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

Optimal. Leaf size=70 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0340432, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {481, 205} \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

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

Mathematica [A]  time = 0.0422226, size = 61, normalized size = 0.87 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}}{b c-a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 55, normalized size = 0.8 \begin{align*} -{\frac{c}{ad-bc}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{a}{ad-bc}\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),x)

[Out]

-c/(a*d-b*c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))+a/(a*d-b*c)/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59029, size = 630, normalized size = 9. \begin{align*} \left [-\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\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 (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\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 (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\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 c - a d\right )}}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right )}{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),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.11334, size = 570, normalized size = 8.14 \begin{align*} \frac{\sqrt{- \frac{a}{b}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac{\sqrt{- \frac{a}{b}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} + \frac{\sqrt{- \frac{c}{d}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac{\sqrt{- \frac{c}{d}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\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),x)

[Out]

sqrt(-a/b)*log(-2*a**2*b*d**3*(-a/b)**(3/2)/(a*d - b*c)**3 + 4*a*b**2*c*d**2*(-a/b)**(3/2)/(a*d - b*c)**3 - a*
d*sqrt(-a/b)/(a*d - b*c) - 2*b**3*c**2*d*(-a/b)**(3/2)/(a*d - b*c)**3 - b*c*sqrt(-a/b)/(a*d - b*c) + x)/(2*(a*
d - b*c)) - sqrt(-a/b)*log(2*a**2*b*d**3*(-a/b)**(3/2)/(a*d - b*c)**3 - 4*a*b**2*c*d**2*(-a/b)**(3/2)/(a*d - b
*c)**3 + a*d*sqrt(-a/b)/(a*d - b*c) + 2*b**3*c**2*d*(-a/b)**(3/2)/(a*d - b*c)**3 + b*c*sqrt(-a/b)/(a*d - b*c)
+ x)/(2*(a*d - b*c)) + sqrt(-c/d)*log(-2*a**2*b*d**3*(-c/d)**(3/2)/(a*d - b*c)**3 + 4*a*b**2*c*d**2*(-c/d)**(3
/2)/(a*d - b*c)**3 - a*d*sqrt(-c/d)/(a*d - b*c) - 2*b**3*c**2*d*(-c/d)**(3/2)/(a*d - b*c)**3 - b*c*sqrt(-c/d)/
(a*d - b*c) + x)/(2*(a*d - b*c)) - sqrt(-c/d)*log(2*a**2*b*d**3*(-c/d)**(3/2)/(a*d - b*c)**3 - 4*a*b**2*c*d**2
*(-c/d)**(3/2)/(a*d - b*c)**3 + a*d*sqrt(-c/d)/(a*d - b*c) + 2*b**3*c**2*d*(-c/d)**(3/2)/(a*d - b*c)**3 + b*c*
sqrt(-c/d)/(a*d - b*c) + x)/(2*(a*d - b*c))

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Giac [B]  time = 1.17806, size = 176, normalized size = 2.51 \begin{align*} \frac{\sqrt{c d}{\left | d \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d + \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{d^{2}{\left | b c - a d \right |}} - \frac{\sqrt{a b}{\left | b \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d - \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2}{\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),x, algorithm="giac")

[Out]

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