3.777 \(\int \frac{1}{x^3 (a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=92 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)}-\frac{1}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.121941, antiderivative size = 92, 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 = {465, 480, 522, 205} \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^4)*(c + d*x^4)),x]

[Out]

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

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [A]  time = 0.203156, size = 169, normalized size = 1.84 \[ \frac{-\frac{b^{3/2} x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/2}}-\frac{b^{3/2} x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{3/2}}+\frac{b}{a}+\frac{d^{3/2} x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{3/2}}+\frac{d^{3/2} x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{3/2}}-\frac{d}{c}}{2 x^2 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^4)*(c + d*x^4)),x]

[Out]

(b/a - d/c - (b^(3/2)*x^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/2) - (b^(3/2)*x^2*ArcTan[1 + (Sqrt[2]*
b^(1/4)*x)/a^(1/4)])/a^(3/2) + (d^(3/2)*x^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(3/2) + (d^(3/2)*x^2*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(3/2))/(2*(-(b*c) + a*d)*x^2)

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Maple [A]  time = 0.008, size = 81, normalized size = 0.9 \begin{align*} -{\frac{{d}^{2}}{2\,c \left ( ad-bc \right ) }\arctan \left ({{x}^{2}d{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{2\,ac{x}^{2}}}+{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) }\arctan \left ({b{x}^{2}{\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^3/(b*x^4+a)/(d*x^4+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.90047, size = 910, normalized size = 9.89 \begin{align*} \left [-\frac{b c x^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} + 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + a d x^{2} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} - 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) + 2 \, b c - 2 \, a d}{4 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}, -\frac{2 \, a d x^{2} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) + b c x^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} + 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + 2 \, b c - 2 \, a d}{4 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}, \frac{2 \, b c x^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - a d x^{2} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} - 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) - 2 \, b c + 2 \, a d}{4 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}, \frac{b c x^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - a d x^{2} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - b c + a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

[-1/4*(b*c*x^2*sqrt(-b/a)*log((b*x^4 + 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + a*d*x^2*sqrt(-d/c)*log((d*x^4 -
2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) + 2*b*c - 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), -1/4*(2*a*d*x^2*sqrt(d/c)*arc
tan(c*sqrt(d/c)/(d*x^2)) + b*c*x^2*sqrt(-b/a)*log((b*x^4 + 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + 2*b*c - 2*a*
d)/((a*b*c^2 - a^2*c*d)*x^2), 1/4*(2*b*c*x^2*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*x^2)) - a*d*x^2*sqrt(-d/c)*log((d
*x^4 - 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) - 2*b*c + 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), 1/2*(b*c*x^2*sqrt(b/a)
*arctan(a*sqrt(b/a)/(b*x^2)) - a*d*x^2*sqrt(d/c)*arctan(c*sqrt(d/c)/(d*x^2)) - b*c + a*d)/((a*b*c^2 - a^2*c*d)
*x^2)]

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Sympy [B]  time = 11.9343, size = 1103, normalized size = 11.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**4+a)/(d*x**4+c),x)

[Out]

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

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Giac [B]  time = 1.20184, size = 508, normalized size = 5.52 \begin{align*} \frac{{\left (\sqrt{c d} a b^{2} c d x^{4}{\left | d \right |} + \sqrt{c d} a b^{2} c^{2}{\left | d \right |} + \sqrt{c d} a^{2} b c d{\left | d \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a b c^{2} + 2 \, a^{2} c d + \sqrt{-16 \, a^{3} b c^{3} d + 4 \,{\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 d^{2} x^{4}{\left | b \right |} + \sqrt{a b} a b c^{2} d{\left | b \right |} + \sqrt{a b} a^{2} c d^{2}{\left | b \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a b c^{2} + 2 \, a^{2} c d - \sqrt{-16 \, a^{3} b c^{3} d + 4 \,{\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}{2 \, a c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

(sqrt(c*d)*a*b^2*c*d*x^4*abs(d) + sqrt(c*d)*a*b^2*c^2*abs(d) + sqrt(c*d)*a^2*b*c*d*abs(d))*arctan(2*x^2/sqrt((
2*a*b*c^2 + 2*a^2*c*d + sqrt(-16*a^3*b*c^3*d + 4*(a*b*c^2 + a^2*c*d)^2))/(a*b*c*d)))/(a*b*c^2*d*abs(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*d^2*x^4*abs(b) + sqr
t(a*b)*a*b*c^2*d*abs(b) + sqrt(a*b)*a^2*c*d^2*abs(b))*arctan(2*x^2/sqrt((2*a*b*c^2 + 2*a^2*c*d - sqrt(-16*a^3*
b*c^3*d + 4*(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) - 1/2/(a*c*x^2)