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

Optimal. Leaf size=460 \[ -\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {1}{a c x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.45, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {480, 584, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {1}{a c x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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 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]

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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.22, size = 385, normalized size = 0.84 \begin {gather*} \frac {\frac {\sqrt {2} b^{5/4} x \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{a^{5/4}}-\frac {\sqrt {2} b^{5/4} x \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{a^{5/4}}-\frac {2 \sqrt {2} b^{5/4} x \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac {2 \sqrt {2} b^{5/4} x \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac {8 b}{a}-\frac {\sqrt {2} d^{5/4} x \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{c^{5/4}}+\frac {\sqrt {2} d^{5/4} x \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{c^{5/4}}+\frac {2 \sqrt {2} d^{5/4} x \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac {2 \sqrt {2} d^{5/4} x \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac {8 d}{c}}{8 a d x-8 b c x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.89, size = 1395, normalized size = 3.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*arcta
n(((-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*(a*b*c - a^2*d)*
x - (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*(a*b*c - a^2*d)
*sqrt((b^3*x^2 - (a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^
2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4)))/b^3))/b) - 4*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c
^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*arctan(((-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^
3 + a^4*c^5*d^4))^(1/4)*(b*c^2 - a*c*d)*x - (-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d
^3 + a^4*c^5*d^4))^(1/4)*(b*c^2 - a*c*d)*sqrt((d^3*x^2 - (b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(-d^5/(b^4*
c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4)))/d^3))/d) + (-b^5/(a^5*b^4*c^4 - 4*a
^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*log(b^4*x + (a^4*b^3*c^3 - 3*a^5*b^2*
c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^
9*d^4))^(3/4)) - (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*
c*x*log(b^4*x - (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d
+ 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(3/4)) - (-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 -
4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*log(d^4*x + (b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*
d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(3/4)) + (-d^5/(b^4*
c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*log(d^4*x - (b^3*c^7 - 3
*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6
*d^3 + a^4*c^5*d^4))^(3/4)) + 4)/(a*c*x)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 488, normalized size = 1.06 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{4 \, {\left (\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{4 \, {\left (\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} - \frac {1}{a c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 331, normalized size = 0.72 \begin {gather*} \frac {\sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a}+\frac {\sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a}+\frac {\sqrt {2}\, b \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a}-\frac {\sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {\sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {\sqrt {2}\, d \ln \left (\frac {x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{8 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {1}{a c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.27, size = 384, normalized size = 0.83 \begin {gather*} -\frac {b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8 \, {\left (a b c - a^{2} d\right )}} + \frac {d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{8 \, {\left (b c^{2} - a c d\right )}} - \frac {1}{a c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sq
rt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sq
rt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(1
/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a*b*c - a^2*
d) + 1/8*d^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x + sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sqr
t(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x - sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c
)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(d)*x^2 + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(
