3.1.41 \(\int \frac {1}{(a+b x^2)^3 (c+d x^2)^2} \, dx\) [41]
Optimal. Leaf size=236 \[ \frac {d (b c-4 a d) (3 b c+a d) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {3 b (b c-3 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{3/2} \left (3 b^2 c^2-14 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^4}-\frac {d^{5/2} (7 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^4} \]
[Out]
1/8*d*(-4*a*d+b*c)*(a*d+3*b*c)*x/a^2/c/(-a*d+b*c)^3/(d*x^2+c)+1/4*b*x/a/(-a*d+b*c)/(b*x^2+a)^2/(d*x^2+c)+3/8*b
*(-3*a*d+b*c)*x/a^2/(-a*d+b*c)^2/(b*x^2+a)/(d*x^2+c)+1/8*b^(3/2)*(35*a^2*d^2-14*a*b*c*d+3*b^2*c^2)*arctan(x*b^
(1/2)/a^(1/2))/a^(5/2)/(-a*d+b*c)^4-1/2*d^(5/2)*(-a*d+7*b*c)*arctan(x*d^(1/2)/c^(1/2))/c^(3/2)/(-a*d+b*c)^4
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Rubi [A]
time = 0.21, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {425, 541, 536,
211} \begin {gather*} \frac {d x (b c-4 a d) (a d+3 b c)}{8 a^2 c \left (c+d x^2\right ) (b c-a d)^3}+\frac {3 b x (b c-3 a d)}{8 a^2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)^2}+\frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (35 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{8 a^{5/2} (b c-a d)^4}-\frac {d^{5/2} (7 b c-a d) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^4}+\frac {b x}{4 a \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x^2)^3*(c + d*x^2)^2),x]
[Out]
(d*(b*c - 4*a*d)*(3*b*c + a*d)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(
c + d*x^2)) + (3*b*(b*c - 3*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + (b^(3/2)*(3*b^2*c^2 - 14*a
*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^4) - (d^(5/2)*(7*b*c - a*d)*ArcTan[(S
qrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^4)
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 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 536
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, 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 )^3 \left (c+d x^2\right )^2} \, dx &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )}-\frac {\int \frac {-3 b c+4 a d-5 b d x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx}{4 a (b c-a d)}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {3 b (b c-3 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {3 b^2 c^2-5 a b c d+8 a^2 d^2+9 b d (b c-3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac {d (b c-4 a d) (3 b c+a d) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {3 b (b c-3 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {2 \left (3 b^3 c^3-11 a b^2 c^2 d+24 a^2 b c d^2-4 a^3 d^3\right )+2 b d (b c-4 a d) (3 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 a^2 c (b c-a d)^3}\\ &=\frac {d (b c-4 a d) (3 b c+a d) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {3 b (b c-3 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (d^3 (7 b c-a d)\right ) \int \frac {1}{c+d x^2} \, dx}{2 c (b c-a d)^4}+\frac {\left (b^2 \left (3 b^2 c^2-14 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 (b c-a d)^4}\\ &=\frac {d (b c-4 a d) (3 b c+a d) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {3 b (b c-3 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{3/2} \left (3 b^2 c^2-14 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^4}-\frac {d^{5/2} (7 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^4}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 197, normalized size = 0.83 \begin {gather*} \frac {1}{8} \left (\frac {2 b^2 x}{a (b c-a d)^2 \left (a+b x^2\right )^2}+\frac {b^2 (-3 b c+11 a d) x}{a^2 (-b c+a d)^3 \left (a+b x^2\right )}-\frac {4 d^3 x}{c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b^{3/2} \left (3 b^2 c^2-14 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (b c-a d)^4}+\frac {4 d^{5/2} (-7 b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x^2)^3*(c + d*x^2)^2),x]
[Out]
((2*b^2*x)/(a*(b*c - a*d)^2*(a + b*x^2)^2) + (b^2*(-3*b*c + 11*a*d)*x)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (4
*d^3*x)/(c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(3*b^2*c^2 - 14*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt
[a]])/(a^(5/2)*(b*c - a*d)^4) + (4*d^(5/2)*(-7*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*(b*c - a*d)^4)
)/8
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Maple [A]
time = 0.22, size = 196, normalized size = 0.83
| | |
| method |
result |
size |
| | |
| default |
\(\frac {b^{2} \left (\frac {\frac {b \left (11 a^{2} d^{2}-14 a b c d +3 b^{2} c^{2}\right ) x^{3}}{8 a^{2}}+\frac {\left (13 a^{2} d^{2}-18 a b c d +5 b^{2} c^{2}\right ) x}{8 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (35 a^{2} d^{2}-14 a b c d +3 b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a d -b c \right )^{4}}+\frac {d^{3} \left (\frac {\left (a d -b c \right ) x}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -7 b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{4}}\) |
\(196\) |
| risch |
\(\text {Expression too large to display}\) |
\(3743\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x^2+a)^3/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
[Out]
b^2/(a*d-b*c)^4*((1/8*b*(11*a^2*d^2-14*a*b*c*d+3*b^2*c^2)/a^2*x^3+1/8*(13*a^2*d^2-18*a*b*c*d+5*b^2*c^2)/a*x)/(
b*x^2+a)^2+1/8*(35*a^2*d^2-14*a*b*c*d+3*b^2*c^2)/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d^3/(a*d-b*c)^4*(1/2
*(a*d-b*c)/c*x/(d*x^2+c)+1/2*(a*d-7*b*c)/c/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 530 vs.
