3.96 \(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^3} \, dx\)
Optimal. Leaf size=313 \[ \frac {b x \left (-3 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{12 a^2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)^3}+\frac {d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}+\frac {d x \sqrt {a+b x^2} \left (9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac {d x}{4 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (b c-a d)}+\frac {b x (3 a d+4 b c)}{12 a c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]
[Out]
-1/4*d*x/c/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^2+1/12*b*(3*a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^2+a)^(3/2)/(d*x
^2+c)+1/8*d^2*(3*a^2*d^2-16*a*b*c*d+48*b^2*c^2)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(5/2)/(-
a*d+b*c)^(9/2)+1/12*b*(-3*a^2*d^2-40*a*b*c*d+8*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(d*x^2+c)/(b*x^2+a)^(1/2)+1/24*d*
(9*a^3*d^3-42*a^2*b*c*d^2-88*a*b^2*c^2*d+16*b^3*c^3)*x*(b*x^2+a)^(1/2)/a^2/c^2/(-a*d+b*c)^4/(d*x^2+c)
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Rubi [A] time = 0.40, antiderivative size = 313, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used =
{414, 527, 12, 377, 208} \[ \frac {d x \sqrt {a+b x^2} \left (-42 a^2 b c d^2+9 a^3 d^3-88 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}+\frac {b x \left (-3 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{12 a^2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)^3}+\frac {d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}-\frac {d x}{4 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (b c-a d)}+\frac {b x (3 a d+4 b c)}{12 a c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]
[Out]
-(d*x)/(4*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^2) + (b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*
x^2)^(3/2)*(c + d*x^2)) + (b*(8*b^2*c^2 - 40*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]*(
c + d*x^2)) + (d*(16*b^3*c^3 - 88*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 9*a^3*d^3)*x*Sqrt[a + b*x^2])/(24*a^2*c^2*(b*
c - a*d)^4*(c + d*x^2)) + (d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt
[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(9/2))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
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 414
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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]
Rule 527
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 )^3} \, dx &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-3 a d-6 b d x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}-\frac {\int \frac {-8 b^2 c^2+24 a b c d-9 a^2 d^2-4 b d (4 b c+3 a d) x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx}{12 a c (b c-a d)^2}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {\int \frac {a d \left (8 b^2 c^2+36 a b c d-9 a^2 d^2\right )+2 b d \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{12 a^2 c (b c-a d)^3}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\int \frac {3 a^2 d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{24 a^2 c^2 (b c-a d)^4}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\left (d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^4}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\left (d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)^4}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}\\ \end {align*}