c^(1/4)*d^(3/4)) + sqrt(2)*log(sqrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(b*c^2 -
a*c*d) - 1/(a*c*x)

________________________________________________________________________________________

mupad [B]  time = 6.08, size = 5962, normalized size = 12.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(((-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^
(1/4)*(x*(4*a^11*b^9*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d
^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(3/4)*(x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*
d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(1024*a^12*b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^5 + 614
4*a^14*b^10*c^18*d^6 - 4096*a^15*b^9*c^17*d^7 + 2048*a^16*b^8*c^16*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*a^18*b^
6*c^14*d^10 - 4096*a^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12*d^12)*1i - 256*a^11*b^12*c^19*d^4 + 768*a^12*b^11*c
^18*d^5 - 768*a^13*b^10*c^17*d^6 + 256*a^14*b^9*c^16*d^7 + 256*a^16*b^7*c^14*d^9 - 768*a^17*b^6*c^13*d^10 + 76
8*a^18*b^5*c^12*d^11 - 256*a^19*b^4*c^11*d^12)*1i) + (-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3
+ 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(x*(4*a^11*b^9*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-d^5/(256
*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(3/4)*(x*(-d^5/(25
6*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(1024*a^12*
b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^5 + 6144*a^14*b^10*c^18*d^6 - 4096*a^15*b^9*c^17*d^7 + 2048*a^16*b^8*c^1
6*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*a^18*b^6*c^14*d^10 - 4096*a^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12*d^12)*
1i + 256*a^11*b^12*c^19*d^4 - 768*a^12*b^11*c^18*d^5 + 768*a^13*b^10*c^17*d^6 - 256*a^14*b^9*c^16*d^7 - 256*a^
16*b^7*c^14*d^9 + 768*a^17*b^6*c^13*d^10 - 768*a^18*b^5*c^12*d^11 + 256*a^19*b^4*c^11*d^12)*1i))/((-d^5/(256*b
^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(x*(4*a^11*b^9
*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^
7*d^2 - 1024*a*b^3*c^8*d))^(3/4)*(x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c
^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(1024*a^12*b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^5 + 6144*a^14*b^10*c^18*d^6
- 4096*a^15*b^9*c^17*d^7 + 2048*a^16*b^8*c^16*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*a^18*b^6*c^14*d^10 - 4096*a
^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12*d^12)*1i - 256*a^11*b^12*c^19*d^4 + 768*a^12*b^11*c^18*d^5 - 768*a^13*b
^10*c^17*d^6 + 256*a^14*b^9*c^16*d^7 + 256*a^16*b^7*c^14*d^9 - 768*a^17*b^6*c^13*d^10 + 768*a^18*b^5*c^12*d^11
- 256*a^19*b^4*c^11*d^12)*1i)*1i - (-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c
^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(x*(4*a^11*b^9*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-d^5/(256*b^4*c^9 + 256*a^
4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(3/4)*(x*(-d^5/(256*b^4*c^9 + 256*a
^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*(1024*a^12*b^12*c^20*d^4 - 4
096*a^13*b^11*c^19*d^5 + 6144*a^14*b^10*c^18*d^6 - 4096*a^15*b^9*c^17*d^7 + 2048*a^16*b^8*c^16*d^8 - 4096*a^17
*b^7*c^15*d^9 + 6144*a^18*b^6*c^14*d^10 - 4096*a^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12*d^12)*1i + 256*a^11*b^1
2*c^19*d^4 - 768*a^12*b^11*c^18*d^5 + 768*a^13*b^10*c^17*d^6 - 256*a^14*b^9*c^16*d^7 - 256*a^16*b^7*c^14*d^9 +
768*a^17*b^6*c^13*d^10 - 768*a^18*b^5*c^12*d^11 + 256*a^19*b^4*c^11*d^12)*1i)*1i))*(-d^5/(256*b^4*c^9 + 256*a
^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4) + atan((a^14*c*d^8*x*(-b^5/(
256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(5/4)*1024i + a
^6*b^8*c^9*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d
^3))^(5/4)*1024i + a^6*b^4*d^5*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*
d^2 - 1024*a^8*b*c*d^3))^(1/4)*4i + a^5*b^5*c*d^4*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d
+ 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*4i - a^7*b^7*c^8*d*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 -
1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(5/4)*4096i - a^13*b*c^2*d^7*x*(-b^5/(256*a^9*d
^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(5/4)*4096i + a^8*b^6*c^
7*d^2*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^
(5/4)*6144i - a^9*b^5*c^6*d^3*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d
^2 - 1024*a^8*b*c*d^3))^(5/4)*4096i + a^10*b^4*c^5*d^4*x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c
^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(5/4)*2048i - a^11*b^3*c^4*d^5*x*(-b^5/(256*a^9*d^4 + 256*a^5
*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(5/4)*4096i + a^12*b^2*c^3*d^6*x*(-b
^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(5/4)*6144i
)/(b^9*c^4 + a^4*b^5*d^4 + a^3*b^6*c*d^3 + a^2*b^7*c^2*d^2 + a*b^8*c^3*d))*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^