\(2 (210) = 420\).
time = 0.52, size = 530, normalized size = 2.25 \begin {gather*} \frac {{\left (3 \, b^{4} c^{2} - 14 \, a b^{3} c d + 35 \, a^{2} b^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}\right )} \sqrt {a b}} - \frac {{\left (7 \, b c d^{3} - a d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{4} c^{2} d - 11 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} x^{5} + {\left (3 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} - 8 \, a^{3} b d^{3}\right )} x^{3} + {\left (5 \, a b^{3} c^{3} - 13 \, a^{2} b^{2} c^{2} d - 4 \, a^{4} d^{3}\right )} x}{8 \, {\left (a^{4} b^{3} c^{5} - 3 \, a^{5} b^{2} c^{4} d + 3 \, a^{6} b c^{3} d^{2} - a^{7} c^{2} d^{3} + {\left (a^{2} b^{5} c^{4} d - 3 \, a^{3} b^{4} c^{3} d^{2} + 3 \, a^{4} b^{3} c^{2} d^{3} - a^{5} b^{2} c d^{4}\right )} x^{6} + {\left (a^{2} b^{5} c^{5} - a^{3} b^{4} c^{4} d - 3 \, a^{4} b^{3} c^{3} d^{2} + 5 \, a^{5} b^{2} c^{2} d^{3} - 2 \, a^{6} b c d^{4}\right )} x^{4} + {\left (2 \, a^{3} b^{4} c^{5} - 5 \, a^{4} b^{3} c^{4} d + 3 \, a^{5} b^{2} c^{3} d^{2} + a^{6} b c^{2} d^{3} - a^{7} c d^{4}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^3/(d*x^2+c)^2,x, algorithm="maxima")
[Out]
1/8*(3*b^4*c^2 - 14*a*b^3*c*d + 35*a^2*b^2*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*
b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4)*sqrt(a*b)) - 1/2*(7*b*c*d^3 - a*d^4)*arctan(d*x/sqrt(c*d))/((b^4*c^5 -
4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4)*sqrt(c*d)) + 1/8*((3*b^4*c^2*d - 11*a*b^3*c*d
^2 - 4*a^2*b^2*d^3)*x^5 + (3*b^4*c^3 - 6*a*b^3*c^2*d - 13*a^2*b^2*c*d^2 - 8*a^3*b*d^3)*x^3 + (5*a*b^3*c^3 - 13
*a^2*b^2*c^2*d - 4*a^4*d^3)*x)/(a^4*b^3*c^5 - 3*a^5*b^2*c^4*d + 3*a^6*b*c^3*d^2 - a^7*c^2*d^3 + (a^2*b^5*c^4*d
- 3*a^3*b^4*c^3*d^2 + 3*a^4*b^3*c^2*d^3 - a^5*b^2*c*d^4)*x^6 + (a^2*b^5*c^5 - a^3*b^4*c^4*d - 3*a^4*b^3*c^3*d
^2 + 5*a^5*b^2*c^2*d^3 - 2*a^6*b*c*d^4)*x^4 + (2*a^3*b^4*c^5 - 5*a^4*b^3*c^4*d + 3*a^5*b^2*c^3*d^2 + a^6*b*c^2
*d^3 - a^7*c*d^4)*x^2)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 788 vs.
