3.4.10 \(\int \frac {x^4}{(a+b x^2)^2 (c+d x^2)^3} \, dx\)
Optimal. Leaf size=207 \[ \frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} \sqrt {d} (b c-a d)^4}+\frac {3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac {x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {3 \sqrt {a} \sqrt {b} (a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 (b c-a d)^4} \]
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Rubi [A] time = 0.28, antiderivative size = 207, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used =
{470, 527, 522, 205} \begin {gather*} \frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} \sqrt {d} (b c-a d)^4}+\frac {3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac {x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {3 \sqrt {a} \sqrt {b} (a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
((b*c + 2*a*d)*x)/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (3*(
b*c + 3*a*d)*x)/(8*(b*c - a*d)^3*(c + d*x^2)) - (3*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2
*(b*c - a*d)^4) + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*Sqrt[d]*(b*c - a*
d)^4)
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 470
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 522
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 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 {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\int \frac {a c+(-2 b c-3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{2 b (b c-a d)}\\ &=\frac {(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\int \frac {6 a b c^2-6 b c (b c+2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 b c (b c-a d)^2}\\ &=\frac {(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 (b c+3 a d) x}{8 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\int \frac {6 a b c^2 (3 b c+a d)-6 b^2 c^2 (b c+3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 b c^2 (b c-a d)^3}\\ &=\frac {(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 (b c+3 a d) x}{8 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(3 a b (b c+a d)) \int \frac {1}{a+b x^2} \, dx}{2 (b c-a d)^4}+\frac {\left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 (b c-a d)^4}\\ &=\frac {(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 (b c+3 a d) x}{8 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 \sqrt {a} \sqrt {b} (b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 (b c-a d)^4}+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} \sqrt {d} (b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 166, normalized size = 0.80 \begin {gather*} \frac {\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}+\frac {2 c x (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {4 a b x (b c-a d)}{a+b x^2}+\frac {x (5 a d+3 b c) (b c-a d)}{c+d x^2}-12 \sqrt {a} \sqrt {b} (a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
((4*a*b*(b*c - a*d)*x)/(a + b*x^2) + (2*c*(b*c - a*d)^2*x)/(c + d*x^2)^2 + ((b*c - a*d)*(3*b*c + 5*a*d)*x)/(c
+ d*x^2) - 12*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*Arc
Tan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]))/(8*(b*c - a*d)^4)
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)^3), x]
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fricas [B] time = 2.90, size = 2859, normalized size = 13.81
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")
[Out]
[1/16*(6*(b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*x^5 + 2*(5*b^3*c^4*d + 9*a*b^2*c^3*d^2 - 9*a^2*b*c^2*
d^3 - 5*a^3*c*d^4)*x^3 + 12*(a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b^2*c^3*d^2 + 3*a*b*
c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*
b)*x - a)/(b*x^2 + a)) - 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4
)*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c
^2*d^2 + 2*a^3*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + 6*(3*a*b^2*c^4*d - 2*a^2
*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*c^7*d - 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^3 - 4*a^4*b*c^4*d^4 + a^5*c^3*
d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4 + 6*a^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a^4*b*c*d^7)*x^6 + (2*b^5*c^6*d
^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4*d^4 - 2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 + a^5*c*d^7)*x^4 + (b^5*c^7*d -
2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3 + 8*a^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a^5*c^2*d^6)*x^2), 1/8*(3*(b^3*
c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*x^5 + (5*b^3*c^4*d + 9*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3 - 5*a^3*c*d^
