3.239 \(\int \frac{1}{x^6 (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=134 \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5} \]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) - (b^(7/2)*ArcTan[(Sq
rt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d))

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Rubi [A]  time = 0.227002, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {480, 583, 522, 205} \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) - (b^(7/2)*ArcTan[(Sq
rt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(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^6 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=-\frac{1}{5 a c x^5}+\frac{\int \frac{-5 (b c+a d)-5 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{5 a c}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{3 a^2 c^2 x^3}-\frac{\int \frac{-15 \left (b^2 c^2+a b c d+a^2 d^2\right )-15 b d (b c+a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{15 a^2 c^2}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{3 a^2 c^2 x^3}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{\int \frac{-15 (b c+a d) \left (b^2 c^2+a^2 d^2\right )-15 b d \left (b^2 c^2+a b c d+a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{15 a^3 c^3}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{3 a^2 c^2 x^3}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}-\frac{b^4 \int \frac{1}{a+b x^2} \, dx}{a^3 (b c-a d)}+\frac{d^4 \int \frac{1}{c+d x^2} \, dx}{c^3 (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{3 a^2 c^2 x^3}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}-\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.122789, size = 135, normalized size = 1.01 \[ \frac{-a^2 d^2-a b c d-b^2 c^2}{a^3 c^3 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (a d-b c)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) + (-(b^2*c^2) - a*b*c*d - a^2*d^2)/(a^3*c^3*x) + (b^(7/2)*ArcTan[
(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(-(b*c) + a*d)) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d))

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Maple [A]  time = 0.012, size = 141, normalized size = 1.1 \begin{align*} -{\frac{{d}^{4}}{{c}^{3} \left ( ad-bc \right ) }\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{5\,ac{x}^{5}}}+{\frac{d}{3\,a{c}^{2}{x}^{3}}}+{\frac{b}{3\,{a}^{2}c{x}^{3}}}-{\frac{{d}^{2}}{a{c}^{3}x}}-{\frac{bd}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}cx}}+{\frac{{b}^{4}}{{a}^{3} \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^6/(b*x^2+a)/(d*x^2+c),x)

[Out]

-1/c^3*d^4/(a*d-b*c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))-1/5/a/c/x^5+1/3/a/c^2/x^3*d+1/3/a^2/c/x^3*b-1/a/c^3/x
*d^2-1/a^2/c^2/x*b*d-1/a^3/c/x*b^2+1/a^3*b^4/(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^6/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.99941, size = 1358, normalized size = 10.13 \begin{align*} \left [-\frac{15 \, b^{3} c^{3} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 15 \, a^{3} d^{3} x^{5} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 6 \, a^{2} b c^{3} - 6 \, a^{3} c^{2} d + 30 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} - 10 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{30 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}, \frac{30 \, a^{3} d^{3} x^{5} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) - 15 \, b^{3} c^{3} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d - 30 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} + 10 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{30 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}, -\frac{30 \, b^{3} c^{3} x^{5} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 15 \, a^{3} d^{3} x^{5} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 6 \, a^{2} b c^{3} - 6 \, a^{3} c^{2} d + 30 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} - 10 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{30 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}, -\frac{15 \, b^{3} c^{3} x^{5} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - 15 \, a^{3} d^{3} x^{5} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) + 3 \, a^{2} b c^{3} - 3 \, a^{3} c^{2} d + 15 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} - 5 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{15 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

[-1/30*(15*b^3*c^3*x^5*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 15*a^3*d^3*x^5*sqrt(-d/c)*
log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 6*a^2*b*c^3 - 6*a^3*c^2*d + 30*(b^3*c^3 - a^3*d^3)*x^4 - 10*
(a*b^2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*d)*x^5), 1/30*(30*a^3*d^3*x^5*sqrt(d/c)*arctan(x*sqrt(d/c))
- 15*b^3*c^3*x^5*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 6*a^2*b*c^3 + 6*a^3*c^2*d - 30*
(b^3*c^3 - a^3*d^3)*x^4 + 10*(a*b^2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*d)*x^5), -1/30*(30*b^3*c^3*x^5
*sqrt(b/a)*arctan(x*sqrt(b/a)) + 15*a^3*d^3*x^5*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 6
*a^2*b*c^3 - 6*a^3*c^2*d + 30*(b^3*c^3 - a^3*d^3)*x^4 - 10*(a*b^2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*
d)*x^5), -1/15*(15*b^3*c^3*x^5*sqrt(b/a)*arctan(x*sqrt(b/a)) - 15*a^3*d^3*x^5*sqrt(d/c)*arctan(x*sqrt(d/c)) +
3*a^2*b*c^3 - 3*a^3*c^2*d + 15*(b^3*c^3 - a^3*d^3)*x^4 - 5*(a*b^2*c^3 - a^3*c*d^2)*x^2)/((a^3*b*c^4 - a^4*c^3*
d)*x^5)]