\(2 (210) = 420\).
time = 2.88, size = 3251, normalized size = 13.78 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^3/(d*x^2+c)^2,x, algorithm="fricas")
[Out]
[1/16*(2*(3*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 7*a^2*b^3*c*d^3 + 4*a^3*b^2*d^4)*x^5 + 2*(3*b^5*c^4 - 9*a*b^4*c^3*d
- 7*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 + 8*a^4*b*d^4)*x^3 + (3*a^2*b^3*c^4 - 14*a^3*b^2*c^3*d + 35*a^4*b*c^2*d
^2 + (3*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 35*a^2*b^3*c*d^3)*x^6 + (3*b^5*c^4 - 8*a*b^4*c^3*d + 7*a^2*b^3*c^2*d^2
+ 70*a^3*b^2*c*d^3)*x^4 + (6*a*b^4*c^4 - 25*a^2*b^3*c^3*d + 56*a^3*b^2*c^2*d^2 + 35*a^4*b*c*d^3)*x^2)*sqrt(-b/
a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 4*(7*a^4*b*c^2*d^2 - a^5*c*d^3 + (7*a^2*b^3*c*d^3 - a^3*b
^2*d^4)*x^6 + (7*a^2*b^3*c^2*d^2 + 13*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*x^4 + (14*a^3*b^2*c^2*d^2 + 5*a^4*b*c*d^3 -
a^5*d^4)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^4*c^4 - 18*a^2*b^3*c^3*d
+ 13*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + 4*a^5*d^4)*x)/(a^4*b^4*c^6 - 4*a^5*b^3*c^5*d + 6*a^6*b^2*c^4*d^2 - 4*a^
7*b*c^3*d^3 + a^8*c^2*d^4 + (a^2*b^6*c^5*d - 4*a^3*b^5*c^4*d^2 + 6*a^4*b^4*c^3*d^3 - 4*a^5*b^3*c^2*d^4 + a^6*b
^2*c*d^5)*x^6 + (a^2*b^6*c^6 - 2*a^3*b^5*c^5*d - 2*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 7*a^6*b^2*c^2*d^4 + 2
*a^7*b*c*d^5)*x^4 + (2*a^3*b^5*c^6 - 7*a^4*b^4*c^5*d + 8*a^5*b^3*c^4*d^2 - 2*a^6*b^2*c^3*d^3 - 2*a^7*b*c^2*d^4
+ a^8*c*d^5)*x^2), 1/16*(2*(3*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 7*a^2*b^3*c*d^3 + 4*a^3*b^2*d^4)*x^5 + 2*(3*b^5*
c^4 - 9*a*b^4*c^3*d - 7*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 + 8*a^4*b*d^4)*x^3 - 8*(7*a^4*b*c^2*d^2 - a^5*c*d^3
+ (7*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (7*a^2*b^3*c^2*d^2 + 13*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*x^4 + (14*a^3*b^2
*c^2*d^2 + 5*a^4*b*c*d^3 - a^5*d^4)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (3*a^2*b^3*c^4 - 14*a^3*b^2*c^3*d + 3
5*a^4*b*c^2*d^2 + (3*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 35*a^2*b^3*c*d^3)*x^6 + (3*b^5*c^4 - 8*a*b^4*c^3*d + 7*a^2
*b^3*c^2*d^2 + 70*a^3*b^2*c*d^3)*x^4 + (6*a*b^4*c^4 - 25*a^2*b^3*c^3*d + 56*a^3*b^2*c^2*d^2 + 35*a^4*b*c*d^3)*
x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(5*a*b^4*c^4 - 18*a^2*b^3*c^3*d + 13*a^3*b