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Mathematica [A] time = 5.66, size = 221, normalized size = 0.71 \[ \frac {1}{24} \left (x \sqrt {a+b x^2} \left (\frac {8 b^3 (2 b c-11 a d)}{a^2 \left (a+b x^2\right ) (b c-a d)^4}-\frac {8 b^3}{a \left (a+b x^2\right )^2 (a d-b c)^3}+\frac {3 d^3 (3 a d-14 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac {6 d^3}{c \left (c+d x^2\right )^2 (b c-a d)^3}\right )+\frac {3 d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tan ^{-1}\left (\frac {x \sqrt {a d-b c}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{5/2} (a d-b c)^{9/2}}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]
[Out]
(x*Sqrt[a + b*x^2]*((-8*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^2)^2) + (8*b^3*(2*b*c - 11*a*d))/(a^2*(b*c - a*d)^4*
(a + b*x^2)) - (6*d^3)/(c*(b*c - a*d)^3*(c + d*x^2)^2) + (3*d^3*(-14*b*c + 3*a*d))/(c^2*(b*c - a*d)^4*(c + d*x
^2))) + (3*d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])
/(c^(5/2)*(-(b*c) + a*d)^(9/2)))/24
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fricas [B] time = 8.34, size = 2250, normalized size = 7.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x, algorithm="fricas")
[Out]
[1/96*(3*(48*a^4*b^2*c^4*d^2 - 16*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + (48*a^2*b^4*c^2*d^4 - 16*a^3*b^3*c*d^5 + 3*a
^4*b^2*d^6)*x^8 + 2*(48*a^2*b^4*c^3*d^3 + 32*a^3*b^3*c^2*d^4 - 13*a^4*b^2*c*d^5 + 3*a^5*b*d^6)*x^6 + (48*a^2*b
^4*c^4*d^2 + 176*a^3*b^3*c^3*d^3 - 13*a^4*b^2*c^2*d^4 - 4*a^5*b*c*d^5 + 3*a^6*d^6)*x^4 + 2*(48*a^3*b^3*c^4*d^2
+ 32*a^4*b^2*c^3*d^3 - 13*a^5*b*c^2*d^4 + 3*a^6*c*d^5)*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)*sq
rt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((16*b^6*c^5*d^2 - 104*a*b^5*c^4*d^3 + 46*a^2*b^4*c^3*d^4 + 51
*a^3*b^3*c^2*d^5 - 9*a^4*b^2*c*d^6)*x^7 + (32*b^6*c^6*d - 184*a*b^5*c^5*d^2 + 8*a^2*b^4*c^4*d^3 + 75*a^3*b^3*c
^3*d^4 + 87*a^4*b^2*c^2*d^5 - 18*a^5*b*c*d^6)*x^5 + (16*b^6*c^7 - 56*a*b^5*c^6*d - 152*a^2*b^4*c^5*d^2 + 96*a^
3*b^3*c^4*d^3 + 84*a^4*b^2*c^3*d^4 + 21*a^5*b*c^2*d^5 - 9*a^6*c*d^6)*x^3 + 3*(8*a*b^5*c^7 - 40*a^2*b^4*c^6*d +
32*a^3*b^3*c^5*d^2 - 16*a^4*b^2*c^4*d^3 + 21*a^5*b*c^3*d^4 - 5*a^6*c^2*d^5)*x)*sqrt(b*x^2 + a))/(a^4*b^5*c^10
- 5*a^5*b^4*c^9*d + 10*a^6*b^3*c^8*d^2 - 10*a^7*b^2*c^7*d^3 + 5*a^8*b*c^6*d^4 - a^9*c^5*d^5 + (a^2*b^7*c^8*d^
2 - 5*a^3*b^6*c^7*d^3 + 10*a^4*b^5*c^6*d^4 - 10*a^5*b^4*c^5*d^5 + 5*a^6*b^3*c^4*d^6 - a^7*b^2*c^3*d^7)*x^8 + 2
*(a^2*b^7*c^9*d - 4*a^3*b^6*c^8*d^2 + 5*a^4*b^5*c^7*d^3 - 5*a^6*b^3*c^5*d^5 + 4*a^7*b^2*c^4*d^6 - a^8*b*c^3*d^
7)*x^6 + (a^2*b^7*c^10 - a^3*b^6*c^9*d - 9*a^4*b^5*c^8*d^2 + 25*a^5*b^4*c^7*d^3 - 25*a^6*b^3*c^6*d^4 + 9*a^7*b
^2*c^5*d^5 + a^8*b*c^4*d^6 - a^9*c^3*d^7)*x^4 + 2*(a^3*b^6*c^10 - 4*a^4*b^5*c^9*d + 5*a^5*b^4*c^8*d^2 - 5*a^7*
b^2*c^6*d^4 + 4*a^8*b*c^5*d^5 - a^9*c^4*d^6)*x^2), -1/48*(3*(48*a^4*b^2*c^4*d^2 - 16*a^5*b*c^3*d^3 + 3*a^6*c^2
*d^4 + (48*a^2*b^4*c^2*d^4 - 16*a^3*b^3*c*d^5 + 3*a^4*b^2*d^6)*x^8 + 2*(48*a^2*b^4*c^3*d^3 + 32*a^3*b^3*c^2*d^
4 - 13*a^4*b^2*c*d^5 + 3*a^5*b*d^6)*x^6 + (48*a^2*b^4*c^4*d^2 + 176*a^3*b^3*c^3*d^3 - 13*a^4*b^2*c^2*d^4 - 4*a
^5*b*c*d^5 + 3*a^6*d^6)*x^4 + 2*(48*a^3*b^3*c^4*d^2 + 32*a^4*b^2*c^3*d^3 - 13*a^5*b*c^2*d^4 + 3*a^6*c*d^5)*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*((16*b^6*c^5*d^2 - 104*a*b^5*c^4*d^3 + 46*a^2*b^4*c^3*d^4 + 51*a^3*b
^3*c^2*d^5 - 9*a^4*b^2*c*d^6)*x^7 + (32*b^6*c^6*d - 184*a*b^5*c^5*d^2 + 8*a^2*b^4*c^4*d^3 + 75*a^3*b^3*c^3*d^4
+ 87*a^4*b^2*c^2*d^5 - 18*a^5*b*c*d^6)*x^5 + (16*b^6*c^7 - 56*a*b^5*c^6*d - 152*a^2*b^4*c^5*d^2 + 96*a^3*b^3*
c^4*d^3 + 84*a^4*b^2*c^3*d^4 + 21*a^5*b*c^2*d^5 - 9*a^6*c*d^6)*x^3 + 3*(8*a*b^5*c^7 - 40*a^2*b^4*c^6*d + 32*a^
