3.6.5 \(\int \frac {x^8}{(a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=457 \[ -\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}+\frac {c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}+\frac {x}{b d} \]

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Rubi [A]  time = 0.44, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {479, 522, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}+\frac {c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}+\frac {x}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/((a + b*x^4)*(c + d*x^4)),x]

[Out]

x/(b*d) - (a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(5/4)*(b*c - a*d)) + (a^(5/4)*ArcTan[
1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(5/4)*(b*c - a*d)) + (c^(5/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^
(1/4)])/(2*Sqrt[2]*d^(5/4)*(b*c - a*d)) - (c^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*d^(5/4)
*(b*c - a*d)) - (a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*b^(5/4)*(b*c - a*d
)) + (a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*b^(5/4)*(b*c - a*d)) + (c^(5/
4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*d^(5/4)*(b*c - a*d)) - (c^(5/4)*Log[Sqrt
[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*d^(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 211

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

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(2*n
- 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q) + 1)), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[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 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 {x^8}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac {x}{b d}-\frac {\int \frac {a c+(b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{b d}\\ &=\frac {x}{b d}+\frac {a^2 \int \frac {1}{a+b x^4} \, dx}{b (b c-a d)}-\frac {c^2 \int \frac {1}{c+d x^4} \, dx}{d (b c-a d)}\\ &=\frac {x}{b d}+\frac {a^{3/2} \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{2 b (b c-a d)}+\frac {a^{3/2} \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{2 b (b c-a d)}-\frac {c^{3/2} \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{2 d (b c-a d)}-\frac {c^{3/2} \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{2 d (b c-a d)}\\ &=\frac {x}{b d}+\frac {a^{3/2} \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^{3/2} (b c-a d)}+\frac {a^{3/2} \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^{3/2} (b c-a d)}-\frac {a^{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} b^{5/4} (b c-a d)}-\frac {a^{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} b^{5/4} (b c-a d)}-\frac {c^{3/2} \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 d^{3/2} (b c-a d)}-\frac {c^{3/2} \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 d^{3/2} (b c-a d)}+\frac {c^{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} d^{5/4} (b c-a d)}+\frac {c^{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} d^{5/4} (b c-a d)}\\ &=\frac {x}{b d}-\frac {a^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}+\frac {a^{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} b^{5/4} (b c-a d)}-\frac {a^{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} b^{5/4} (b c-a d)}-\frac {c^{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} d^{5/4} (b c-a d)}+\frac {c^{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} d^{5/4} (b c-a d)}\\ &=\frac {x}{b d}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {a^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} d^{5/4} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 377, normalized size = 0.82 \begin {gather*} \frac {-\frac {\sqrt {2} a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{b^{5/4}}+\frac {\sqrt {2} a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{b^{5/4}}-\frac {2 \sqrt {2} a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{5/4}}+\frac {2 \sqrt {2} a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{b^{5/4}}-\frac {8 a x}{b}+\frac {\sqrt {2} c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{d^{5/4}}-\frac {\sqrt {2} c^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{d^{5/4}}+\frac {2 \sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{d^{5/4}}-\frac {2 \sqrt {2} c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{d^{5/4}}+\frac {8 c x}{d}}{8 b c-8 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/((a + b*x^4)*(c + d*x^4)),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^8/((a + b*x^4)*(c + d*x^4)),x]

[Out]

IntegrateAlgebraic[x^8/((a + b*x^4)*(c + d*x^4)), x]

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fricas [B]  time = 0.72, size = 1378, normalized size = 3.02

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 469, normalized size = 1.03 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \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} b^{3} c - \sqrt {2} a b^{2} d\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \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} b^{3} c - \sqrt {2} a b^{2} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \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 d^{2} - \sqrt {2} a d^{3}\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \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 d^{2} - \sqrt {2} a d^{3}\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} b^{3} c - \sqrt {2} a b^{2} d\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} b^{3} c - \sqrt {2} a b^{2} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \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 d^{2} - \sqrt {2} a d^{3}\right )}} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \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 d^{2} - \sqrt {2} a d^{3}\right )}} + \frac {x}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 328, normalized size = 0.72 \begin {gather*} -\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (a d -b c \right ) b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (a d -b c \right ) b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \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 ) b}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{4 \left (a d -b c \right ) d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{4 \left (a d -b c \right ) d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, c \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 ) d}+\frac {x}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^4+a)/(d*x^4+c),x)

[Out]

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

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maxima [A]  time = 1.22, size = 375, normalized size = 0.82 \begin {gather*} \frac {\frac {2 \, \sqrt {2} a^{\frac {3}{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}}} + \frac {2 \, \sqrt {2} a^{\frac {3}{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}}} + \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}}{8 \, {\left (b^{2} c - a b d\right )}} - \frac {\frac {2 \, \sqrt {2} c^{\frac {3}{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}}} + \frac {2 \, \sqrt {2} c^{\frac {3}{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}}} + \frac {\sqrt {2} c^{\frac {5}{4}} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{d^{\frac {1}{4}}} - \frac {\sqrt {2} c^{\frac {5}{4}} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{d^{\frac {1}{4}}}}{8 \, {\left (b c d - a d^{2}\right )}} + \frac {x}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")