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Sympy [B]  time = 21.3415, size = 1504, normalized size = 11.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**2+a)/(d*x**2+c),x)

[Out]

-sqrt(-b**7/a**7)*log(x + (-a**13*c**7*d**6*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 + 2*a**12*b*c**8*d**5*(-b**7/a*
*7)**(3/2)/(a*d - b*c)**3 - a**11*b**2*c**9*d**4*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 - a**11*d**11*sqrt(-b**7/a
**7)/(a*d - b*c) - a**9*b**4*c**11*d**2*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 + 2*a**8*b**5*c**12*d*(-b**7/a**7)*
*(3/2)/(a*d - b*c)**3 - a**7*b**6*c**13*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 - b**11*c**11*sqrt(-b**7/a**7)/(a*d
- b*c))/(a**6*b**4*d**10 + a**5*b**5*c*d**9 + a**4*b**6*c**2*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d**6
+ a*b**9*c**5*d**5 + b**10*c**6*d**4))/(2*(a*d - b*c)) + sqrt(-b**7/a**7)*log(x + (a**13*c**7*d**6*(-b**7/a**
7)**(3/2)/(a*d - b*c)**3 - 2*a**12*b*c**8*d**5*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 + a**11*b**2*c**9*d**4*(-b**
7/a**7)**(3/2)/(a*d - b*c)**3 + a**11*d**11*sqrt(-b**7/a**7)/(a*d - b*c) + a**9*b**4*c**11*d**2*(-b**7/a**7)**
(3/2)/(a*d - b*c)**3 - 2*a**8*b**5*c**12*d*(-b**7/a**7)**(3/2)/(a*d - b*c)**3 + a**7*b**6*c**13*(-b**7/a**7)**
(3/2)/(a*d - b*c)**3 + b**11*c**11*sqrt(-b**7/a**7)/(a*d - b*c))/(a**6*b**4*d**10 + a**5*b**5*c*d**9 + a**4*b*
*6*c**2*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d**6 + a*b**9*c**5*d**5 + b**10*c**6*d**4))/(2*(a*d - b*c)
) - sqrt(-d**7/c**7)*log(x + (-a**13*c**7*d**6*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + 2*a**12*b*c**8*d**5*(-d**7
/c**7)**(3/2)/(a*d - b*c)**3 - a**11*b**2*c**9*d**4*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 - a**11*d**11*sqrt(-d**
7/c**7)/(a*d - b*c) - a**9*b**4*c**11*d**2*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + 2*a**8*b**5*c**12*d*(-d**7/c**
7)**(3/2)/(a*d - b*c)**3 - a**7*b**6*c**13*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 - b**11*c**11*sqrt(-d**7/c**7)/(
a*d - b*c))/(a**6*b**4*d**10 + a**5*b**5*c*d**9 + a**4*b**6*c**2*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d
**6 + a*b**9*c**5*d**5 + b**10*c**6*d**4))/(2*(a*d - b*c)) + sqrt(-d**7/c**7)*log(x + (a**13*c**7*d**6*(-d**7/
c**7)**(3/2)/(a*d - b*c)**3 - 2*a**12*b*c**8*d**5*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + a**11*b**2*c**9*d**4*(-
d**7/c**7)**(3/2)/(a*d - b*c)**3 + a**11*d**11*sqrt(-d**7/c**7)/(a*d - b*c) + a**9*b**4*c**11*d**2*(-d**7/c**7
)**(3/2)/(a*d - b*c)**3 - 2*a**8*b**5*c**12*d*(-d**7/c**7)**(3/2)/(a*d - b*c)**3 + a**7*b**6*c**13*(-d**7/c**7
)**(3/2)/(a*d - b*c)**3 + b**11*c**11*sqrt(-d**7/c**7)/(a*d - b*c))/(a**6*b**4*d**10 + a**5*b**5*c*d**9 + a**4
*b**6*c**2*d**8 + a**3*b**7*c**3*d**7 + a**2*b**8*c**4*d**6 + a*b**9*c**5*d**5 + b**10*c**6*d**4))/(2*(a*d - b
*c)) - (3*a**2*c**2 + x**4*(15*a**2*d**2 + 15*a*b*c*d + 15*b**2*c**2) + x**2*(-5*a**2*c*d - 5*a*b*c**2))/(15*a
**3*c**3*x**5)