3*b^3*c^5*d^2 - 16*a^4*b^2*c^4*d^3 + 21*a^5*b*c^3*d^4 - 5*a^6*c^2*d^5)*x)*sqrt(b*x^2 + a))/(a^4*b^5*c^10 - 5*a
^5*b^4*c^9*d + 10*a^6*b^3*c^8*d^2 - 10*a^7*b^2*c^7*d^3 + 5*a^8*b*c^6*d^4 - a^9*c^5*d^5 + (a^2*b^7*c^8*d^2 - 5*
a^3*b^6*c^7*d^3 + 10*a^4*b^5*c^6*d^4 - 10*a^5*b^4*c^5*d^5 + 5*a^6*b^3*c^4*d^6 - a^7*b^2*c^3*d^7)*x^8 + 2*(a^2*
b^7*c^9*d - 4*a^3*b^6*c^8*d^2 + 5*a^4*b^5*c^7*d^3 - 5*a^6*b^3*c^5*d^5 + 4*a^7*b^2*c^4*d^6 - a^8*b*c^3*d^7)*x^6
+ (a^2*b^7*c^10 - a^3*b^6*c^9*d - 9*a^4*b^5*c^8*d^2 + 25*a^5*b^4*c^7*d^3 - 25*a^6*b^3*c^6*d^4 + 9*a^7*b^2*c^5
*d^5 + a^8*b*c^4*d^6 - a^9*c^3*d^7)*x^4 + 2*(a^3*b^6*c^10 - 4*a^4*b^5*c^9*d + 5*a^5*b^4*c^8*d^2 - 5*a^7*b^2*c^
6*d^4 + 4*a^8*b*c^5*d^5 - a^9*c^4*d^6)*x^2)]
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giac [B] time = 3.76, size = 1010, normalized size = 3.23 \[ \frac {{\left (\frac {{\left (2 \, b^{10} c^{5} - 19 \, a b^{9} c^{4} d + 56 \, a^{2} b^{8} c^{3} d^{2} - 74 \, a^{3} b^{7} c^{2} d^{3} + 46 \, a^{4} b^{6} c d^{4} - 11 \, a^{5} b^{5} d^{5}\right )} x^{2}}{a^{2} b^{9} c^{8} - 8 \, a^{3} b^{8} c^{7} d + 28 \, a^{4} b^{7} c^{6} d^{2} - 56 \, a^{5} b^{6} c^{5} d^{3} + 70 \, a^{6} b^{5} c^{4} d^{4} - 56 \, a^{7} b^{4} c^{3} d^{5} + 28 \, a^{8} b^{3} c^{2} d^{6} - 8 \, a^{9} b^{2} c d^{7} + a^{10} b d^{8}} + \frac {3 \, {\left (a b^{9} c^{5} - 8 \, a^{2} b^{8} c^{4} d + 22 \, a^{3} b^{7} c^{3} d^{2} - 28 \, a^{4} b^{6} c^{2} d^{3} + 17 \, a^{5} b^{5} c d^{4} - 4 \, a^{6} b^{4} d^{5}\right )}}{a^{2} b^{9} c^{8} - 8 \, a^{3} b^{8} c^{7} d + 28 \, a^{4} b^{7} c^{6} d^{2} - 56 \, a^{5} b^{6} c^{5} d^{3} + 70 \, a^{6} b^{5} c^{4} d^{4} - 56 \, a^{7} b^{4} c^{3} d^{5} + 28 \, a^{8} b^{3} c^{2} d^{6} - 8 \, a^{9} b^{2} c d^{7} + a^{10} b d^{8}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{\frac {5}{2}} c^{2} d^{2} - 16 \, a b^{\frac {3}{2}} c d^{3} + 3 \, a^{2} \sqrt {b} d^{4}\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 )}{8 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c d^{4} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{5} + 112 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} d^{2} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d^{3} + 66 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{4} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{5} + 88 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d^{3} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{4} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{5} + 14 \, a^{4} b^{\frac {3}{2}} c d^{4} - 3 \, a^{5} \sqrt {b} d^{5}}{4 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\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 )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x, algorithm="giac")
[Out]
1/3*((2*b^10*c^5 - 19*a*b^9*c^4*d + 56*a^2*b^8*c^3*d^2 - 74*a^3*b^7*c^2*d^3 + 46*a^4*b^6*c*d^4 - 11*a^5*b^5*d^
5)*x^2/(a^2*b^9*c^8 - 8*a^3*b^8*c^7*d + 28*a^4*b^7*c^6*d^2 - 56*a^5*b^6*c^5*d^3 + 70*a^6*b^5*c^4*d^4 - 56*a^7*
b^4*c^3*d^5 + 28*a^8*b^3*c^2*d^6 - 8*a^9*b^2*c*d^7 + a^10*b*d^8) + 3*(a*b^9*c^5 - 8*a^2*b^8*c^4*d + 22*a^3*b^7
*c^3*d^2 - 28*a^4*b^6*c^2*d^3 + 17*a^5*b^5*c*d^4 - 4*a^6*b^4*d^5)/(a^2*b^9*c^8 - 8*a^3*b^8*c^7*d + 28*a^4*b^7*
c^6*d^2 - 56*a^5*b^6*c^5*d^3 + 70*a^6*b^5*c^4*d^4 - 56*a^7*b^4*c^3*d^5 + 28*a^8*b^3*c^2*d^6 - 8*a^9*b^2*c*d^7
+ a^10*b*d^8))*x/(b*x^2 + a)^(3/2) - 1/8*(48*b^(5/2)*c^2*d^2 - 16*a*b^(3/2)*c*d^3 + 3*a^2*sqrt(b)*d^4)*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^4*c^6 - 4*a*b^3*c^5*d + 6*