^2*c^2*d^2 - 4*a^4*b*c*d^3 + 4*a^5*d^4)*x)/(a^4*b^4*c^6 - 4*a^5*b^3*c^5*d + 6*a^6*b^2*c^4*d^2 - 4*a^7*b*c^3*d^
3 + a^8*c^2*d^4 + (a^2*b^6*c^5*d - 4*a^3*b^5*c^4*d^2 + 6*a^4*b^4*c^3*d^3 - 4*a^5*b^3*c^2*d^4 + a^6*b^2*c*d^5)*
x^6 + (a^2*b^6*c^6 - 2*a^3*b^5*c^5*d - 2*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 7*a^6*b^2*c^2*d^4 + 2*a^7*b*c*d
^5)*x^4 + (2*a^3*b^5*c^6 - 7*a^4*b^4*c^5*d + 8*a^5*b^3*c^4*d^2 - 2*a^6*b^2*c^3*d^3 - 2*a^7*b*c^2*d^4 + a^8*c*d
^5)*x^2), 1/8*((3*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 7*a^2*b^3*c*d^3 + 4*a^3*b^2*d^4)*x^5 + (3*b^5*c^4 - 9*a*b^4*c
^3*d - 7*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 + 8*a^4*b*d^4)*x^3 + (3*a^2*b^3*c^4 - 14*a^3*b^2*c^3*d + 35*a^4*b*c
^2*d^2 + (3*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 35*a^2*b^3*c*d^3)*x^6 + (3*b^5*c^4 - 8*a*b^4*c^3*d + 7*a^2*b^3*c^2*
d^2 + 70*a^3*b^2*c*d^3)*x^4 + (6*a*b^4*c^4 - 25*a^2*b^3*c^3*d + 56*a^3*b^2*c^2*d^2 + 35*a^4*b*c*d^3)*x^2)*sqrt
(b/a)*arctan(x*sqrt(b/a)) - 2*(7*a^4*b*c^2*d^2 - a^5*c*d^3 + (7*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (7*a^2*b^3*
c^2*d^2 + 13*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*x^4 + (14*a^3*b^2*c^2*d^2 + 5*a^4*b*c*d^3 - a^5*d^4)*x^2)*sqrt(-d/c)
*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + (5*a*b^4*c^4 - 18*a^2*b^3*c^3*d + 13*a^3*b^2*c^2*d^2 - 4*a^
4*b*c*d^3 + 4*a^5*d^4)*x)/(a^4*b^4*c^6 - 4*a^5*b^3*c^5*d + 6*a^6*b^2*c^4*d^2 - 4*a^7*b*c^3*d^3 + a^8*c^2*d^4 +
(a^2*b^6*c^5*d - 4*a^3*b^5*c^4*d^2 + 6*a^4*b^4*c^3*d^3 - 4*a^5*b^3*c^2*d^4 + a^6*b^2*c*d^5)*x^6 + (a^2*b^6*c^
6 - 2*a^3*b^5*c^5*d - 2*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 7*a^6*b^2*c^2*d^4 + 2*a^7*b*c*d^5)*x^4 + (2*a^3*
b^5*c^6 - 7*a^4*b^4*c^5*d + 8*a^5*b^3*c^4*d^2 - 2*a^6*b^2*c^3*d^3 - 2*a^7*b*c^2*d^4 + a^8*c*d^5)*x^2), 1/8*((3
*b^5*c^3*d - 14*a*b^4*c^2*d^2 + 7*a^2*b^3*c*d^3 + 4*a^3*b^2*d^4)*x^5 + (3*b^5*c^4 - 9*a*b^4*c^3*d - 7*a^2*b^3*
c^2*d^2 + 5*a^3*b^2*c*d^3 + 8*a^4*b*d^4)*x^3 + (3*a^2*b^3*c^4 - 14*a^3*b^2*c^3*d + 35*a^4*b*c^2*d^2 + (3*b^5*c
^3*d - 14*a*b^4*c^2*d^2 + 35*a^2*b^3*c*d^3)*x^6 + (3*b^5*c^4 - 8*a*b^4*c^3*d + 7*a^2*b^3*c^2*d^2 + 70*a^3*b^2*
c*d^3)*x^4 + (6*a*b^4*c^4 - 25*a^2*b^3*c^3*d + 56*a^3*b^2*c^2*d^2 + 35*a^4*b*c*d^3)*x^2)*sqrt(b/a)*arctan(x*sq
rt(b/a)) - 4*(7*a^4*b*c^2*d^2 - a^5*c*d^3 + (7*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (7*a^2*b^3*c^2*d^2 + 13*a^3*
b^2*c*d^3 - 2*a^4*b*d^4)*x^4 + (14*a^3*b^2*c^2*d^2 + 5*a^4*b*c*d^3 - a^5*d^4)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c
