3.237 \(\int \frac{1}{x^4 (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=100 \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3} \]

[Out]

-1/(3*a*c*x^3) + (b*c + a*d)/(a^2*c^2*x) + (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(b*c - a*d)) - (d^(5
/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d))

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Rubi [A]  time = 0.176916, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {480, 583, 522, 205} \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(3*a*c*x^3) + (b*c + a*d)/(a^2*c^2*x) + (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(b*c - a*d)) - (d^(5
/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(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 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=-\frac{1}{3 a c x^3}+\frac{\int \frac{-3 (b c+a d)-3 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{3 a c}\\ &=-\frac{1}{3 a c x^3}+\frac{b c+a d}{a^2 c^2 x}-\frac{\int \frac{-3 \left (b^2 c^2+a b c d+a^2 d^2\right )-3 b d (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{3 a^2 c^2}\\ &=-\frac{1}{3 a c x^3}+\frac{b c+a d}{a^2 c^2 x}+\frac{b^3 \int \frac{1}{a+b x^2} \, dx}{a^2 (b c-a d)}-\frac{d^3 \int \frac{1}{c+d x^2} \, dx}{c^2 (b c-a d)}\\ &=-\frac{1}{3 a c x^3}+\frac{b c+a d}{a^2 c^2 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (b c-a d)}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.122165, size = 101, normalized size = 1.01 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (a d-b c)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(3*a*c*x^3) + (b*c + a*d)/(a^2*c^2*x) - (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)) - (d
^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d))

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Maple [A]  time = 0.011, size = 98, normalized size = 1. \begin{align*}{\frac{{d}^{3}}{{c}^{2} \left ( ad-bc \right ) }\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{3\,ac{x}^{3}}}+{\frac{d}{a{c}^{2}x}}+{\frac{b}{{a}^{2}cx}}-{\frac{{b}^{3}}{{a}^{2} \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^4/(b*x^2+a)/(d*x^2+c),x)

[Out]

1/c^2*d^3/(a*d-b*c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))-1/3/a/c/x^3+1/a/c^2/x*d+1/a^2/c/x*b-1/a^2*b^3/(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^4/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7505, size = 1131, normalized size = 11.31 \begin{align*} \left [-\frac{3 \, b^{2} c^{2} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 3 \, a^{2} d^{2} x^{3} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 2 \, a b c^{2} - 2 \, a^{2} c d - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}, -\frac{6 \, a^{2} d^{2} x^{3} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) + 3 \, b^{2} c^{2} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 2 \, a b c^{2} - 2 \, a^{2} c d - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}, \frac{6 \, b^{2} c^{2} x^{3} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - 3 \, a^{2} d^{2} x^{3} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 2 \, a b c^{2} + 2 \, a^{2} c d + 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}, \frac{3 \, b^{2} c^{2} x^{3} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - 3 \, a^{2} d^{2} x^{3} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) - a b c^{2} + a^{2} c d + 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{3 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

[-1/6*(3*b^2*c^2*x^3*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 3*a^2*d^2*x^3*sqrt(-d/c)*log
((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*a*b*c^2 - 2*a^2*c*d - 6*(b^2*c^2 - a^2*d^2)*x^2)/((a^2*b*c^3
- a^3*c^2*d)*x^3), -1/6*(6*a^2*d^2*x^3*sqrt(d/c)*arctan(x*sqrt(d/c)) + 3*b^2*c^2*x^3*sqrt(-b/a)*log((b*x^2 - 2
*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*a*b*c^2 - 2*a^2*c*d - 6*(b^2*c^2 - a^2*d^2)*x^2)/((a^2*b*c^3 - a^3*c^2*d
)*x^3), 1/6*(6*b^2*c^2*x^3*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*a^2*d^2*x^3*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d
/c) - c)/(d*x^2 + c)) - 2*a*b*c^2 + 2*a^2*c*d + 6*(b^2*c^2 - a^2*d^2)*x^2)/((a^2*b*c^3 - a^3*c^2*d)*x^3), 1/3*
(3*b^2*c^2*x^3*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*a^2*d^2*x^3*sqrt(d/c)*arctan(x*sqrt(d/c)) - a*b*c^2 + a^2*c*d
+ 3*(b^2*c^2 - a^2*d^2)*x^2)/((a^2*b*c^3 - a^3*c^2*d)*x^3)]

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Sympy [B]  time = 9.50538, size = 1353, normalized size = 13.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)/(d*x**2+c),x)

