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(x*b^(1/2)/a^(1/2))/a^(3/2)/(-a*d+b*c)+d^(3/2)*arctan(x*d^(1/2)/c^(1/2))/c^(3/2)/(-a*d+
b*c)

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Rubi [A]  time = 0.09, antiderivative size = 81, 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 = {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 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 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]

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*}

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Mathematica [A]  time = 0.09, 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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fricas [A]  time = 0.55, size = 384, normalized size = 4.74 \[ \left [-\frac {b c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + a d x \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x}, \frac {2 \, a d x \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) - b c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b c + 2 \, a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x}, -\frac {2 \, b c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + a d x \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x}, -\frac {b c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - a d x \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + b c - a d}{{\left (a b c^{2} - a^{2} c d\right )} x}\right ] \]

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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giac [A]  time = 0.35, size = 75, normalized size = 0.93 \[ -\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b c - a^{2} d\right )} \sqrt {a b}} + \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c^{2} - a c d\right )} \sqrt {c d}} - \frac {1}{a c x} \]

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]

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

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maple [A]  time = 0.01, size = 76, normalized size = 0.94 \[ \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \sqrt {a b}\, a}-\frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \sqrt {c d}\, c}-\frac {1}{a c x} \]

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

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maxima [A]  time = 2.39, size = 75, normalized size = 0.93 \[ -\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b c - a^{2} d\right )} \sqrt {a b}} + \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c^{2} - a c d\right )} \sqrt {c d}} - \frac {1}{a c x} \]

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]

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

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mupad [B]  time = 0.63, size = 338, normalized size = 4.17 \[ \frac {\ln \left (a^3\,c^5\,d^4-b^3\,c^8\,d+a^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+b^3\,c^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,b\,c^4-2\,a\,c^3\,d}-\frac {\ln \left (b^3\,c^8\,d-a^3\,c^5\,d^4+a^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+b^3\,c^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,\left (b\,c^4-a\,c^3\,d\right )}-\frac {1}{a\,c\,x}-\frac {\ln \left (a^8\,b\,d^3-a^5\,b^4\,c^3+c^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+a^6\,d^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,\left (a^4\,d-a^3\,b\,c\right )}+\frac {\ln \left (a^5\,b^4\,c^3-a^8\,b\,d^3+c^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+a^6\,d^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,a^4\,d-2\,a^3\,b\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(a^3*c^5*d^4 - b^3*c^8*d + a^3*x*(-c^3*d^3)^(3/2) + b^3*c^6*x*(-c^3*d^3)^(1/2))*(-c^3*d^3)^(1/2))/(2*b*c^4
- 2*a*c^3*d) - (log(b^3*c^8*d - a^3*c^5*d^4 + a^3*x*(-c^3*d^3)^(3/2) + b^3*c^6*x*(-c^3*d^3)^(1/2))*(-c^3*d^3)
^(1/2))/(2*(b*c^4 - a*c^3*d)) - 1/(a*c*x) - (log(a^8*b*d^3 - a^5*b^4*c^3 + c^3*x*(-a^3*b^3)^(3/2) + a^6*d^3*x*
(-a^3*b^3)^(1/2))*(-a^3*b^3)^(1/2))/(2*(a^4*d - a^3*b*c)) + (log(a^5*b^4*c^3 - a^8*b*d^3 + c^3*x*(-a^3*b^3)^(3
/2) + a^6*d^3*x*(-a^3*b^3)^(1/2))*(-a^3*b^3)^(1/2))/(2*a^4*d - 2*a^3*b*c)

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sympy [B]  time = 12.23, size = 1093, normalized size = 13.49 \[ \text {result too large to display} \]

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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