3.786 \(\int \frac{1}{x^6 (a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=479 \[ \frac{b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{9/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{d^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{1}{5 a c x^5} \]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(a^2*c^2*x) - (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(9/4
)*(b*c - a*d)) + (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)) + (d^(9/4)*
ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(9/4)*(b*c - a*d)) - (d^(9/4)*ArcTan[1 + (Sqrt[2]*d^(1/4
)*x)/c^(1/4)])/(2*Sqrt[2]*c^(9/4)*(b*c - a*d)) + (b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^
2])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)) - (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt
[2]*a^(9/4)*(b*c - a*d)) - (d^(9/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(9/4)
*(b*c - a*d)) + (d^(9/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d
))

________________________________________________________________________________________

Rubi [A]  time = 0.596542, antiderivative size = 479, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {480, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac{b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{9/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{d^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{1}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^4)*(c + d*x^4)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(a^2*c^2*x) - (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(9/4
)*(b*c - a*d)) + (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)) + (d^(9/4)*
ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(9/4)*(b*c - a*d)) - (d^(9/4)*ArcTan[1 + (Sqrt[2]*d^(1/4
)*x)/c^(1/4)])/(2*Sqrt[2]*c^(9/4)*(b*c - a*d)) + (b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^
2])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)) - (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt
[2]*a^(9/4)*(b*c - a*d)) - (d^(9/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(9/4)
*(b*c - a*d)) + (d^(9/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(9/4)*(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 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=-\frac{1}{5 a c x^5}+\frac{\int \frac{-5 (b c+a d)-5 b d x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{5 a c}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}-\frac{\int \frac{x^2 \left (-5 \left (b^2 c^2+a b c d+a^2 d^2\right )-5 b d (b c+a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{5 a^2 c^2}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}-\frac{\int \left (-\frac{5 b^3 c^2 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac{5 a^2 d^3 x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx}{5 a^2 c^2}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}+\frac{b^3 \int \frac{x^2}{a+b x^4} \, dx}{a^2 (b c-a d)}-\frac{d^3 \int \frac{x^2}{c+d x^4} \, dx}{c^2 (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}-\frac{b^{5/2} \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{2 a^2 (b c-a d)}+\frac{b^{5/2} \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{2 a^2 (b c-a d)}+\frac{d^{5/2} \int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx}{2 c^2 (b c-a d)}-\frac{d^{5/2} \int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx}{2 c^2 (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}+\frac{b^2 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 a^2 (b c-a d)}+\frac{b^2 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 a^2 (b c-a d)}+\frac{b^{9/4} \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt{2} a^{9/4} (b c-a d)}+\frac{b^{9/4} \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{d^2 \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 c^2 (b c-a d)}-\frac{d^2 \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 c^2 (b c-a d)}-\frac{d^{9/4} \int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{4 \sqrt{2} c^{9/4} (b c-a d)}-\frac{d^{9/4} \int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{4 \sqrt{2} c^{9/4} (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}+\frac{b^{9/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{d^{9/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{b^{9/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}-\frac{d^{9/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{a^2 c^2 x}-\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{b^{9/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{d^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{d^{9/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{9/4} (b c-a d)}+\frac{b^{9/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{d^{9/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{9/4} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.253502, size = 428, normalized size = 0.89 \[ \frac{-\frac{40 b^2 x^4}{a^2}-\frac{5 \sqrt{2} b^{9/4} x^5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{9/4}}+\frac{5 \sqrt{2} b^{9/4} x^5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{9/4}}+\frac{10 \sqrt{2} b^{9/4} x^5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{9/4}}-\frac{10 \sqrt{2} b^{9/4} x^5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{9/4}}+\frac{8 b}{a}+\frac{40 d^2 x^4}{c^2}+\frac{5 \sqrt{2} d^{9/4} x^5 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{9/4}}-\frac{5 \sqrt{2} d^{9/4} x^5 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{9/4}}-\frac{10 \sqrt{2} d^{9/4} x^5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{9/4}}+\frac{10 \sqrt{2} d^{9/4} x^5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{9/4}}-\frac{8 d}{c}}{40 x^5 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^4)*(c + d*x^4)),x]

[Out]

((8*b)/a - (8*d)/c - (40*b^2*x^4)/a^2 + (40*d^2*x^4)/c^2 + (10*Sqrt[2]*b^(9/4)*x^5*ArcTan[1 - (Sqrt[2]*b^(1/4)
*x)/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*b^(9/4)*x^5*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(9/4) - (10*Sqrt[2]
*d^(9/4)*x^5*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(9/4) + (10*Sqrt[2]*d^(9/4)*x^5*ArcTan[1 + (Sqrt[2]*d^
(1/4)*x)/c^(1/4)])/c^(9/4) - (5*Sqrt[2]*b^(9/4)*x^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^
(9/4) + (5*Sqrt[2]*b^(9/4)*x^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(9/4) + (5*Sqrt[2]*d^
(9/4)*x^5*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(9/4) - (5*Sqrt[2]*d^(9/4)*x^5*Log[Sqrt[c]
+ Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(9/4))/(40*(-(b*c) + a*d)*x^5)