4)*x^3 + 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c
^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*a^3*c
*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + 6*(a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b
^2*c^3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2)*sqrt(-a*b)*log(
(b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 3*(3*a*b^2*c^4*d - 2*a^2*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*c^7*d
- 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^3 - 4*a^4*b*c^4*d^4 + a^5*c^3*d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4 + 6*a
^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a^4*b*c*d^7)*x^6 + (2*b^5*c^6*d^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4*d^4 -
2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 + a^5*c*d^7)*x^4 + (b^5*c^7*d - 2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3 + 8*a
^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a^5*c^2*d^6)*x^2), 1/16*(6*(b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4
)*x^5 + 2*(5*b^3*c^4*d + 9*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3 - 5*a^3*c*d^4)*x^3 - 24*(a*b*c^4*d + a^2*c^3*d^2 +
(b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b^2*c^3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 +
2*a^2*c^2*d^3)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d
^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^
4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*a^3*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 +
c)) + 6*(3*a*b^2*c^4*d - 2*a^2*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*c^7*d - 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^
3 - 4*a^4*b*c^4*d^4 + a^5*c^3*d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4 + 6*a^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a
^4*b*c*d^7)*x^6 + (2*b^5*c^6*d^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4*d^4 - 2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 +
a^5*c*d^7)*x^4 + (b^5*c^7*d - 2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3 + 8*a^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a
^5*c^2*d^6)*x^2), 1/8*(3*(b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*x^5 + (5*b^3*c^4*d + 9*a*b^2*c^3*d^2
- 9*a^2*b*c^2*d^3 - 5*a^3*c*d^4)*x^3 - 12*(a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b^2*c^
3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2)*sqrt(a*b)*arctan(sqr
t(a*b)*x/a) + 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*
b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*
a^3*c*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + 3*(3*a*b^2*c^4*d - 2*a^2*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*
c^7*d - 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^3 - 4*a^4*b*c^4*d^4 + a^5*c^3*d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4
+ 6*a^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a^4*b*c*d^7)*x^6 + (2*b^5*c^6*d^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4
*d^4 - 2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 + a^5*c*d^7)*x^4 + (b^5*c^7*d - 2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3
+ 8*a^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a^5*c^2*d^6)*x^2)]
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giac [A] time = 0.41, size = 301, normalized size = 1.45 \begin {gather*} \frac {a b x}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} - \frac {3 \, {\left (a b^{2} c + a^{2} b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b}} + \frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {c d}} + \frac {3 \, b c d x^{3} + 5 \, a d^{2} x^{3} + 5 \, b c^{2} x + 3 \, a c d x}{8 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")
[Out]
1/2*a*b*x/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) - 3/2*(a*b^2*c + a^2*b*d)*arctan(b
*x/sqrt(a*b))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*b)) + 3/8*(b^2*c
^2 + 6*a*b*c*d + a^2*d^2)*arctan(d*x/sqrt(c*d))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3
+ a^4*d^4)*sqrt(c*d)) + 1/8*(3*b*c*d*x^3 + 5*a*d^2*x^3 + 5*b*c^2*x + 3*a*c*d*x)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*
a^2*b*c*d^2 - a^3*d^3)*(d*x^2 + c)^2)
