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

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

[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)

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Rubi [A]  time = 0.0631776, 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} \[ \frac{x}{2 \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^2}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d} (b c-a d)^2} \]

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 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 ) \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 = 0.13119, size = 90, normalized size = 0.87 \[ \frac{\frac{x (b c-a d)}{c+d x^2}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d}}-2 \sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^2} \]

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)

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

[Out]

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

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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)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.12539, size = 1485, normalized size = 14.28 \begin{align*} \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{align*}

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)]

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Sympy [B]  time = 7.07189, 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)/(d*x**2+c)**2,x)

[Out]

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

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Giac [A]  time = 1.16073, size = 149, normalized size = 1.43 \begin{align*} -\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{align*}

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))