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Maple [A]  time = 0.013, size = 365, normalized size = 0.8 \begin{align*}{\frac{{d}^{2}\sqrt{2}}{8\,{c}^{2} \left ( ad-bc \right ) }\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+{\frac{{d}^{2}\sqrt{2}}{4\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+{\frac{{d}^{2}\sqrt{2}}{4\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{{b}^{2}\sqrt{2}}{8\,{a}^{2} \left ( ad-bc \right ) }\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}\sqrt{2}}{4\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}\sqrt{2}}{4\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{1}{5\,ac{x}^{5}}}+{\frac{d}{a{c}^{2}x}}+{\frac{b}{{a}^{2}cx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^4+a)/(d*x^4+c),x)

[Out]

1/8*d^2/c^2/(a*d-b*c)/(1/d*c)^(1/4)*2^(1/2)*ln((x^2-(1/d*c)^(1/4)*x*2^(1/2)+(1/d*c)^(1/2))/(x^2+(1/d*c)^(1/4)*
x*2^(1/2)+(1/d*c)^(1/2)))+1/4*d^2/c^2/(a*d-b*c)/(1/d*c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/d*c)^(1/4)*x+1)+1/4*d^
2/c^2/(a*d-b*c)/(1/d*c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/d*c)^(1/4)*x-1)-1/8*b^2/a^2/(a*d-b*c)/(a/b)^(1/4)*2^(1
/2)*ln((x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))-1/4*b^2/a^2/(a*d-b*c)/
(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)-1/4*b^2/a^2/(a*d-b*c)/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a/b)^(1/4)*x-1)-1/5/a/c/x^5+1/a/c^2/x*d+1/a^2/c/x*b

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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^4+a)/(d*x^4+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 36.764, size = 2942, normalized size = 6.14 \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^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

1/20*(20*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*a^2*c^
2*x^5*arctan(((-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*(
a^2*b*c - a^3*d)*x - (-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^
(1/4)*(a^2*b*c - a^3*d)*sqrt((b^5*x^2 - (a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2)*sqrt(-b^9/(a^9*b^4*c^4 - 4*a^10*
b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4)))/b^5))/b^2) - 20*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d
+ 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*a^2*c^2*x^5*arctan(((-d^9/(b^4*c^13 - 4*a*b^3*c
^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*(b*c^3 - a*c^2*d)*x - (-d^9/(b^4*c^13 - 4*
a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*(b*c^3 - a*c^2*d)*sqrt((d^5*x^2 - (
b^2*c^7 - 2*a*b*c^6*d + a^2*c^5*d^2)*sqrt(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*
d^3 + a^4*c^9*d^4)))/d^5))/d^2) + 5*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^
3 + a^13*d^4))^(1/4)*a^2*c^2*x^5*log(b^7*x + (a^7*b^3*c^3 - 3*a^8*b^2*c^2*d + 3*a^9*b*c*d^2 - a^10*d^3)*(-b^9/
(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) - 5*(-b^9/(a^9*b^4*c
^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*a^2*c^2*x^5*log(b^7*x - (a^7*b^
3*c^3 - 3*a^8*b^2*c^2*d + 3*a^9*b*c*d^2 - a^10*d^3)*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2
- 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) - 5*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10
*d^3 + a^4*c^9*d^4))^(1/4)*a^2*c^2*x^5*log(d^7*x + (b^3*c^10 - 3*a*b^2*c^9*d + 3*a^2*b*c^8*d^2 - a^3*c^7*d^3)*
(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(3/4)) + 5*(-d^9/(b^4
*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*a^2*c^2*x^5*log(d^7*x - (
b^3*c^10 - 3*a*b^2*c^9*d + 3*a^2*b*c^8*d^2 - a^3*c^7*d^3)*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^
2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(3/4)) + 20*(b*c + a*d)*x^4 - 4*a*c)/(a^2*c^2*x^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**4+a)/(d*x**4+c),x)

[Out]

Timed out

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Giac [C]  time = 3.52955, size = 1737, normalized size = 3.63 \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^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

