3.3.97 \(\int \frac {1}{x^4 (a+b x^2)^2 (c+d x^2)} \, dx\)

Optimal. Leaf size=189 \[ \frac {b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^2}-\frac {5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac {-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2}+\frac {b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]

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Rubi [A]  time = 0.28, antiderivative size = 189, 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 = {472, 583, 522, 205} \begin {gather*} \frac {-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac {b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^2}-\frac {5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2}+\frac {b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

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

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 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && 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 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int \frac {-5 b c+2 a d-5 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\int \frac {-3 \left (5 b^2 c^2-2 a b c d-2 a^2 d^2\right )-3 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{6 a^2 c (b c-a d)}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int \frac {-3 \left (5 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right )-3 b d \left (5 b^2 c^2-2 a b c d-2 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{6 a^3 c^2 (b c-a d)}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {d^4 \int \frac {1}{c+d x^2} \, dx}{c^2 (b c-a d)^2}+\frac {\left (b^3 (5 b c-7 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^2}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^2}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 142, normalized size = 0.75 \begin {gather*} -\frac {b^{5/2} (7 a d-5 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (a d-b c)^2}-\frac {b^3 x}{2 a^3 \left (a+b x^2\right ) (a d-b c)}+\frac {a d+2 b c}{a^3 c^2 x}-\frac {1}{3 a^2 c x^3}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-1/3*1/(a^2*c*x^3) + (2*b*c + a*d)/(a^3*c^2*x) - (b^3*x)/(2*a^3*(-(b*c) + a*d)*(a + b*x^2)) - (b^(5/2)*(-5*b*c
+ 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(-(b*c) + a*d)^2) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c
^(5/2)*(b*c - a*d)^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)), x]

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fricas [A]  time = 3.48, size = 1281, normalized size = 6.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.38, size = 165, normalized size = 0.87 \begin {gather*} \frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt {c d}} + \frac {b^{3} x}{2 \, {\left (a^{3} b c - a^{4} d\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (5 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {a b}} + \frac {6 \, b c x^{2} + 3 \, a d x^{2} - a c}{3 \, a^{3} c^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")

[Out]

d^4*arctan(d*x/sqrt(c*d))/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*sqrt(c*d)) + 1/2*b^3*x/((a^3*b*c - a^4*d)*(b*
x^2 + a)) + 1/2*(5*b^4*c - 7*a*b^3*d)*arctan(b*x/sqrt(a*b))/((a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt(a*b))
+ 1/3*(6*b*c*x^2 + 3*a*d*x^2 - a*c)/(a^3*c^2*x^3)

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maple [A]  time = 0.02, size = 191, normalized size = 1.01 \begin {gather*} -\frac {b^{3} d x}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a^{2}}-\frac {7 b^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {a b}\, a^{2}}+\frac {b^{4} c x}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{4} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {a b}\, a^{3}}+\frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}\, c^{2}}+\frac {d}{a^{2} c^{2} x}+\frac {2 b}{a^{3} c x}-\frac {1}{3 a^{2} c \,x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^2/(d*x^2+c),x)

[Out]

-1/2*b^3/a^2/(a*d-b*c)^2*x/(b*x^2+a)*d+1/2*b^4/a^3/(a*d-b*c)^2*x/(b*x^2+a)*c-7/2*b^3/a^2/(a*d-b*c)^2/(a*b)^(1/
2)*arctan(1/(a*b)^(1/2)*b*x)*d+5/2*b^4/a^3/(a*d-b*c)^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c+1/c^2*d^4/(a*d-
b*c)^2/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)-1/3/a^2/c/x^3+1/a^2/c^2/x*d+2/a^3/c/x*b

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maxima [A]  time = 2.46, size = 236, normalized size = 1.25 \begin {gather*} \frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (5 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {a b}} - \frac {2 \, a^{2} b c^{2} - 2 \, a^{3} c d - 3 \, {\left (5 \, b^{3} c^{2} - 2 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} x^{2}}{6 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{5} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

d^4*arctan(d*x/sqrt(c*d))/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*sqrt(c*d)) + 1/2*(5*b^4*c - 7*a*b^3*d)*arctan
(b*x/sqrt(a*b))/((a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt(a*b)) - 1/6*(2*a^2*b*c^2 - 2*a^3*c*d - 3*(5*b^3*c^
2 - 2*a*b^2*c*d - 2*a^2*b*d^2)*x^4 - 2*(5*a*b^2*c^2 - 2*a^2*b*c*d - 3*a^3*d^2)*x^2)/((a^3*b^2*c^3 - a^4*b*c^2*
d)*x^5 + (a^4*b*c^3 - a^5*c^2*d)*x^3)

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mupad [B]  time = 1.23, size = 4654, normalized size = 24.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^2*(c + d*x^2)),x)

[Out]

