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

Optimal. Leaf size=147 \[ -\frac {x}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {d x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt {b} (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} (b c-a d)^3}-\frac {\sqrt {d} (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} (b c-a d)^3} \]

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Rubi [A]  time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {471, 527, 522, 205} \begin {gather*} -\frac {x}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {d x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt {b} (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} (b c-a d)^3}-\frac {\sqrt {d} (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {c-3 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 (b c-a d)}\\ &=-\frac {d x}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {2 c (b c+a d)-4 b c d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^2}\\ &=-\frac {d x}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(d (3 b c+a d)) \int \frac {1}{c+d x^2} \, dx}{2 (b c-a d)^3}+\frac {(b (b c+3 a d)) \int \frac {1}{a+b x^2} \, dx}{2 (b c-a d)^3}\\ &=-\frac {d x}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {b} (b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} (b c-a d)^3}-\frac {\sqrt {d} (3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 137, normalized size = 0.93 \begin {gather*} \frac {1}{2} \left (-\frac {b x}{\left (a+b x^2\right ) (b c-a d)^2}-\frac {d x}{\left (c+d x^2\right ) (b c-a d)^2}-\frac {\sqrt {b} (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (a d-b c)^3}-\frac {\sqrt {d} (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.95, size = 1387, normalized size = 9.44

