3.235 \(\int \frac{1}{x^2 (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=81 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)}-\frac{1}{a c x} \]

[Out]

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

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Rubi [A]  time = 0.0855364, antiderivative size = 81, 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 = {480, 522, 205} \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)*(c + d*x^2)),x]

[Out]

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

Rule 480

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

Mathematica [A]  time = 0.0830965, size = 76, normalized size = 0.94 \[ \frac{-\frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{b}{a}+\frac{d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}+\frac{d}{c}}{b c x-a d x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 76, normalized size = 0.9 \begin{align*} -{\frac{{d}^{2}}{c \left ( ad-bc \right ) }\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{acx}}+{\frac{{b}^{2}}{a \left ( ad-bc \right ) }\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(1/x^2/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65387, size = 819, normalized size = 10.11 \begin{align*} \left [-\frac{b c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + a d x \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x}, \frac{2 \, a d x \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) - b c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b c + 2 \, a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x}, -\frac{2 \, b c x \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + a d x \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x}, -\frac{b c x \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - a d x \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) + b c - a d}{{\left (a b c^{2} - a^{2} c d\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.76874, size = 1093, normalized size = 13.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**2+a)/(d*x**2+c),x)

[Out]

-sqrt(-b**3/a**3)*log(x + (-a**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + 2*a**6*b*c**4*d**3*(-b**3/a**3
)**(3/2)/(a*d - b*c)**3 - 2*a**5*b**2*c**5*d**2*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**5*d**5*sqrt(-b**3/a**3
)/(a*d - b*c) + 2*a**4*b**3*c**6*d*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**3*b**4*c**7*(-b**3/a**3)**(3/2)/(a*
d - b*c)**3 - b**5*c**5*sqrt(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d**2))/(2*(a
*d - b*c)) + sqrt(-b**3/a**3)*log(x + (a**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - 2*a**6*b*c**4*d**3*
(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + 2*a**5*b**2*c**5*d**2*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + a**5*d**5*sqrt
(-b**3/a**3)/(a*d - b*c) - 2*a**4*b**3*c**6*d*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + a**3*b**4*c**7*(-b**3/a**3)
**(3/2)/(a*d - b*c)**3 + b**5*c**5*sqrt(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d
**2))/(2*(a*d - b*c)) - sqrt(-d**3/c**3)*log(x + (-a**7*c**3*d**4*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + 2*a**6*
b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - 2*a**5*b**2*c**5*d**2*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - a*
*5*d**5*sqrt(-d**3/c**3)/(a*d - b*c) + 2*a**4*b**3*c**6*d*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - a**3*b**4*c**7*
(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - b**5*c**5*sqrt(-d**3/c**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 +
b**4*c**2*d**2))/(2*(a*d - b*c)) + sqrt(-d**3/c**3)*log(x + (a**7*c**3*d**4*(-d**3/c**3)**(3/2)/(a*d - b*c)**
3 - 2*a**6*b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + 2*a**5*b**2*c**5*d**2*(-d**3/c**3)**(3/2)/(a*d - b
*c)**3 + a**5*d**5*sqrt(-d**3/c**3)/(a*d - b*c) - 2*a**4*b**3*c**6*d*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + a**3
*b**4*c**7*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + b**5*c**5*sqrt(-d**3/c**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b*
*3*c*d**3 + b**4*c**2*d**2))/(2*(a*d - b*c)) - 1/(a*c*x)

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Giac [B]  time = 1.65034, size = 520, normalized size = 6.42 \begin{align*} \frac{{\left (\sqrt{c d} a b^{2} c^{2}{\left | d \right |} + \sqrt{c d} a^{2} b c d{\left | d \right |} - \sqrt{c d} b{\left | a b c^{2} - a^{2} c d \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a b c^{2} + a^{2} c d + \sqrt{-4 \, a^{3} b c^{3} d +{\left (a b c^{2} + a^{2} c d\right )}^{2}}}{a b c d}}}\right )}{a b c^{2} d{\left | a b c^{2} - a^{2} c d \right |} + a^{2} c d^{2}{\left | a b c^{2} - a^{2} c d \right |} +{\left (a b c^{2} - a^{2} c d\right )}^{2} d} - \frac{{\left (\sqrt{a b} a b c^{2} d{\left | b \right |} + \sqrt{a b} a^{2} c d^{2}{\left | b \right |} + \sqrt{a b} d{\left | a b c^{2} - a^{2} c d \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a b c^{2} + a^{2} c d - \sqrt{-4 \, a^{3} b c^{3} d +{\left (a b c^{2} + a^{2} c d\right )}^{2}}}{a b c d}}}\right )}{a b^{2} c^{2}{\left | a b c^{2} - a^{2} c d \right |} + a^{2} b c d{\left | a b c^{2} - a^{2} c d \right |} -{\left (a b c^{2} - a^{2} c d\right )}^{2} b} - \frac{1}{a c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")

[Out]

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