((x^2*(3*a*d + 5*b*c))/(3*a^2*c^2) - 1/(3*a*c) + (x^4*(2*a^2*b*d^2 - 5*b^3*c^2 + 2*a*b^2*c*d))/(2*a^3*c^2*(a*d
- b*c)))/(a*x^3 + b*x^5) - (atan(((((x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5344*a^11*b^10*c^13
*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6*c^9*d^9 - 192*a^1
6*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12))/2 + ((-c^5*d^7)^(1/2)*((x*(-c^5*d^7)^(1/2)*(25
6*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*d^5 + 3584*a^20*b^
5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(b^2*c^7 + a^2*c^5*
d^2 - 2*a*b*c^6*d)) - 160*a^12*b^11*c^17*d^2 + 1024*a^13*b^10*c^16*d^3 - 2720*a^14*b^9*c^15*d^4 + 3840*a^15*b^
8*c^14*d^5 - 3104*a^16*b^7*c^13*d^6 + 1600*a^17*b^6*c^12*d^7 - 864*a^18*b^5*c^11*d^8 + 640*a^19*b^4*c^10*d^9 -
320*a^20*b^3*c^9*d^10 + 64*a^21*b^2*c^8*d^11))/(2*(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d)))*(-c^5*d^7)^(1/2)*1i
)/(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d) + (((x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5344*a^11*b^
10*c^13*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6*c^9*d^9 -
192*a^16*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12))/2 + ((-c^5*d^7)^(1/2)*((x*(-c^5*d^7)^(1
/2)*(256*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*d^5 + 3584*
a^20*b^5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(b^2*c^7 + a
^2*c^5*d^2 - 2*a*b*c^6*d)) + 160*a^12*b^11*c^17*d^2 - 1024*a^13*b^10*c^16*d^3 + 2720*a^14*b^9*c^15*d^4 - 3840*
a^15*b^8*c^14*d^5 + 3104*a^16*b^7*c^13*d^6 - 1600*a^17*b^6*c^12*d^7 + 864*a^18*b^5*c^11*d^8 - 640*a^19*b^4*c^1
0*d^9 + 320*a^20*b^3*c^9*d^10 - 64*a^21*b^2*c^8*d^11))/(2*(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d)))*(-c^5*d^7)^(
1/2)*1i)/(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d))/((((x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5344*
a^11*b^10*c^13*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6*c^9
*d^9 - 192*a^16*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12))/2 + ((-c^5*d^7)^(1/2)*((x*(-c^5*
d^7)^(1/2)*(256*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*d^5
+ 3584*a^20*b^5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(b^2*
c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d)) - 160*a^12*b^11*c^17*d^2 + 1024*a^13*b^10*c^16*d^3 - 2720*a^14*b^9*c^15*d^4
+ 3840*a^15*b^8*c^14*d^5 - 3104*a^16*b^7*c^13*d^6 + 1600*a^17*b^6*c^12*d^7 - 864*a^18*b^5*c^11*d^8 + 640*a^19*
b^4*c^10*d^9 - 320*a^20*b^3*c^9*d^10 + 64*a^21*b^2*c^8*d^11))/(2*(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d)))*(-c^5
*d^7)^(1/2))/(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d) - (((x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5
344*a^11*b^10*c^13*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6
*c^9*d^9 - 192*a^16*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12))/2 + ((-c^5*d^7)^(1/2)*((x*(-
c^5*d^7)^(1/2)*(256*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*
d^5 + 3584*a^20*b^5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(
b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d)) + 160*a^12*b^11*c^17*d^2 - 1024*a^13*b^10*c^16*d^3 + 2720*a^14*b^9*c^15*
d^4 - 3840*a^15*b^8*c^14*d^5 + 3104*a^16*b^7*c^13*d^6 - 1600*a^17*b^6*c^12*d^7 + 864*a^18*b^5*c^11*d^8 - 640*a
^19*b^4*c^10*d^9 + 320*a^20*b^3*c^9*d^10 - 64*a^21*b^2*c^8*d^11))/(2*(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d)))*(
-c^5*d^7)^(1/2))/(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d) - 400*a^9*b^10*c^11*d^6 + 1520*a^10*b^9*c^10*d^7 - 1904
*a^11*b^8*c^9*d^8 + 624*a^12*b^7*c^8*d^9 + 384*a^13*b^6*c^7*d^10 - 224*a^14*b^5*c^6*d^11))*(-c^5*d^7)^(1/2)*1i
)/(b^2*c^7 + a^2*c^5*d^2 - 2*a*b*c^6*d) + (atan((((7*a*d - 5*b*c)*(x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c
^14*d^4 + 5344*a^11*b^10*c^13*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 +
64*a^15*b^6*c^9*d^9 - 192*a^16*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12) + ((7*a*d - 5*b*c)