[Out]

sqrt(-b**5/a**5)*log(x + (-a**10*c**5*d**5*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + 2*a**9*b*c**6*d**4*(-b**5/a**5
)**(3/2)/(a*d - b*c)**3 - a**8*b**2*c**7*d**3*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - a**8*d**8*sqrt(-b**5/a**5)/
(a*d - b*c) - a**7*b**3*c**8*d**2*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + 2*a**6*b**4*c**9*d*(-b**5/a**5)**(3/2)/
(a*d - b*c)**3 - a**5*b**5*c**10*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - b**8*c**8*sqrt(-b**5/a**5)/(a*d - b*c))/
(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c**2*d**5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d - b*c))
- sqrt(-b**5/a**5)*log(x + (a**10*c**5*d**5*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - 2*a**9*b*c**6*d**4*(-b**5/a*
*5)**(3/2)/(a*d - b*c)**3 + a**8*b**2*c**7*d**3*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + a**8*d**8*sqrt(-b**5/a**5
)/(a*d - b*c) + a**7*b**3*c**8*d**2*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - 2*a**6*b**4*c**9*d*(-b**5/a**5)**(3/2
)/(a*d - b*c)**3 + a**5*b**5*c**10*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + b**8*c**8*sqrt(-b**5/a**5)/(a*d - b*c)
)/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c**2*d**5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d - b*c
)) + sqrt(-d**5/c**5)*log(x + (-a**10*c**5*d**5*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 + 2*a**9*b*c**6*d**4*(-d**5
/c**5)**(3/2)/(a*d - b*c)**3 - a**8*b**2*c**7*d**3*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - a**8*d**8*sqrt(-d**5/c
**5)/(a*d - b*c) - a**7*b**3*c**8*d**2*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 + 2*a**6*b**4*c**9*d*(-d**5/c**5)**(
3/2)/(a*d - b*c)**3 - a**5*b**5*c**10*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - b**8*c**8*sqrt(-d**5/c**5)/(a*d - b
*c))/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c**2*d**5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d -
b*c)) - sqrt(-d**5/c**5)*log(x + (a**10*c**5*d**5*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - 2*a**9*b*c**6*d**4*(-d*
*5/c**5)**(3/2)/(a*d - b*c)**3 + a**8*b**2*c**7*d**3*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 + a**8*d**8*sqrt(-d**5
/c**5)/(a*d - b*c) + a**7*b**3*c**8*d**2*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - 2*a**6*b**4*c**9*d*(-d**5/c**5)*
*(3/2)/(a*d - b*c)**3 + a**5*b**5*c**10*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 + b**8*c**8*sqrt(-d**5/c**5)/(a*d -
b*c))/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c**2*d**5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d
- b*c)) + (-a*c + x**2*(3*a*d + 3*b*c))/(3*a**2*c**2*x**3)

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Giac [B]  time = 1.28634, size = 728, normalized size = 7.28 \begin{align*} -\frac{{\left (\sqrt{c d} a^{2} b^{3} c^{4}{\left | d \right |} + \sqrt{c d} a^{4} b c^{2} d^{2}{\left | d \right |} - \sqrt{c d} b^{2} c{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | d \right |} - \sqrt{c d} a b d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{2} b c^{3} + a^{3} c^{2} d + \sqrt{-4 \, a^{5} b c^{5} d +{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b c^{3} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} c^{2} d^{2}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} +{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} d} + \frac{{\left (\sqrt{a b} a^{2} b^{2} c^{4} d{\left | b \right |} + \sqrt{a b} a^{4} c^{2} d^{3}{\left | b \right |} + \sqrt{a b} b c d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | b \right |} + \sqrt{a b} a d^{2}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{2} b c^{3} + a^{3} c^{2} d - \sqrt{-4 \, a^{5} b c^{5} d +{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b^{2} c^{3}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} b c^{2} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} -{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} b} + \frac{3 \, b c x^{2} + 3 \, a d x^{2} - a c}{3 \, a^{2} c^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")

[Out]

-(sqrt(c*d)*a^2*b^3*c^4*abs(d) + sqrt(c*d)*a^4*b*c^2*d^2*abs(d) - sqrt(c*d)*b^2*c*abs(a^2*b*c^3 - a^3*c^2*d)*a
bs(d) - sqrt(c*d)*a*b*d*abs(a^2*b*c^3 - a^3*c^2*d)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((a^2*b*c^3 + a^3*c^2*d +
sqrt(-4*a^5*b*c^5*d + (a^2*b*c^3 + a^3*c^2*d)^2))/(a^2*b*c^2*d)))/(a^2*b*c^3*d*abs(a^2*b*c^3 - a^3*c^2*d) + a^
3*c^2*d^2*abs(a^2*b*c^3 - a^3*c^2*d) + (a^2*b*c^3 - a^3*c^2*d)^2*d) + (sqrt(a*b)*a^2*b^2*c^4*d*abs(b) + sqrt(a
*b)*a^4*c^2*d^3*abs(b) + sqrt(a*b)*b*c*d*abs(a^2*b*c^3 - a^3*c^2*d)*abs(b) + sqrt(a*b)*a*d^2*abs(a^2*b*c^3 - a
^3*c^2*d)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((a^2*b*c^3 + a^3*c^2*d - sqrt(-4*a^5*b*c^5*d + (a^2*b*c^3 + a^3*c^
2*d)^2))/(a^2*b*c^2*d)))/(a^2*b^2*c^3*abs(a^2*b*c^3 - a^3*c^2*d) + a^3*b*c^2*d*abs(a^2*b*c^3 - a^3*c^2*d) - (a
^2*b*c^3 - a^3*c^2*d)^2*b) + 1/3*(3*b*c*x^2 + 3*a*d*x^2 - a*c)/(a^2*c^2*x^3)