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maple [B] time = 0.02, size = 388, normalized size = 1.87 \begin {gather*} -\frac {5 a^{2} d^{3} x^{3}}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}+\frac {a b c \,d^{2} x^{3}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}+\frac {3 b^{2} c^{2} d \,x^{3}}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {3 a^{2} c \,d^{2} x}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {a b \,c^{2} d x}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}+\frac {5 b^{2} c^{3} x}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {a^{2} b d x}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right )}-\frac {3 a^{2} b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{4} \sqrt {a b}}+\frac {3 a^{2} d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \left (a d -b c \right )^{4} \sqrt {c d}}+\frac {a \,b^{2} c x}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right )}-\frac {3 a \,b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{4} \sqrt {a b}}+\frac {9 a b c d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{4 \left (a d -b c \right )^{4} \sqrt {c d}}+\frac {3 b^{2} c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \left (a d -b c \right )^{4} \sqrt {c d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)
[Out]
-1/2*a^2*b/(a*d-b*c)^4*x/(b*x^2+a)*d+1/2*a*b^2/(a*d-b*c)^4*x/(b*x^2+a)*c-3/2*a^2*b/(a*d-b*c)^4/(a*b)^(1/2)*arc
tan(1/(a*b)^(1/2)*b*x)*d-3/2*a*b^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c-5/8/(a*d-b*c)^4/(d*x^2+
c)^2*x^3*a^2*d^3+1/4/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*b*c*d^2+3/8/(a*d-b*c)^4/(d*x^2+c)^2*x^3*b^2*c^2*d-3/8/(a*d-
b*c)^4/(d*x^2+c)^2*a^2*c*d^2*x-1/4/(a*d-b*c)^4/(d*x^2+c)^2*a*b*c^2*d*x+5/8/(a*d-b*c)^4/(d*x^2+c)^2*b^2*c^3*x+3
/8/(a*d-b*c)^4/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*a^2*d^2+9/4/(a*d-b*c)^4/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*
d*x)*a*b*c*d+3/8/(a*d-b*c)^4/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*b^2*c^2
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maxima [B] time = 2.54, size = 443, normalized size = 2.14 \begin {gather*} -\frac {3 \, {\left (a b^{2} c + a^{2} b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b}} + \frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {c d}} + \frac {3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x^{5} + {\left (5 \, b^{2} c^{2} + 14 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + a^{2} c d\right )} x}{8 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")
[Out]
-3/2*(a*b^2*c + a^2*b*d)*arctan(b*x/sqrt(a*b))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)*sqrt(a*b)) + 3/8*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*arctan(d*x/sqrt(c*d))/((b^4*c^4 - 4*a*b^3*c^3*d + 6
*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(c*d)) + 1/8*(3*(b^2*c*d + 3*a*b*d^2)*x^5 + (5*b^2*c^2 + 14*a*
b*c*d + 5*a^2*d^2)*x^3 + 3*(3*a*b*c^2 + a^2*c*d)*x)/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d
^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^
2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*
a^4*c*d^4)*x^2)
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mupad [B] time = 2.93, size = 7515, normalized size = 36.30
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/((a + b*x^2)^2*(c + d*x^2)^3),x)
[Out]
(atan(((((x*(9*b^7*c^4*d + 153*a^4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*a^2*b^5*c^2*d^3))/(32
*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c
*d^5)) - (3*(-a*b)^(1/2)*(((3*a^10*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^10)/2 - (69*a^2*b^10*c
^8*d^3)/2 + 114*a^3*b^9*c^7*d^4 - 210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6*c^4*d^7 + 42*a^7*b^5
*c^3*d^8 + 6*a^8*b^4*c^2*d^9)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d
^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) - (3*x*(-a
*b)^(1/2)*(a*d + b*c)*(256*a^9*b^2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a^8*b^3*c*d^10 + 5120*
a^2*b^9*c^7*d^4 - 7168*a^3*b^8*c^6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 7168*a^6*b^5*c^3*d^8 +
5120*a^7*b^4*c^2*d^9))/(128*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)*(a^6*d^6 +
b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)))*(a*
d + b*c))/(4*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(-a*b)^(1/2)*(a*d + b*c
)*3i)/(4*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (((x*(9*b^7*c^4*d + 153*a^
4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*a^2*b^5*c^2*d^3))/(32*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*
c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)) + (3*(-a*b)^(1/2)*(((3*a^1
0*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^10)/2 - (69*a^2*b^10*c^8*d^3)/2 + 114*a^3*b^9*c^7*d^4 -
210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6*c^4*d^7 + 42*a^7*b^5*c^3*d^8 + 6*a^8*b^4*c^2*d^9)/(a^
9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6
*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) + (3*x*(-a*b)^(1/2)*(a*d + b*c)*(256*a^9*b^
2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a^8*b^3*c*d^10 + 5120*a^2*b^9*c^7*d^4 - 7168*a^3*b^8*c^