*(-a^7*b^5)^(1/2)*(2048*a^13*b^10*c^16*d^3 - 320*a^12*b^11*c^17*d^2 - 5440*a^14*b^9*c^15*d^4 + 7680*a^15*b^8*c
^14*d^5 - 6208*a^16*b^7*c^13*d^6 + 3200*a^17*b^6*c^12*d^7 - 1728*a^18*b^5*c^11*d^8 + 1280*a^19*b^4*c^10*d^9 -
640*a^20*b^3*c^9*d^10 + 128*a^21*b^2*c^8*d^11 + (x*(7*a*d - 5*b*c)*(-a^7*b^5)^(1/2)*(256*a^15*b^10*c^18*d^2 -
1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*d^5 + 3584*a^20*b^5*c^13*d^7 - 3584*a^21*
b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d))))/(
4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d)))*(-a^7*b^5)^(1/2)*1i)/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d)) + ((7
*a*d - 5*b*c)*(x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5344*a^11*b^10*c^13*d^5 - 6112*a^12*b^9*c^
12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6*c^9*d^9 - 192*a^16*b^5*c^8*d^10 + 192*a^
17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12) + ((7*a*d - 5*b*c)*(-a^7*b^5)^(1/2)*(320*a^12*b^11*c^17*d^2 - 2048*a^1
3*b^10*c^16*d^3 + 5440*a^14*b^9*c^15*d^4 - 7680*a^15*b^8*c^14*d^5 + 6208*a^16*b^7*c^13*d^6 - 3200*a^17*b^6*c^1
2*d^7 + 1728*a^18*b^5*c^11*d^8 - 1280*a^19*b^4*c^10*d^9 + 640*a^20*b^3*c^9*d^10 - 128*a^21*b^2*c^8*d^11 + (x*(
7*a*d - 5*b*c)*(-a^7*b^5)^(1/2)*(256*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 35
84*a^18*b^7*c^15*d^5 + 3584*a^20*b^5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2
*c^10*d^10))/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d))))/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d)))*(-a^7*b^5)
^(1/2)*1i)/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d)))/(400*a^9*b^10*c^11*d^6 - 1520*a^10*b^9*c^10*d^7 + 1904*a
^11*b^8*c^9*d^8 - 624*a^12*b^7*c^8*d^9 - 384*a^13*b^6*c^7*d^10 + 224*a^14*b^5*c^6*d^11 - ((7*a*d - 5*b*c)*(x*(
400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5344*a^11*b^10*c^13*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13
*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6*c^9*d^9 - 192*a^16*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 -
64*a^18*b^3*c^6*d^12) + ((7*a*d - 5*b*c)*(-a^7*b^5)^(1/2)*(2048*a^13*b^10*c^16*d^3 - 320*a^12*b^11*c^17*d^2 -
5440*a^14*b^9*c^15*d^4 + 7680*a^15*b^8*c^14*d^5 - 6208*a^16*b^7*c^13*d^6 + 3200*a^17*b^6*c^12*d^7 - 1728*a^18*
b^5*c^11*d^8 + 1280*a^19*b^4*c^10*d^9 - 640*a^20*b^3*c^9*d^10 + 128*a^21*b^2*c^8*d^11 + (x*(7*a*d - 5*b*c)*(-a
^7*b^5)^(1/2)*(256*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*d
^5 + 3584*a^20*b^5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 + 1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(a
^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d))))/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d)))*(-a^7*b^5)^(1/2))/(4*(a^9*d^
2 + a^7*b^2*c^2 - 2*a^8*b*c*d)) + ((7*a*d - 5*b*c)*(x*(400*a^9*b^12*c^15*d^3 - 2320*a^10*b^11*c^14*d^4 + 5344*
a^11*b^10*c^13*d^5 - 6112*a^12*b^9*c^12*d^6 + 3472*a^13*b^8*c^11*d^7 - 784*a^14*b^7*c^10*d^8 + 64*a^15*b^6*c^9
*d^9 - 192*a^16*b^5*c^8*d^10 + 192*a^17*b^4*c^7*d^11 - 64*a^18*b^3*c^6*d^12) + ((7*a*d - 5*b*c)*(-a^7*b^5)^(1/
2)*(320*a^12*b^11*c^17*d^2 - 2048*a^13*b^10*c^16*d^3 + 5440*a^14*b^9*c^15*d^4 - 7680*a^15*b^8*c^14*d^5 + 6208*
a^16*b^7*c^13*d^6 - 3200*a^17*b^6*c^12*d^7 + 1728*a^18*b^5*c^11*d^8 - 1280*a^19*b^4*c^10*d^9 + 640*a^20*b^3*c^
9*d^10 - 128*a^21*b^2*c^8*d^11 + (x*(7*a*d - 5*b*c)*(-a^7*b^5)^(1/2)*(256*a^15*b^10*c^18*d^2 - 1536*a^16*b^9*c
^17*d^3 + 3584*a^17*b^8*c^16*d^4 - 3584*a^18*b^7*c^15*d^5 + 3584*a^20*b^5*c^13*d^7 - 3584*a^21*b^4*c^12*d^8 +
1536*a^22*b^3*c^11*d^9 - 256*a^23*b^2*c^10*d^10))/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d))))/(4*(a^9*d^2 + a^
7*b^2*c^2 - 2*a^8*b*c*d)))*(-a^7*b^5)^(1/2))/(4*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d))))*(7*a*d - 5*b*c)*(-a^7
*b^5)^(1/2)*1i)/(2*(a^9*d^2 + a^7*b^2*c^2 - 2*a^8*b*c*d))

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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(1/x**4/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

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