4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*2i + atan((b^5*c^6*d^4*x*(-d^5/(256*b
^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*4i + a*b^8*c^1
4*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4
)*1024i + a^9*c^6*d^8*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 102
4*a*b^3*c^8*d))^(5/4)*1024i + a*b^4*c^5*d^5*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536
*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*4i - a^2*b^7*c^13*d*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*
a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4)*4096i - a^8*b*c^7*d^7*x*(-d^5/(256*b^4*c^9 + 2
56*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4)*4096i + a^3*b^6*c^12*d^2
*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4)
*6144i - a^4*b^5*c^11*d^3*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 -
1024*a*b^3*c^8*d))^(5/4)*4096i + a^5*b^4*c^10*d^4*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3
+ 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4)*2048i - a^6*b^3*c^9*d^5*x*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*
d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4)*4096i + a^7*b^2*c^8*d^6*x*(-d^5/(25
6*b^4*c^9 + 256*a^4*c^5*d^4 - 1024*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(5/4)*6144i)/(a^4
*d^9 + b^4*c^4*d^5 + a*b^3*c^3*d^6 + a^2*b^2*c^2*d^7 + a^3*b*c*d^8))*(-d^5/(256*b^4*c^9 + 256*a^4*c^5*d^4 - 10
24*a^3*b*c^6*d^3 + 1536*a^2*b^2*c^7*d^2 - 1024*a*b^3*c^8*d))^(1/4)*2i + 2*atan(((-b^5/(256*a^9*d^4 + 256*a^5*b
^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*(x*(4*a^11*b^9*c^12*d^8 + 4*a^12
*b^8*c^11*d^9) - (-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b
*c*d^3))^(3/4)*(x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*
b*c*d^3))^(1/4)*(1024*a^12*b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^5 + 6144*a^14*b^10*c^18*d^6 - 4096*a^15*b^9*c
^17*d^7 + 2048*a^16*b^8*c^16*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*a^18*b^6*c^14*d^10 - 4096*a^19*b^5*c^13*d^11
+ 1024*a^20*b^4*c^12*d^12)*1i - 256*a^11*b^12*c^19*d^4 + 768*a^12*b^11*c^18*d^5 - 768*a^13*b^10*c^17*d^6 + 256
*a^14*b^9*c^16*d^7 + 256*a^16*b^7*c^14*d^9 - 768*a^17*b^6*c^13*d^10 + 768*a^18*b^5*c^12*d^11 - 256*a^19*b^4*c^
11*d^12)*1i) + (-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c
*d^3))^(1/4)*(x*(4*a^11*b^9*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*
b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(3/4)*(x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6
*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*(1024*a^12*b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^
5 + 6144*a^14*b^10*c^18*d^6 - 4096*a^15*b^9*c^17*d^7 + 2048*a^16*b^8*c^16*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*
a^18*b^6*c^14*d^10 - 4096*a^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12*d^12)*1i + 256*a^11*b^12*c^19*d^4 - 768*a^12
*b^11*c^18*d^5 + 768*a^13*b^10*c^17*d^6 - 256*a^14*b^9*c^16*d^7 - 256*a^16*b^7*c^14*d^9 + 768*a^17*b^6*c^13*d^
10 - 768*a^18*b^5*c^12*d^11 + 256*a^19*b^4*c^11*d^12)*1i))/((-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^
3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*(x*(4*a^11*b^9*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-b
^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(3/4)*(x*(-
b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*(102
4*a^12*b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^5 + 6144*a^14*b^10*c^18*d^6 - 4096*a^15*b^9*c^17*d^7 + 2048*a^16*
b^8*c^16*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*a^18*b^6*c^14*d^10 - 4096*a^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12
*d^12)*1i - 256*a^11*b^12*c^19*d^4 + 768*a^12*b^11*c^18*d^5 - 768*a^13*b^10*c^17*d^6 + 256*a^14*b^9*c^16*d^7 +
256*a^16*b^7*c^14*d^9 - 768*a^17*b^6*c^13*d^10 + 768*a^18*b^5*c^12*d^11 - 256*a^19*b^4*c^11*d^12)*1i)*1i - (-
b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*(x*(
4*a^11*b^9*c^12*d^8 + 4*a^12*b^8*c^11*d^9) - (-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536*
a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(3/4)*(x*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536
*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4)*(1024*a^12*b^12*c^20*d^4 - 4096*a^13*b^11*c^19*d^5 + 6144*a^14*b^1
0*c^18*d^6 - 4096*a^15*b^9*c^17*d^7 + 2048*a^16*b^8*c^16*d^8 - 4096*a^17*b^7*c^15*d^9 + 6144*a^18*b^6*c^14*d^1
0 - 4096*a^19*b^5*c^13*d^11 + 1024*a^20*b^4*c^12*d^12)*1i + 256*a^11*b^12*c^19*d^4 - 768*a^12*b^11*c^18*d^5 +
768*a^13*b^10*c^17*d^6 - 256*a^14*b^9*c^16*d^7 - 256*a^16*b^7*c^14*d^9 + 768*a^17*b^6*c^13*d^10 - 768*a^18*b^5
*c^12*d^11 + 256*a^19*b^4*c^11*d^12)*1i)*1i))*(-b^5/(256*a^9*d^4 + 256*a^5*b^4*c^4 - 1024*a^6*b^3*c^3*d + 1536
*a^7*b^2*c^2*d^2 - 1024*a^8*b*c*d^3))^(1/4) - 1/(a*c*x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________