6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 7168*a^6*b^5*c^3*d^8 + 5120*a^7*b^4*c^2*d^9))/(128*(a^4*
d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 2
0*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)))*(a*d + b*c))/(4*(a^4*d^4 + b^4*c^4 +
6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(-a*b)^(1/2)*(a*d + b*c)*3i)/(4*(a^4*d^4 + b^4*c^4 + 6*a
^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))/(((81*a^5*b^3*d^5)/64 + (297*a^4*b^4*c*d^4)/32 + (135*a^2*b^
6*c^3*d^2)/32 + (189*a^3*b^5*c^2*d^3)/16 + (27*a*b^7*c^4*d)/64)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a
^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8
*c^8*d - 9*a^8*b*c*d^8) - (3*((x*(9*b^7*c^4*d + 153*a^4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*
a^2*b^5*c^2*d^3))/(32*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*
b^5*c^5*d - 6*a^5*b*c*d^5)) - (3*(-a*b)^(1/2)*(((3*a^10*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^1
0)/2 - (69*a^2*b^10*c^8*d^3)/2 + 114*a^3*b^9*c^7*d^4 - 210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6
*c^4*d^7 + 42*a^7*b^5*c^3*d^8 + 6*a^8*b^4*c^2*d^9)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^
3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^
8*b*c*d^8) - (3*x*(-a*b)^(1/2)*(a*d + b*c)*(256*a^9*b^2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a
^8*b^3*c*d^10 + 5120*a^2*b^9*c^7*d^4 - 7168*a^3*b^8*c^6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 71
68*a^6*b^5*c^3*d^8 + 5120*a^7*b^4*c^2*d^9))/(128*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^
3*b*c*d^3)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d -
6*a^5*b*c*d^5)))*(a*d + b*c))/(4*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(-
a*b)^(1/2)*(a*d + b*c))/(4*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (3*((x*(
9*b^7*c^4*d + 153*a^4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*a^2*b^5*c^2*d^3))/(32*(a^6*d^6 + b
^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)) + (3*(
-a*b)^(1/2)*(((3*a^10*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^10)/2 - (69*a^2*b^10*c^8*d^3)/2 + 1
14*a^3*b^9*c^7*d^4 - 210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6*c^4*d^7 + 42*a^7*b^5*c^3*d^8 + 6*
a^8*b^4*c^2*d^9)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*
b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) + (3*x*(-a*b)^(1/2)*(a*
d + b*c)*(256*a^9*b^2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a^8*b^3*c*d^10 + 5120*a^2*b^9*c^7*d
^4 - 7168*a^3*b^8*c^6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 7168*a^6*b^5*c^3*d^8 + 5120*a^7*b^4*
c^2*d^9))/(128*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)*(a^6*d^6 + b^6*c^6 + 15
*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)))*(a*d + b*c))/(4*
(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(-a*b)^(1/2)*(a*d + b*c))/(4*(a^4*d^
4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))))*(-a*b)^(1/2)*(a*d + b*c)*3i)/(2*(a^4*d^4 +
b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - ((3*x^5*(3*a*b*d^2 + b^2*c*d))/(8*(a^3*d^3 -
b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (3*x*(3*a*b*c^2 + a^2*c*d))/(8*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*
a*b*c*d)) + (x^3*(5*a^2*d^2 + 5*b^2*c^2 + 14*a*b*c*d))/(8*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(a*c^2
+ x^2*(b*c^2 + 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) + b*d^2*x^6) + (atan(((((x*(9*b^7*c^4*d + 153*a^4*b^3*d^5 + 1
08*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*a^2*b^5*c^2*d^3))/(32*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*
a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)) - (3*(-c*d)^(1/2)*(((3*a^10*b^2*d^11)/2
+ (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^10)/2 - (69*a^2*b^10*c^8*d^3)/2 + 114*a^3*b^9*c^7*d^4 - 210*a^4*b^8*
c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6*c^4*d^7 + 42*a^7*b^5*c^3*d^8 + 6*a^8*b^4*c^2*d^9)/(a^9*d^9 - b^9*c
^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6
+ 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) - (3*x*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d)*(256
*a^9*b^2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a^8*b^3*c*d^10 + 5120*a^2*b^9*c^7*d^4 - 7168*a^3
*b^8*c^6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 7168*a^6*b^5*c^3*d^8 + 5120*a^7*b^4*c^2*d^9))/(51
2*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)*(a^6*d^6 + b^6*c^6 + 15*a^2*
b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)))*(a^2*d^2 + b^2*c^2 +
6*a*b*c*d))/(16*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)))*(-c*d)^(1/2)
*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d)*3i)/(16*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^