a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 + a^4*c^2*d^4)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/4*(24*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*b^(5/2)*c^2*d^3 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d^4 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^
6*a^2*sqrt(b)*d^5 + 112*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3*d^2 - 136*(sqrt(b)*x - sqrt(b*x^2 + a))^4*
a*b^(5/2)*c^2*d^3 + 66*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^4 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a
^3*sqrt(b)*d^5 + 88*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d^3 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a
^3*b^(3/2)*c*d^4 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d^5 + 14*a^4*b^(3/2)*c*d^4 - 3*a^5*sqrt(b)*d^
5)/((b^4*c^6 - 4*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 + a^4*c^2*d^4)*((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)^2)
________________________________________________________________________________________
maple [B] time = 0.03, size = 4495, normalized size = 14.36 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x)
[Out]
3/16/c^2/(a*d-b*c)/(x-(-c*d)^(1/2)/d)/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/
d)^(3/2)+3/16/c^2/(a*d-b*c)/(x+(-c*d)^(1/2)/d)/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(
a*d-b*c)/d)^(3/2)+1/16/(-c*d)^(1/2)/c/(a*d-b*c)/(x-(-c*d)^(1/2)/d)^2/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x
-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-7/16/c*b/(a*d-b*c)^2/(x-(-c*d)^(1/2)/d)/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*
d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-15/16/c^2*d*(-c*d)^(1/2)*b/(a*d-b*c)^3/((a*d-b*c)/d)^(1/2)*
ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)
^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))+5/48/(-c*d)^(1/2)/c*d*b/(a*d-b*c)^2/((x-
(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)+5/16/(-c*d)^(1/2)/c*d^2*b/(a*d-b*
c)^3/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)+3/16/c^2*b/(a*d-b*c)/a/(
(x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x+3/8/c^2*b/(a*d-b*c)/a^2/((x-
(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x-3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-
b*c)^2/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x-(
-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-7/16/c*b/(a*d
-b*c)^2/(x+(-c*d)^(1/2)/d)/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-35
/48/(-c*d)^(1/2)*d*b^2/(a*d-b*c)^3/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^
(3/2)-35/48*b^3/(a*d-b*c)^3/a/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)
*x-35/24*b^3/(a*d-b*c)^3/a^2/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*
x-35/16/(-c*d)^(1/2)*d^2*b^2/(a*d-b*c)^4/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*
c)/d)^(1/2)+5/16/c^2*(-c*d)^(1/2)*b/(a*d-b*c)^2/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+
(a*d-b*c)/d)^(3/2)+3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^2/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/
2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a
*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))+15/16/c^2*d*(-c*d)^(1/2)*b/(a*d-b*c)^3/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^
(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)-15/16/c^2*d*(-c*d)^(1/2)*b/(a*d-b*c)^3/((x+(-c*d)^(1/2)/d)^2*b
-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)+3/16/c^2*b/(a*d-b*c)/a/((x+(-c*d)^(1/2)/d)^2*b-2*(-c
*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x+3/8/c^2*b/(a*d-b*c)/a^2/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)
