalized size = 1.08 \begin {gather*} \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) \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(3448\) vs. \(2(178)=356\).
time = 0.08, size = 3449, normalized size = 17.07

method	result	size
default	\(\text {Expression too large to display}\)	\(3449\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (178) = 356\).
time = 1.43, size = 1440, normalized size = 7.13 \begin {gather*} \left [\frac {3 \, {\left (6 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (6 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{6} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} + 11 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{4} + {\left (12 \, a^{3} b^{2} c^{2} d^{2} + 4 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (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}\right )} x^{5} + 2 \, {\left (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}\right )} x^{3} + 3 \, {\left (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}\right )} x\right )} \sqrt {b x^{2} + a}}{24 \, {\left (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} + {\left (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}\right )} x^{6} + {\left (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}\right )} x^{4} + {\left (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}\right )} x^{2}\right )}}, -\frac {3 \, {\left (6 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (6 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{6} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} + 11 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{4} + {\left (12 \, a^{3} b^{2} c^{2} d^{2} + 4 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (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}\right )} x^{5} + 2 \, {\left (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}\right )} x^{3} + 3 \, {\left (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}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (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} + {\left (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}\right )} x^{6} + {\left (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}\right )} x^{4} + {\left (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}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (178) = 356\).
time = 1.15, size = 620, normalized size = 3.07 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________