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

Optimal. Leaf size=109 \[ -\frac {\sqrt {a} (3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {470, 522, 205} \[ -\frac {\sqrt {a} (3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 95, normalized size = 0.87 \[ \frac {\frac {\sqrt {a} (a d-3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {a x (b c-a d)}{b \left (a+b x^2\right )}+\frac {2 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 726, normalized size = 6.66 \[ \left [-\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (a b c - a^{2} d\right )} x}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - {\left (a b c - a^{2} d\right )} x}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {4 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, {\left (a b c - a^{2} d\right )} x}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (a b c - a^{2} d\right )} x}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.30, size = 122, normalized size = 1.12 \[ \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} - \frac {{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {a b}} + \frac {a x}{2 \, {\left (b^{2} c - a b d\right )} {\left (b x^{2} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 144, normalized size = 1.32 \[ -\frac {a^{2} d x}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) b}+\frac {a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {a b}\, b}+\frac {a c x}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right )}-\frac {3 a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {a b}}+\frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2/(a*d-b*c)^2/b*x/(b*x^2+a)*d+1/2*a/(a*d-b*c)^2*x/(b*x^2+a)*c+1/2*a^2/(a*d-b*c)^2/b/(a*b)^(1/2)*arctan(
1/(a*b)^(1/2)*b*x)*d-3/2*a/(a*d-b*c)^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c+c^2/(a*d-b*c)^2/(c*d)^(1/2)*arc
tan(1/(c*d)^(1/2)*d*x)

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maxima [A]  time = 2.43, size = 133, normalized size = 1.22 \[ \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {a x}{2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )}} - \frac {{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.06, size = 3558, normalized size = 32.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((((-c^3*d)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*
b^3*c^2*d^5)/(2*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)) - (x*(-c^3*d)^(1/2)*(16*a^5*b^3*d^7 +
16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^2*d^5))/(8*(b^3*c^2
+ a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) - (x
*(a^4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(4*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(-c^3*d
)^(1/2)*1i)/(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2) - ((((-c^3*d)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^
2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(2*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^
2*d)) + (x*(-c^3*d)^(1/2)*(16*a^5*b^3*d^7 + 16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*
c^3*d^4 + 32*a^3*b^5*c^2*d^5))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(
2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) + (x*(a^4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(4*(b
^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(-c^3*d)^(1/2)*1i)/(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))/(((a^3*c^2*d^3)/2
- (5*a^2*b*c^3*d^2)/2 + 3*a*b^2*c^4*d)/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d) + ((((-c^3*d)^(
1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(2*(b^4
*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)) - (x*(-c^3*d)^(1/2)*(16*a^5*b^3*d^7 + 16*b^8*c^5*d^2 - 48
*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^2*d^5))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^
2*c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) - (x*(a^4*d^5 + 4*b^4*c^
4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(4*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(-c^3*d)^(1/2))/(a^2*d^3 +
b^2*c^2*d - 2*a*b*c*d^2) + ((((-c^3*d)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*
b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(2*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)) + (x*(-c^3*d)^(1/
2)*(16*a^5*b^3*d^7 + 16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^
2*d^5))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(2*(a^2*d^3 + b^2*c^2*d
- 2*a*b*c*d^2)) + (x*(a^4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(4*(b^3*c^2 + a^2*b*d^2 - 2*
a*b^2*c*d)))*(-c^3*d)^(1/2))/(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)))*(-c^3*d)^(1/2)*1i)/(a^2*d^3 + b^2*c^2*d - 2
*a*b*c*d^2) - (atan(((((x*(a^4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(2*(b^3*c^2 + a^2*b*d^2
- 2*a*b^2*c*d)) - ((-a*b^3)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^
4 - 8*a^4*b^3*c^2*d^5)/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d) - (x*(-a*b^3)^(1/2)*(a*d - 3*b*
c)*(16*a^5*b^3*d^7 + 16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^
2*d^5))/(8*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(a*d - 3*b*c))/(4*(b^5*
c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)))*(-a*b^3)^(1/2)*(a*d - 3*b*c)*1i)/(4*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d))
+ (((x*(a^4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(2*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)) +
((-a*b^3)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d
^5)/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d) + (x*(-a*b^3)^(1/2)*(a*d - 3*b*c)*(16*a^5*b^3*d^7
+ 16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^2*d^5))/(8*(b^5*c^2
+ a^2*b^3*d^2 - 2*a*b^4*c*d)*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(a*d - 3*b*c))/(4*(b^5*c^2 + a^2*b^3*d^2 -
2*a*b^4*c*d)))*(-a*b^3)^(1/2)*(a*d - 3*b*c)*1i)/(4*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)))/(((a^3*c^2*d^3)/2
- (5*a^2*b*c^3*d^2)/2 + 3*a*b^2*c^4*d)/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d) - (((x*(a^4*d^5
+ 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(2*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)) - ((-a*b^3)^(1/2)
*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(b^4*c^3 -
a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d) - (x*(-a*b^3)^(1/2)*(a*d - 3*b*c)*(16*a^5*b^3*d^7 + 16*b^8*c^5*d^
2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^2*d^5))/(8*(b^5*c^2 + a^2*b^3*d^2
- 2*a*b^4*c*d)*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(a*d - 3*b*c))/(4*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)))
*(-a*b^3)^(1/2)*(a*d - 3*b*c))/(4*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)) + (((x*(a^4*d^5 + 4*b^4*c^4*d + 9*a^2
*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(2*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)) + ((-a*b^3)^(1/2)*((2*a*b^6*c^5*d^2 + 2
*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*
c*d^2 - 3*a*b^3*c^2*d) + (x*(-a*b^3)^(1/2)*(a*d - 3*b*c)*(16*a^5*b^3*d^7 + 16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 -
48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 32*a^3*b^5*c^2*d^5))/(8*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)*(b^3*c^
2 + a^2*b*d^2 - 2*a*b^2*c*d)))*(a*d - 3*b*c))/(4*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d)))*(-a*b^3)^(1/2)*(a*d -
3*b*c))/(4*(b^5*c^2 + a^2*b^3*d^2 - 2*a*b^4*c*d))))*(-a*b^3)^(1/2)*(a*d - 3*b*c)*1i)/(2*(b^5*c^2 + a^2*b^3*d^
2 - 2*a*b^4*c*d)) - (a*x)/(2*b*(a + b*x^2)*(a*d - b*c))

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

[Out]

Timed out

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