-1/4*I*2^(1/4)*(1/2)^(1/4)*(-(b^13*c^13 - 4*a*b^12*c^12*d + 6*a^2*b^11*c^11*d^2 - 4*a^3*b^10*c^10*d^3 + a^4*b^
9*c^9*d^4)/(a^9*b^8*c^17 - 8*a^10*b^7*c^16*d + 28*a^11*b^6*c^15*d^2 - 56*a^12*b^5*c^14*d^3 + 70*a^13*b^4*c^13*
d^4 - 56*a^14*b^3*c^12*d^5 + 28*a^15*b^2*c^11*d^6 - 8*a^16*b*c^10*d^7 + a^17*c^9*d^8))^(1/4)*log(8^(3/4)*a + 4
*I*2^(1/4)*(-a^3*b)^(1/4)*x) + 1/4*I*2^(1/4)*(1/2)^(1/4)*(-(b^13*c^13 - 4*a*b^12*c^12*d + 6*a^2*b^11*c^11*d^2
- 4*a^3*b^10*c^10*d^3 + a^4*b^9*c^9*d^4)/(a^9*b^8*c^17 - 8*a^10*b^7*c^16*d + 28*a^11*b^6*c^15*d^2 - 56*a^12*b^
5*c^14*d^3 + 70*a^13*b^4*c^13*d^4 - 56*a^14*b^3*c^12*d^5 + 28*a^15*b^2*c^11*d^6 - 8*a^16*b*c^10*d^7 + a^17*c^9
*d^8))^(1/4)*log(8^(3/4)*a - 4*I*2^(1/4)*(-a^3*b)^(1/4)*x) - 1/4*I*2^(1/4)*(1/2)^(1/4)*(-(a^9*b^4*c^4*d^9 - 4*
a^10*b^3*c^3*d^10 + 6*a^11*b^2*c^2*d^11 - 4*a^12*b*c*d^12 + a^13*d^13)/(a^9*b^8*c^17 - 8*a^10*b^7*c^16*d + 28*
a^11*b^6*c^15*d^2 - 56*a^12*b^5*c^14*d^3 + 70*a^13*b^4*c^13*d^4 - 56*a^14*b^3*c^12*d^5 + 28*a^15*b^2*c^11*d^6
- 8*a^16*b*c^10*d^7 + a^17*c^9*d^8))^(1/4)*log(8^(3/4)*c + 4*I*2^(1/4)*(-c^3*d)^(1/4)*x) + 1/4*I*2^(1/4)*(1/2)
^(1/4)*(-(a^9*b^4*c^4*d^9 - 4*a^10*b^3*c^3*d^10 + 6*a^11*b^2*c^2*d^11 - 4*a^12*b*c*d^12 + a^13*d^13)/(a^9*b^8*
c^17 - 8*a^10*b^7*c^16*d + 28*a^11*b^6*c^15*d^2 - 56*a^12*b^5*c^14*d^3 + 70*a^13*b^4*c^13*d^4 - 56*a^14*b^3*c^
12*d^5 + 28*a^15*b^2*c^11*d^6 - 8*a^16*b*c^10*d^7 + a^17*c^9*d^8))^(1/4)*log(8^(3/4)*c - 4*I*2^(1/4)*(-c^3*d)^
(1/4)*x) - 1/4*2^(1/4)*(1/2)^(1/4)*(-(b^13*c^13 - 4*a*b^12*c^12*d + 6*a^2*b^11*c^11*d^2 - 4*a^3*b^10*c^10*d^3
+ a^4*b^9*c^9*d^4)/(a^9*b^8*c^17 - 8*a^10*b^7*c^16*d + 28*a^11*b^6*c^15*d^2 - 56*a^12*b^5*c^14*d^3 + 70*a^13*b
^4*c^13*d^4 - 56*a^14*b^3*c^12*d^5 + 28*a^15*b^2*c^11*d^6 - 8*a^16*b*c^10*d^7 + a^17*c^9*d^8))^(1/4)*log(abs(8
^(3/4)*a + 4*2^(1/4)*(-a^3*b)^(1/4)*x)) + 1/4*2^(1/4)*(1/2)^(1/4)*(-(b^13*c^13 - 4*a*b^12*c^12*d + 6*a^2*b^11*
c^11*d^2 - 4*a^3*b^10*c^10*d^3 + a^4*b^9*c^9*d^4)/(a^9*b^8*c^17 - 8*a^10*b^7*c^16*d + 28*a^11*b^6*c^15*d^2 - 5
6*a^12*b^5*c^14*d^3 + 70*a^13*b^4*c^13*d^4 - 56*a^14*b^3*c^12*d^5 + 28*a^15*b^2*c^11*d^6 - 8*a^16*b*c^10*d^7 +
a^17*c^9*d^8))^(1/4)*log(abs(8^(3/4)*a - 4*2^(1/4)*(-a^3*b)^(1/4)*x)) + 1/5*(5*b*c*x^4 + 5*a*d*x^4 - a*c)/(a^
2*c^2*x^5)