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

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

[Out]

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

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Rubi [A]  time = 0.0646264, antiderivative size = 104, normalized size of antiderivative = 1., 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} \[ -\frac{x}{2 \left (a+b x^2\right ) (b c-a d)}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} (b c-a d)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{(b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.135417, size = 104, normalized size = 1. \[ \frac{x}{2 \left (a+b x^2\right ) (a d-b c)}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} (a d-b c)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{(b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 134, normalized size = 1.3 \begin{align*} -{\frac{cd}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{axd}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{bcx}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{ad}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{bc}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.99292, size = 1485, normalized size = 14.28 \begin{align*} \left [-\frac{{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) - 2 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) + 2 \,{\left (a b^{2} c - a^{2} b d\right )} x}{4 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} +{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, \frac{{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) -{\left (a b^{2} c - a^{2} b d\right )} x}{2 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} +{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac{4 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \,{\left (a b^{2} c - a^{2} b d\right )} x}{4 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} +{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, \frac{{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - 2 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (a b^{2} c - a^{2} b d\right )} x}{2 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} +{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 7.14263, size = 1530, normalized size = 14.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(2*a**2*d - 2*a*b*c + x**2*(2*a*b*d - 2*b**2*c)) + sqrt(-1/(a*b))*(a*d + b*c)*log(x + (-3*a**6*b*d**5*(-1/(a
*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) + 11*a**5*b**2*c*d**4*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d -
b*c)**6) - 7*a**4*b**3*c**2*d**3*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)**6 + 3*a**3*b**4*c**3*d**2*(-1/
(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)**6 - a**3*d**3*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2) + a**2*b
**5*c**4*d*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) - 11*a**2*b*c*d**2*sqrt(-1/(a*b))*(a*d + b*c)/(
2*(a*d - b*c)**2) - a*b**6*c**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) - 3*a*b**2*c**2*d*sqrt(-1/
(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2) - b**3*c**3*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2))/(a*d**2 + b*c
*d))/(4*(a*d - b*c)**2) - sqrt(-1/(a*b))*(a*d + b*c)*log(x + (3*a**6*b*d**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(
2*(a*d - b*c)**6) - 11*a**5*b**2*c*d**4*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) + 7*a**4*b**3*c**2
*d**3*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)**6 - 3*a**3*b**4*c**3*d**2*(-1/(a*b))**(3/2)*(a*d + b*c)**3
/(a*d - b*c)**6 + a**3*d**3*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2) - a**2*b**5*c**4*d*(-1/(a*b))**(3/2)
*(a*d + b*c)**3/(2*(a*d - b*c)**6) + 11*a**2*b*c*d**2*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2) + a*b**6*c
**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) + 3*a*b**2*c**2*d*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d -
b*c)**2) + b**3*c**3*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2))/(a*d**2 + b*c*d))/(4*(a*d - b*c)**2) + sq
rt(-c*d)*log(x + (-12*a**6*b*d**5*(-c*d)**(3/2)/(a*d - b*c)**6 + 44*a**5*b**2*c*d**4*(-c*d)**(3/2)/(a*d - b*c)
**6 - 56*a**4*b**3*c**2*d**3*(-c*d)**(3/2)/(a*d - b*c)**6 + 24*a**3*b**4*c**3*d**2*(-c*d)**(3/2)/(a*d - b*c)**
6 - a**3*d**3*sqrt(-c*d)/(a*d - b*c)**2 + 4*a**2*b**5*c**4*d*(-c*d)**(3/2)/(a*d - b*c)**6 - 11*a**2*b*c*d**2*s
qrt(-c*d)/(a*d - b*c)**2 - 4*a*b**6*c**5*(-c*d)**(3/2)/(a*d - b*c)**6 - 3*a*b**2*c**2*d*sqrt(-c*d)/(a*d - b*c)
**2 - b**3*c**3*sqrt(-c*d)/(a*d - b*c)**2)/(a*d**2 + b*c*d))/(2*(a*d - b*c)**2) - sqrt(-c*d)*log(x + (12*a**6*
b*d**5*(-c*d)**(3/2)/(a*d - b*c)**6 - 44*a**5*b**2*c*d**4*(-c*d)**(3/2)/(a*d - b*c)**6 + 56*a**4*b**3*c**2*d**
3*(-c*d)**(3/2)/(a*d - b*c)**6 - 24*a**3*b**4*c**3*d**2*(-c*d)**(3/2)/(a*d - b*c)**6 + a**3*d**3*sqrt(-c*d)/(a
*d - b*c)**2 - 4*a**2*b**5*c**4*d*(-c*d)**(3/2)/(a*d - b*c)**6 + 11*a**2*b*c*d**2*sqrt(-c*d)/(a*d - b*c)**2 +
4*a*b**6*c**5*(-c*d)**(3/2)/(a*d - b*c)**6 + 3*a*b**2*c**2*d*sqrt(-c*d)/(a*d - b*c)**2 + b**3*c**3*sqrt(-c*d)/
(a*d - b*c)**2)/(a*d**2 + b*c*d))/(2*(a*d - b*c)**2)

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Giac [A]  time = 1.14471, size = 149, normalized size = 1.43 \begin{align*} -\frac{c d \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 (b c + a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} - \frac{x}{2 \,{\left (b x^{2} + a\right )}{\left (b c - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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