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.36, size = 196, normalized size = 1.33 \begin {gather*} \frac {{\left (b^{2} c + 3 \, a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{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 )} \sqrt {a b}} - \frac {{\left (3 \, b c d + a d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{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 )} \sqrt {c d}} - \frac {2 \, b d x^{3} + b c x + a d x}{2 \, {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 222, normalized size = 1.51 \begin {gather*} -\frac {a b d x}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}-\frac {3 a b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{3} \sqrt {a b}}-\frac {a \,d^{2} x}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {a \,d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \left (a d -b c \right )^{3} \sqrt {c d}}+\frac {b^{2} c x}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}-\frac {b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{3} \sqrt {a b}}+\frac {b c d x}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {3 b c d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \left (a d -b c \right )^{3} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.42, size = 249, normalized size = 1.69 \begin {gather*} \frac {{\left (b^{2} c + 3 \, a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{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 )} \sqrt {a b}} - \frac {{\left (3 \, b c d + a d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{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 )} \sqrt {c d}} - \frac {2 \, b d x^{3} + {\left (b c + a d\right )} x}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.67, size = 5236, normalized size = 35.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-a*b)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*d^3 + 6*a*b^4*c*d^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)*((2*a^7*b^2*d^9 + 2*b^9*c^7*d^2 - 10*a*b^8*c^6*d^3 - 10*a^
6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^5 - 10*a^4*b^5*c^3*d^6 + 18*a^5*b^4*c^2*d^7)/(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) - (x*(-a
*b)^(1/2)*(3*a*d + b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d^2 - 40*a*b^8*c^6*d^3 - 40*a^6*b^3*c*d^8 + 72*a^2*b^7*c^5*
d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a^5*b^4*c^2*d^7))/(4*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d
- 3*a^3*b*c*d^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*a*d + b*c))/(4*
(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2)))*(3*a*d + b*c)*1i)/(4*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^
2*c^2*d - 3*a^3*b*c*d^2)) + ((-a*b)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*d^3 + 6*a*b^4*c*d^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)*((2*a^7*b^2*d^9 + 2*b^9*c^7*d^2 - 10*
a*b^8*c^6*d^3 - 10*a^6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^5 - 10*a^4*b^5*c^3*d^6 + 18*a^5*b^4*c
^2*d^7)/(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) + (x*(-a*b)^(1/2)*(3*a*d + b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d^2 - 40*a*b^8*c^6*d^3 - 40*a^6*b^3*c*
d^8 + 72*a^2*b^7*c^5*d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a^5*b^4*c^2*d^7))/(4*(a^4*d^3 - a*b^3*
c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^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*a*d + b*c))/(4*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2)))*(3*a*d + b*c)*1i)/(4*(a^4*d^3 -
a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2)))/(((3*a^2*b^3*d^5)/2 + (3*b^5*c^2*d^3)/2 + 5*a*b^4*c*d^4)/(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*b)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*d^3 + 6*a*b^4*c*d^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)*((2*a^7*b^2*d^9 + 2*b^9*c^7*d^2 - 10*a*b^8*c^6*d^3 - 10*a^6*b^3
*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^5 - 10*a^4*b^5*c^3*d^6 + 18*a^5*b^4*c^2*d^7)/(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) - (x*(-a*b)^(
1/2)*(3*a*d + b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d^2 - 40*a*b^8*c^6*d^3 - 40*a^6*b^3*c*d^8 + 72*a^2*b^7*c^5*d^4 -
40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a^5*b^4*c^2*d^7))/(4*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*
a^3*b*c*d^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*a*d + b*c))/(4*(a^4*
d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2)))*(3*a*d + b*c))/(4*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d
- 3*a^3*b*c*d^2)) + ((-a*b)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*d^3 + 6*a*b^4*c*d^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)*((2*a^7*b^2*d^9 + 2*b^9*c^7*d^2 - 10*a*b^8*c^
6*d^3 - 10*a^6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^5 - 10*a^4*b^5*c^3*d^6 + 18*a^5*b^4*c^2*d^7)/
(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) + (x*(-a*b)^(1/2)*(3*a*d + b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d^2 - 40*a*b^8*c^6*d^3 - 40*a^6*b^3*c*d^8 + 72
*a^2*b^7*c^5*d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a^5*b^4*c^2*d^7))/(4*(a^4*d^3 - a*b^3*c^3 + 3*
a^2*b^2*c^2*d - 3*a^3*b*c*d^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*a*
d + b*c))/(4*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2)))*(3*a*d + b*c))/(4*(a^4*d^3 - a*b^3*c^3
+ 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2))))*(-a*b)^(1/2)*(3*a*d + b*c)*1i)/(2*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d
- 3*a^3*b*c*d^2)) - ((x*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^3)/(a^2*d^2 + b^2*c^2 - 2*a
*b*c*d))/(a*c + x^2*(a*d + b*c) + b*d*x^4) + (atan((((-c*d)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*d^3 + 6*a*b^4
*c*d^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) - ((-c*d)^(1/2)*((2*a^7*b^2*d
^9 + 2*b^9*c^7*d^2 - 10*a*b^8*c^6*d^3 - 10*a^6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^5 - 10*a^4*b^
5*c^3*d^6 + 18*a^5*b^4*c^2*d^7)/(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) - (x*(-c*d)^(1/2)*(a*d + 3*b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d^2 - 40*a*b^8
*c^6*d^3 - 40*a^6*b^3*c*d^8 + 72*a^2*b^7*c^5*d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a^5*b^4*c^2*d^
7))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(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*d + 3*b*c))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)))*(a*d +
3*b*c)*1i)/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)) + ((-c*d)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*
b^5*c^2*d^3 + 6*a*b^4*c*d^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) + ((-c*d
)^(1/2)*((2*a^7*b^2*d^9 + 2*b^9*c^7*d^2 - 10*a*b^8*c^6*d^3 - 10*a^6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^
6*c^4*d^5 - 10*a^4*b^5*c^3*d^6 + 18*a^5*b^4*c^2*d^7)/(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) + (x*(-c*d)^(1/2)*(a*d + 3*b*c)*(8*a^7*b^2*d^9 + 8*b
^9*c^7*d^2 - 40*a*b^8*c^6*d^3 - 40*a^6*b^3*c*d^8 + 72*a^2*b^7*c^5*d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^
6 + 72*a^5*b^4*c^2*d^7))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(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*d + 3*b*c))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a
*b^2*c^3*d)))*(a*d + 3*b*c)*1i)/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)))/(((3*a^2*b^3*d^5)
/2 + (3*b^5*c^2*d^3)/2 + 5*a*b^4*c*d^4)/(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) - ((-c*d)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*d^3 + 6*a*b^4*c*d^
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) - ((-c*d)^(1/2)*((2*a^7*b^2*d^9 +
2*b^9*c^7*d^2 - 10*a*b^8*c^6*d^3 - 10*a^6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^5 - 10*a^4*b^5*c^3
*d^6 + 18*a^5*b^4*c^2*d^7)/(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) - (x*(-c*d)^(1/2)*(a*d + 3*b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d^2 - 40*a*b^8*c^6*
d^3 - 40*a^6*b^3*c*d^8 + 72*a^2*b^7*c^5*d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a^5*b^4*c^2*d^7))/(
4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(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*d + 3*b*c))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)))*(a*d + 3*b*c
))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)) + ((-c*d)^(1/2)*((x*(5*a^2*b^3*d^5 + 5*b^5*c^2*
d^3 + 6*a*b^4*c*d^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) + ((-c*d)^(1/2)*
((2*a^7*b^2*d^9 + 2*b^9*c^7*d^2 - 10*a*b^8*c^6*d^3 - 10*a^6*b^3*c*d^8 + 18*a^2*b^7*c^5*d^4 - 10*a^3*b^6*c^4*d^
5 - 10*a^4*b^5*c^3*d^6 + 18*a^5*b^4*c^2*d^7)/(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) + (x*(-c*d)^(1/2)*(a*d + 3*b*c)*(8*a^7*b^2*d^9 + 8*b^9*c^7*d
^2 - 40*a*b^8*c^6*d^3 - 40*a^6*b^3*c*d^8 + 72*a^2*b^7*c^5*d^4 - 40*a^3*b^6*c^4*d^5 - 40*a^4*b^5*c^3*d^6 + 72*a
^5*b^4*c^2*d^7))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(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*d + 3*b*c))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3
*d)))*(a*d + 3*b*c))/(4*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d))))*(-c*d)^(1/2)*(a*d + 3*b*c)*
1i)/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*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(x**2/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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