)) + (5*a*b^4*c^4 - 18*a^2*b^3*c^3*d + 13*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + 4*a^5*d^4)*x)/(a^4*b^4*c^6 - 4*a^5
*b^3*c^5*d + 6*a^6*b^2*c^4*d^2 - 4*a^7*b*c^3*d^3 + a^8*c^2*d^4 + (a^2*b^6*c^5*d - 4*a^3*b^5*c^4*d^2 + 6*a^4*b^
4*c^3*d^3 - 4*a^5*b^3*c^2*d^4 + a^6*b^2*c*d^5)*x^6 + (a^2*b^6*c^6 - 2*a^3*b^5*c^5*d - 2*a^4*b^4*c^4*d^2 + 8*a^
5*b^3*c^3*d^3 - 7*a^6*b^2*c^2*d^4 + 2*a^7*b*c*d^5)*x^4 + (2*a^3*b^5*c^6 - 7*a^4*b^4*c^5*d + 8*a^5*b^3*c^4*d^2
- 2*a^6*b^2*c^3*d^3 - 2*a^7*b*c^2*d^4 + a^8*c*d^5)*x^2)]
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x**2+a)**3/(d*x**2+c)**2,x)
[Out]
Timed out
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Giac [A]
time = 1.17, size = 333, normalized size = 1.41 \begin {gather*} -\frac {d^{3} x}{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 )} {\left (d x^{2} + c\right )}} + \frac {{\left (3 \, b^{4} c^{2} - 14 \, a b^{3} c d + 35 \, a^{2} b^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}\right )} \sqrt {a b}} - \frac {{\left (7 \, b c d^{3} - a d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt {c d}} + \frac {3 \, b^{4} c x^{3} - 11 \, a b^{3} d x^{3} + 5 \, a b^{3} c x - 13 \, a^{2} b^{2} d x}{8 \, {\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 )} {\left (b x^{2} + a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^3/(d*x^2+c)^2,x, algorithm="giac")
[Out]
-1/2*d^3*x/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*x^2 + c)) + 1/8*(3*b^4*c^2 - 14*a*b^3*c
*d + 35*a^2*b^2*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3
+ a^6*d^4)*sqrt(a*b)) - 1/2*(7*b*c*d^3 - a*d^4)*arctan(d*x/sqrt(c*d))/((b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c
^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4)*sqrt(c*d)) + 1/8*(3*b^4*c*x^3 - 11*a*b^3*d*x^3 + 5*a*b^3*c*x - 13*a^2*b^
2*d*x)/((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)*(b*x^2 + a)^2)
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Mupad [B]
time = 7.85, size = 2500, normalized size = 10.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b*x^2)^3*(c + d*x^2)^2),x)
[Out]
((x^5*(4*a^2*b^2*d^3 - 3*b^4*c^2*d + 11*a*b^3*c*d^2))/(8*a^2*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*
d^2)) + (x*(4*a^3*d^3 - 5*b^3*c^3 + 13*a*b^2*c^2*d))/(8*a*c*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*
x^3*(8*a^3*d^3 - 3*b^3*c^3 + 6*a*b^2*c^2*d + 13*a^2*b*c*d^2))/(8*a^2*c*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*