2*c^3*d^3)) + (((x*(9*b^7*c^4*d + 153*a^4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*a^2*b^5*c^2*d^
3))/(32*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*
a^5*b*c*d^5)) + (3*(-c*d)^(1/2)*(((3*a^10*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^10)/2 - (69*a^2
*b^10*c^8*d^3)/2 + 114*a^3*b^9*c^7*d^4 - 210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6*c^4*d^7 + 42*
a^7*b^5*c^3*d^8 + 6*a^8*b^4*c^2*d^9)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^
5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) + (
3*x*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d)*(256*a^9*b^2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 -
1792*a^8*b^3*c*d^10 + 5120*a^2*b^9*c^7*d^4 - 7168*a^3*b^8*c^6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^
7 - 7168*a^6*b^5*c^3*d^8 + 5120*a^7*b^4*c^2*d^9))/(512*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*
d^4 + 6*a^2*b^2*c^3*d^3)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6
*a*b^5*c^5*d - 6*a^5*b*c*d^5)))*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/(16*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2
- 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)))*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d)*3i)/(16*(a^4*c*d^5 + b^4
*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)))/(((81*a^5*b^3*d^5)/64 + (297*a^4*b^4*c*d^4)/
32 + (135*a^2*b^6*c^3*d^2)/32 + (189*a^3*b^5*c^2*d^3)/16 + (27*a*b^7*c^4*d)/64)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^
7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c
^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) - (3*((x*(9*b^7*c^4*d + 153*a^4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*
b^4*c*d^4 + 486*a^2*b^5*c^2*d^3))/(32*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^
2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)) - (3*(-c*d)^(1/2)*(((3*a^10*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (
15*a^9*b^3*c*d^10)/2 - (69*a^2*b^10*c^8*d^3)/2 + 114*a^3*b^9*c^7*d^4 - 210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d
^6 - 147*a^6*b^6*c^4*d^7 + 42*a^7*b^5*c^3*d^8 + 6*a^8*b^4*c^2*d^9)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 8
4*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*
b^8*c^8*d - 9*a^8*b*c*d^8) - (3*x*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d)*(256*a^9*b^2*d^11 + 256*b^11*c^
9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a^8*b^3*c*d^10 + 5120*a^2*b^9*c^7*d^4 - 7168*a^3*b^8*c^6*d^5 + 3584*a^4*b^7
*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 7168*a^6*b^5*c^3*d^8 + 5120*a^7*b^4*c^2*d^9))/(512*(a^4*c*d^5 + b^4*c^5*d -
4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^
3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)))*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/(16*(a^4*c*d^5
+ b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)))*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b
*c*d))/(16*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)) + (3*((x*(9*b^7*c^
4*d + 153*a^4*b^3*d^5 + 108*a*b^6*c^3*d^2 + 396*a^3*b^4*c*d^4 + 486*a^2*b^5*c^2*d^3))/(32*(a^6*d^6 + b^6*c^6 +
15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)) + (3*(-c*d)^(1
/2)*(((3*a^10*b^2*d^11)/2 + (9*a*b^11*c^9*d^2)/2 - (15*a^9*b^3*c*d^10)/2 - (69*a^2*b^10*c^8*d^3)/2 + 114*a^3*b
^9*c^7*d^4 - 210*a^4*b^8*c^6*d^5 + 231*a^5*b^7*c^5*d^6 - 147*a^6*b^6*c^4*d^7 + 42*a^7*b^5*c^3*d^8 + 6*a^8*b^4*
c^2*d^9)/(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*
d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8) + (3*x*(-c*d)^(1/2)*(a^2*d^2 +
b^2*c^2 + 6*a*b*c*d)*(256*a^9*b^2*d^11 + 256*b^11*c^9*d^2 - 1792*a*b^10*c^8*d^3 - 1792*a^8*b^3*c*d^10 + 5120*a
^2*b^9*c^7*d^4 - 7168*a^3*b^8*c^6*d^5 + 3584*a^4*b^7*c^5*d^6 + 3584*a^5*b^6*c^4*d^7 - 7168*a^6*b^5*c^3*d^8 + 5
120*a^7*b^4*c^2*d^9))/(512*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3)*(a^
6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5
)))*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/(16*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^
2*c^3*d^3)))*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/(16*(a^4*c*d^5 + b^4*c^5*d - 4*a*b^3*c^4*d^2 - 4*a^
3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3))))*(-c*d)^(1/2)*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d)*3i)/(8*(a^4*c*d^5 + b^4*c^5*d
- 4*a*b^3*c^4*d^2 - 4*a^3*b*c^2*d^4 + 6*a^2*b^2*c^3*d^3))
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sympy [F(-1)] 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(x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
Timed out
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