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Giac [B]  time = 1.30495, size = 890, normalized size = 6.64 \begin{align*} \frac{{\left (\sqrt{c d} a^{3} b^{4} c^{6}{\left | d \right |} + \sqrt{c d} a^{6} b c^{3} d^{3}{\left | d \right |} - \sqrt{c d} b^{3} c^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | d \right |} - \sqrt{c d} a b^{2} c d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | d \right |} - \sqrt{c d} a^{2} b d^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{3} b c^{4} + a^{4} c^{3} d + \sqrt{-4 \, a^{7} b c^{7} d +{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )}^{2}}}{a^{3} b c^{3} d}}}\right )}{a^{3} b c^{4} d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} + a^{4} c^{3} d^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} +{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )}^{2} d} - \frac{{\left (\sqrt{a b} a^{3} b^{3} c^{6} d{\left | b \right |} + \sqrt{a b} a^{6} c^{3} d^{4}{\left | b \right |} + \sqrt{a b} b^{2} c^{2} d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | b \right |} + \sqrt{a b} a b c d^{2}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | b \right |} + \sqrt{a b} a^{2} d^{3}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{3} b c^{4} + a^{4} c^{3} d - \sqrt{-4 \, a^{7} b c^{7} d +{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )}^{2}}}{a^{3} b c^{3} d}}}\right )}{a^{3} b^{2} c^{4}{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} + a^{4} b c^{3} d{\left | a^{3} b c^{4} - a^{4} c^{3} d \right |} -{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )}^{2} b} - \frac{15 \, b^{2} c^{2} x^{4} + 15 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} - 5 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, a^{3} c^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")

[Out]

(sqrt(c*d)*a^3*b^4*c^6*abs(d) + sqrt(c*d)*a^6*b*c^3*d^3*abs(d) - sqrt(c*d)*b^3*c^2*abs(a^3*b*c^4 - a^4*c^3*d)*
abs(d) - sqrt(c*d)*a*b^2*c*d*abs(a^3*b*c^4 - a^4*c^3*d)*abs(d) - sqrt(c*d)*a^2*b*d^2*abs(a^3*b*c^4 - a^4*c^3*d
)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((a^3*b*c^4 + a^4*c^3*d + sqrt(-4*a^7*b*c^7*d + (a^3*b*c^4 + a^4*c^3*d)^2))
/(a^3*b*c^3*d)))/(a^3*b*c^4*d*abs(a^3*b*c^4 - a^4*c^3*d) + a^4*c^3*d^2*abs(a^3*b*c^4 - a^4*c^3*d) + (a^3*b*c^4
- a^4*c^3*d)^2*d) - (sqrt(a*b)*a^3*b^3*c^6*d*abs(b) + sqrt(a*b)*a^6*c^3*d^4*abs(b) + sqrt(a*b)*b^2*c^2*d*abs(
a^3*b*c^4 - a^4*c^3*d)*abs(b) + sqrt(a*b)*a*b*c*d^2*abs(a^3*b*c^4 - a^4*c^3*d)*abs(b) + sqrt(a*b)*a^2*d^3*abs(
a^3*b*c^4 - a^4*c^3*d)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((a^3*b*c^4 + a^4*c^3*d - sqrt(-4*a^7*b*c^7*d + (a^3*b
*c^4 + a^4*c^3*d)^2))/(a^3*b*c^3*d)))/(a^3*b^2*c^4*abs(a^3*b*c^4 - a^4*c^3*d) + a^4*b*c^3*d*abs(a^3*b*c^4 - a^
4*c^3*d) - (a^3*b*c^4 - a^4*c^3*d)^2*b) - 1/15*(15*b^2*c^2*x^4 + 15*a*b*c*d*x^4 + 15*a^2*d^2*x^4 - 5*a*b*c^2*x
^2 - 5*a^2*c*d*x^2 + 3*a^2*c^2)/(a^3*c^3*x^5)