c*d)))/(a^2*c + x^2*(a^2*d + 2*a*b*c) + x^4*(b^2*c + 2*a*b*d) + b^2*d*x^6) - (atan(((((x*(16*a^6*b^3*d^9 + 9*b
^9*c^6*d^3 - 84*a*b^8*c^5*d^4 - 224*a^5*b^4*c*d^8 + 406*a^2*b^7*c^4*d^5 - 980*a^3*b^6*c^3*d^6 + 2009*a^4*b^5*c
^2*d^7))/(32*(a^4*b^6*c^8 + a^10*c^2*d^6 - 6*a^5*b^5*c^7*d - 6*a^9*b*c^3*d^5 + 15*a^6*b^4*c^6*d^2 - 20*a^7*b^3
*c^5*d^3 + 15*a^8*b^2*c^4*d^4)) - (((2*a^13*b^2*c*d^13 - (3*a^2*b^13*c^12*d^2)/2 + (35*a^3*b^12*c^11*d^3)/2 -
98*a^4*b^11*c^10*d^4 + 336*a^5*b^10*c^9*d^5 - 765*a^6*b^9*c^8*d^6 + 1197*a^7*b^8*c^7*d^7 - 1302*a^8*b^7*c^6*d^
8 + 978*a^9*b^6*c^5*d^9 - (987*a^10*b^5*c^4*d^10)/2 + (315*a^11*b^4*c^3*d^11)/2 - 28*a^12*b^3*c^2*d^12)/(a^4*b
^9*c^11 - a^13*c^2*d^9 - 9*a^5*b^8*c^10*d + 9*a^12*b*c^3*d^8 + 36*a^6*b^7*c^9*d^2 - 84*a^7*b^6*c^8*d^3 + 126*a
^8*b^5*c^7*d^4 - 126*a^9*b^4*c^6*d^5 + 84*a^10*b^3*c^5*d^6 - 36*a^11*b^2*c^4*d^7) - (x*(-a^5*b^3)^(1/2)*(35*a^
2*d^2 + 3*b^2*c^2 - 14*a*b*c*d)*(256*a^4*b^11*c^11*d^2 - 1792*a^5*b^10*c^10*d^3 + 5120*a^6*b^9*c^9*d^4 - 7168*
a^7*b^8*c^8*d^5 + 3584*a^8*b^7*c^7*d^6 + 3584*a^9*b^6*c^6*d^7 - 7168*a^10*b^5*c^5*d^8 + 5120*a^11*b^4*c^4*d^9
- 1792*a^12*b^3*c^3*d^10 + 256*a^13*b^2*c^2*d^11))/(512*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c
^2*d^2 - 4*a^8*b*c*d^3)*(a^4*b^6*c^8 + a^10*c^2*d^6 - 6*a^5*b^5*c^7*d - 6*a^9*b*c^3*d^5 + 15*a^6*b^4*c^6*d^2 -
20*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4)))*(-a^5*b^3)^(1/2)*(35*a^2*d^2 + 3*b^2*c^2 - 14*a*b*c*d))/(16*(a^9*d
^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3)))*(-a^5*b^3)^(1/2)*(35*a^2*d^2 + 3*b^2
*c^2 - 14*a*b*c*d)*1i)/(16*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3)) + ((
(x*(16*a^6*b^3*d^9 + 9*b^9*c^6*d^3 - 84*a*b^8*c^5*d^4 - 224*a^5*b^4*c*d^8 + 406*a^2*b^7*c^4*d^5 - 980*a^3*b^6*
c^3*d^6 + 2009*a^4*b^5*c^2*d^7))/(32*(a^4*b^6*c^8 + a^10*c^2*d^6 - 6*a^5*b^5*c^7*d - 6*a^9*b*c^3*d^5 + 15*a^6*
b^4*c^6*d^2 - 20*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4)) + (((2*a^13*b^2*c*d^13 - (3*a^2*b^13*c^12*d^2)/2 + (35
*a^3*b^12*c^11*d^3)/2 - 98*a^4*b^11*c^10*d^4 + 336*a^5*b^10*c^9*d^5 - 765*a^6*b^9*c^8*d^6 + 1197*a^7*b^8*c^7*d
^7 - 1302*a^8*b^7*c^6*d^8 + 978*a^9*b^6*c^5*d^9 - (987*a^10*b^5*c^4*d^10)/2 + (315*a^11*b^4*c^3*d^11)/2 - 28*a
^12*b^3*c^2*d^12)/(a^4*b^9*c^11 - a^13*c^2*d^9 - 9*a^5*b^8*c^10*d + 9*a^12*b*c^3*d^8 + 36*a^6*b^7*c^9*d^2 - 84
*a^7*b^6*c^8*d^3 + 126*a^8*b^5*c^7*d^4 - 126*a^9*b^4*c^6*d^5 + 84*a^10*b^3*c^5*d^6 - 36*a^11*b^2*c^4*d^7) + (x