[Out]

1/8*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/sqrt(
sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(3/2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*s
qrt(b)))/sqrt(sqrt(a)*sqrt(b)) + sqrt(2)*a^(5/4)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/b^(1/4
) - sqrt(2)*a^(5/4)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/b^(1/4))/(b^2*c - a*b*d) - 1/8*(2*s
qrt(2)*c^(3/2)*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)) + 2*sqrt(2)*c^(3/2)*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(2)*c^(5/4)*log(sqrt(d)*x^2 + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/d^(1/4) - sqrt
(2)*c^(5/4)*log(sqrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/d^(1/4))/(b*c*d - a*d^2) + x/(b*d)

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mupad [B]  time = 5.63, size = 6361, normalized size = 13.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/((a + b*x^4)*(c + d*x^4)),x)

[Out]

atan(((-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1
/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (4*x*(-a^5/(256*b^9*c^4 + 256*a
^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a
^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9))/(b*d)
)*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)
- (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*1i - (-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 15
36*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5
))/(b*d) + (4*x*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*
c^3*d))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7
*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9))/(b*d))*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a
^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4) + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*1i)/((-a^5/(256*b^9*c^4 +
256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*(((16*(a^3*b^6*c^9 + a
^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (4*x*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*
c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b
^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9))/(b*d))*(-a^5/(256*b^9*c^4 + 256
*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4) - (4*x*(a^4*b^4*c^8 + a^8*
c^4*d^4))/(b*d)) + (-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b
^8*c^3*d))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (4*x*(-a^5/(256*b^
9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(3/4)*(256*a^3*b^9*c^
8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^
3*d^9))/(b*d))*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c
^3*d))^(1/4) + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))))*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*
c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*2i - 2*atan(((-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 10
24*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*
c^8*d - a^8*b*c^4*d^5))/(b*d) - (x*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^
2*d^2 - 1024*a*b^8*c^3*d))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^
6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b*d))*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*
a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*1i + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))
- (-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*
(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (x*(-a^5/(256*b^9*c^4 + 256*a^4*b^5
*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8
*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b*d))*(
-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*1i
- (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d)))/((-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*
a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/
(b*d) - (x*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d
))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*
c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b*d))*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2
*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*1i + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*1i + (-a^5/(256*b^9*c^4
+ 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*(((16*(a^3*b^6*c^9 +
a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (x*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c
*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^
7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b*d))*(-a^5/(256*b^9*c^4 + 2
56*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 - 1024*a*b^8*c^3*d))^(1/4)*1i - (4*x*(a^4*b^4*c^8 +
a^8*c^4*d^4))/(b*d))*1i))*(-a^5/(256*b^9*c^4 + 256*a^4*b^5*d^4 - 1024*a^3*b^6*c*d^3 + 1536*a^2*b^7*c^2*d^2 -
1024*a*b^8*c^3*d))^(1/4) + atan(((-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*
d^7 - 1024*a^3*b*c*d^8))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (4*x
*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(
256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 +
256*a^8*b^4*c^3*d^9))/(b*d))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7
- 1024*a^3*b*c*d^8))^(1/4) - (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*1i - (-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^
5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^
4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (4*x*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^
2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 51
2*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9))/(b*d))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 -
1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4) + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d)
)*1i)/((-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(
1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (4*x*(-c^5/(256*a^4*d^9 + 256*
b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*
a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9))/(b*d
))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)
- (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d)) + (-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536
*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))
/(b*d) + (4*x*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*
d^8))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b
^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9))/(b*d))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2
*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4) + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))))*(-c^5/(256*a^4*d^9 + 256*
b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*2i - 2*atan(((-c^5/(256*a^4
*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*(((16*(a^3*b^6*c
^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (x*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^
3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a
^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b*d))*(-c^5/(256*a^4*d^
9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*1i + (4*x*(a^4*b^4*
c^8 + a^8*c^4*d^4))/(b*d)) - (-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7
- 1024*a^3*b*c*d^8))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (x*(-c^5
/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(256*a^
3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^
8*b^4*c^3*d^9)*4i)/(b*d))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1
024*a^3*b*c*d^8))^(1/4)*1i - (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d)))/((-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 -
1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b
^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (x*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2
*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6
*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b*d))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 10
24*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4)*1i + (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d
))*1i + (-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^
(1/4)*(((16*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (x*(-c^5/(256*a^4*d^9 + 256*b
^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(3/4)*(256*a^3*b^9*c^8*d^4 - 768*a
^4*b^8*c^7*d^5 + 512*a^5*b^7*c^6*d^6 + 512*a^6*b^6*c^5*d^7 - 768*a^7*b^5*c^4*d^8 + 256*a^8*b^4*c^3*d^9)*4i)/(b
*d))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/
4)*1i - (4*x*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*1i))*(-c^5/(256*a^4*d^9 + 256*b^4*c^4*d^5 - 1024*a*b^3*c^3*d^
6 + 1536*a^2*b^2*c^2*d^7 - 1024*a^3*b*c*d^8))^(1/4) + x/(b*d)

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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(x**8/(b*x**4+a)/(d*x**4+c),x)

[Out]

Timed out

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