^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x-35/16*d*b^3/(a*d-b*c)^4/a/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^
(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x+35/16/(-c*d)^(1/2)*d^2*b^2/(a*d-b*c)^4/((a*d-b*c)/d)^(1/2)*l
n((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)
^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))-3/8/c*b^2/(a*d-b*c)^2/a/((x+(-c*d)^(1/2)
/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x-3/4/c*b^2/(a*d-b*c)^2/a^2/((x+(-c*d)^(1/2)/
d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x-5/48/(-c*d)^(1/2)/c*d*b/(a*d-b*c)^2/((x+(-c*
d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-5/16/(-c*d)^(1/2)/c*d^2*b/(a*d-b*c)^3
/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)-35/16*d*b^3/(a*d-b*c)^4/a/((
x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x-35/16/(-c*d)^(1/2)*d^2*b^2/(a
*d-b*c)^4/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((
x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-3/8/c*b^2/
(a*d-b*c)^2/a/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x-3/4/c*b^2/(a*
d-b*c)^2/a^2/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x+15/16/c^2*d*(-
c*d)^(1/2)*b/(a*d-b*c)^3/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-
b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2
)/d))+5/8/c*d*b^2/(a*d-b*c)^3/a/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/
2)*x+5/16/(-c*d)^(1/2)/c*d^2*b/(a*d-b*c)^3/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a
*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/
2))/(x+(-c*d)^(1/2)/d))+5/8/c*d*b^2/(a*d-b*c)^3/a/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/
d+(a*d-b*c)/d)^(1/2)*x-5/16/(-c*d)^(1/2)/c*d^2*b/(a*d-b*c)^3/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^
(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d
+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-3/16/c^2*d/(a*d-b*c)^2/a/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-
c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x*b-3/16/c^2*d/(a*d-b*c)^2/a/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-
c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x*b+35/48/(-c*d)^(1/2)*d*b^2/(a*d-b*c)^3/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)
^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-35/48*b^3/(a*d-b*c)^3/a/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2
)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x-35/24*b^3/(a*d-b*c)^3/a^2/((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)
*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x+35/16/(-c*d)^(1/2)*d^2*b^2/(a*d-b*c)^4/((x-(-c*d)^(1/2)/d)^2*b+2*
(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)+1/16/(-c*d)^(1/2)/c^2/(a*d-b*c)*d/((x-(-c*d)^(1/2)/d)^2
*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)+3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^2/((x-(-c*d)^(
1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)-1/16/(-c*d)^(1/2)/c^2/(a*d-b*c)*d/((x+(-c
*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^
2/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)-5/16/c^2*(-c*d)^(1/2)*b/(a*
d-b*c)^2/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-1/16/(-c*d)^(1/2)/c/
(a*d-b*c)/(x+(-c*d)^(1/2)/d)^2/((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2
)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^3), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x)
[Out]
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^3), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**3,x)
[Out]
Timed out
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