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

Optimal. Leaf size=104 \[ \frac {x}{2 \left (c+d x^2\right ) (b c-a d)}-\frac {\sqrt {a} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^2}+\frac {(a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 104, 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 = {471, 522, 205} \begin {gather*} \frac {x}{2 \left (c+d x^2\right ) (b c-a d)}-\frac {\sqrt {a} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^2}+\frac {(a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 90, normalized size = 0.87 \begin {gather*} \frac {\frac {x (b c-a d)}{c+d x^2}+\frac {(a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-2 \sqrt {a} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.05, size = 705, normalized size = 6.78 \begin {gather*} \left [\frac {2 \, {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x}{4 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (b c^{2} d - a c d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, -\frac {4 \, {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 2 \, {\left (b c^{2} d - a c d^{2}\right )} x}{4 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, -\frac {2 \, {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b c^{2} d - a c d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.32, size = 110, normalized size = 1.06 \begin {gather*} -\frac {a b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {x}{2 \, {\left (d x^{2} + c\right )} {\left (b c - a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 134, normalized size = 1.29 \begin {gather*} -\frac {a b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{2} \sqrt {a b}}-\frac {a d x}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {a d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {c d}}+\frac {b c x}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {b c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.45, size = 119, normalized size = 1.14 \begin {gather*} -\frac {a b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {x}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.78, size = 3154, normalized size = 30.33

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)/(d*x**2+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________