*(-a^5*b^3)^(1/2)*(35*a^2*d^2 + 3*b^2*c^2 - 14*a*b*c*d)*(256*a^4*b^11*c^11*d^2 - 1792*a^5*b^10*c^10*d^3 + 5120
*a^6*b^9*c^9*d^4 - 7168*a^7*b^8*c^8*d^5 + 3584*a^8*b^7*c^7*d^6 + 3584*a^9*b^6*c^6*d^7 - 7168*a^10*b^5*c^5*d^8
+ 5120*a^11*b^4*c^4*d^9 - 1792*a^12*b^3*c^3*d^10 + 256*a^13*b^2*c^2*d^11))/(512*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6
*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3)*(a^4*b^6*c^8 + a^10*c^2*d^6 - 6*a^5*b^5*c^7*d - 6*a^9*b*c^3*d^
5 + 15*a^6*b^4*c^6*d^2 - 20*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4)))*(-a^5*b^3)^(1/2)*(35*a^2*d^2 + 3*b^2*c^2 -
14*a*b*c*d))/(16*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3)))*(-a^5*b^3)^(
1/2)*(35*a^2*d^2 + 3*b^2*c^2 - 14*a*b*c*d)*1i)/(16*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^
2 - 4*a^8*b*c*d^3)))/(((35*a^5*b^4*d^10)/16 + (63*b^9*c^5*d^5)/64 - (267*a*b^8*c^4*d^6)/32 - (651*a^4*b^5*c*d^
9)/64 + (451*a^2*b^7*c^3*d^7)/16 - (1275*a^3*b^6*c^2*d^8)/32)/(a^4*b^9*c^11 - a^13*c^2*d^9 - 9*a^5*b^8*c^10*d
+ 9*a^12*b*c^3*d^8 + 36*a^6*b^7*c^9*d^2 - 84*a^7*b^6*c^8*d^3 + 126*a^8*b^5*c^7*d^4 - 126*a^9*b^4*c^6*d^5 + 84*
a^10*b^3*c^5*d^6 - 36*a^11*b^2*c^4*d^7) - (((x*(16*a^6*b^3*d^9 + 9*b^9*c^6*d^3 - 84*a*b^8*c^5*d^4 - 224*a^5*b^
4*c*d^8 + 406*a^2*b^7*c^4*d^5 - 980*a^3*b^6*c^3*d^6 + 2009*a^4*b^5*c^2*d^7))/(32*(a^4*b^6*c^8 + a^10*c^2*d^6 -
6*a^5*b^5*c^7*d - 6*a^9*b*c^3*d^5 + 15*a^6*b^4*c^6*d^2 - 20*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4)) - (((2*a^1
3*b^2*c*d^13 - (3*a^2*b^13*c^12*d^2)/2 + (35*a^3*b^12*c^11*d^3)/2 - 98*a^4*b^11*c^10*d^4 + 336*a^5*b^10*c^9*d^
5 - 765*a^6*b^9*c^8*d^6 + 1197*a^7*b^8*c^7*d^7 - 1302*a^8*b^7*c^6*d^8 + 978*a^9*b^6*c^5*d^9 - (987*a^10*b^5*c^
4*d^10)/2 + (315*a^11*b^4*c^3*d^11)/2 - 28*a^12*b^3*c^2*d^12)/(a^4*b^9*c^11 - a^13*c^2*d^9 - 9*a^5*b^8*c^10*d
+ 9*a^12*b*c^3*d^8 + 36*a^6*b^7*c^9*d^2 - 84*a^7*b^6*c^8*d^3 + 126*a^8*b^5*c^7*d^4 - 126*a^9*b^4*c^6*d^5 + 84*
a^10*b^3*c^5*d^6 - 36*a^11*b^2*c^4*d^7) - (x*(-a^5*b^3)^(1/2)*(35*a^2*d^2 + 3*b^2*c^2 - 14*a*b*c*d)*(256*a^4*b
^11*c^11*d^2 - 1792*a^5*b^10*c^10*d^3 + 5120*a^6*b^9*c^9*d^4 - 7168*a^7*b^8*c^8*d^5 + 3584*a^8*b^7*c^7*d^6 + 3
584*a^9*b^6*c^6*d^7 - 7168*a^10*b^5*c^5*d^8 + 5120*a^11*b^4*c^4*d^9 - 1792*a^12*b^3*c^3*d^10 + 256*a^13*b^2*c^
2*d^11))/(512*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^...
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