3.488 \(\int \frac{x^3 (c+d x+e x^2+f x^3)}{a+b x^4} \, dx\)

Optimal. Leaf size=321 \[ \frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}+\frac{c \log \left (a+b x^4\right )}{4 b}+\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b} \]

[Out]

(d*x)/b + (e*x^2)/(2*b) + (f*x^3)/(3*b) - (Sqrt[a]*e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*b^(3/2)) + (a^(1/4)*(Sq
rt[b]*d + Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(7/4)) - (a^(1/4)*(Sqrt[b]*d + Sqrt
[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(7/4)) + (a^(1/4)*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqr
t[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*b^(7/4)) - (a^(1/4)*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*b^(7/4)) + (c*Log[a + b*x^4])/(4*b)

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Rubi [A]  time = 0.333813, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {1831, 1252, 774, 635, 205, 260, 1280, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}+\frac{c \log \left (a+b x^4\right )}{4 b}+\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4),x]

[Out]

(d*x)/b + (e*x^2)/(2*b) + (f*x^3)/(3*b) - (Sqrt[a]*e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*b^(3/2)) + (a^(1/4)*(Sq
rt[b]*d + Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(7/4)) - (a^(1/4)*(Sqrt[b]*d + Sqrt
[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(7/4)) + (a^(1/4)*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqr
t[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*b^(7/4)) - (a^(1/4)*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*b^(7/4)) + (c*Log[a + b*x^4])/(4*b)

Rule 1831

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[((c*x)^(m + ii)*(Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2)))/(c^ii*(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 774

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/c, x] + Dist[1
/c, Int[(c*d*f - a*e*g + c*(e*f + d*g)*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1280

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(e*f*(f*x)^(m - 1)*
(a + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*
(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] &
& IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1168

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

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{x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx &=\int \left (\frac{x^3 \left (c+e x^2\right )}{a+b x^4}+\frac{x^4 \left (d+f x^2\right )}{a+b x^4}\right ) \, dx\\ &=\int \frac{x^3 \left (c+e x^2\right )}{a+b x^4} \, dx+\int \frac{x^4 \left (d+f x^2\right )}{a+b x^4} \, dx\\ &=\frac{f x^3}{3 b}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (c+e x)}{a+b x^2} \, dx,x,x^2\right )-\frac{\int \frac{x^2 \left (3 a f-3 b d x^2\right )}{a+b x^4} \, dx}{3 b}\\ &=\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b}+\frac{\int \frac{-3 a b d-3 a b f x^2}{a+b x^4} \, dx}{3 b^2}+\frac{\operatorname{Subst}\left (\int \frac{-a e+b c x}{a+b x^2} \, dx,x,x^2\right )}{2 b}\\ &=\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{x}{a+b x^2} \, dx,x,x^2\right )-\frac{(a e) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 b}-\frac{\left (\sqrt{a} \left (\sqrt{b} d-\sqrt{a} f\right )\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{2 b^2}-\frac{\left (\sqrt{a} \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{2 b^2}\\ &=\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b}-\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}+\frac{c \log \left (a+b x^4\right )}{4 b}+\frac{\left (\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right )\right ) \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^{7/4}}+\frac{\left (\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right )\right ) \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^{7/4}}-\frac{\left (\sqrt{a} \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}-\frac{\left (\sqrt{a} \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}\\ &=\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b}-\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}+\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}+\frac{c \log \left (a+b x^4\right )}{4 b}-\frac{\left (\sqrt [4]{a} \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \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^{7/4}}+\frac{\left (\sqrt [4]{a} \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \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^{7/4}}\\ &=\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b}-\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}+\frac{\sqrt [4]{a} \left (\sqrt{b} d+\sqrt{a} f\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{b} d+\sqrt{a} f\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} b^{7/4}}+\frac{c \log \left (a+b x^4\right )}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.173454, size = 311, normalized size = 0.97 \[ \frac{-3 \sqrt{2} \left (a^{3/4} f-\sqrt [4]{a} \sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+3 \sqrt{2} \left (a^{3/4} f-\sqrt [4]{a} \sqrt{b} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+6 b^{3/4} c \log \left (a+b x^4\right )+6 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (2 \sqrt [4]{a} \sqrt [4]{b} e+\sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )-6 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-2 \sqrt [4]{a} \sqrt [4]{b} e+\sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )+24 b^{3/4} d x+12 b^{3/4} e x^2+8 b^{3/4} f x^3}{24 b^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4),x]

[Out]

(24*b^(3/4)*d*x + 12*b^(3/4)*e*x^2 + 8*b^(3/4)*f*x^3 + 6*a^(1/4)*(Sqrt[2]*Sqrt[b]*d + 2*a^(1/4)*b^(1/4)*e + Sq
rt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 6*a^(1/4)*(Sqrt[2]*Sqrt[b]*d - 2*a^(1/4)*b^(1/4)*e
+ Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*(-(a^(1/4)*Sqrt[b]*d) + a^(3/4)*f)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*(-(a^(1/4)*Sqrt[b]*d) + a^(3/4)*f)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 6*b^(3/4)*c*Log[a + b*x^4])/(24*b^(7/4))

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Maple [A]  time = 0.004, size = 325, normalized size = 1. \begin{align*}{\frac{f{x}^{3}}{3\,b}}+{\frac{e{x}^{2}}{2\,b}}+{\frac{dx}{b}}-{\frac{d\sqrt{2}}{4\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{d\sqrt{2}}{8\,b}\sqrt [4]{{\frac{a}{b}}}\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{d\sqrt{2}}{4\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{ae}{2\,b}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{af\sqrt{2}}{8\,{b}^{2}}\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{af\sqrt{2}}{4\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{af\sqrt{2}}{4\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{c\ln \left ( b{x}^{4}+a \right ) }{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a),x)

[Out]

1/3*f*x^3/b+1/2*e*x^2/b+d*x/b-1/4/b*d*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)-1/8/b*d*(1/b*a)^
(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))-1/4/
b*d*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)-1/2/b*a*e/(a*b)^(1/2)*arctan(x^2*(b/a)^(1/2))-1/8/
b^2*a*f/(1/b*a)^(1/4)*2^(1/2)*ln((x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b
*a)^(1/2)))-1/4/b^2*a*f/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)-1/4/b^2*a*f/(1/b*a)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)+1/4*c*ln(b*x^4+a)/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(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 12.1739, size = 886, normalized size = 2.76 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} b^{7} - 256 t^{3} b^{6} c + t^{2} \left (64 a b^{4} d f + 32 a b^{4} e^{2} + 96 b^{5} c^{2}\right ) + t \left (- 16 a^{2} b^{2} e f^{2} - 32 a b^{3} c d f - 16 a b^{3} c e^{2} + 16 a b^{3} d^{2} e - 16 b^{4} c^{3}\right ) + a^{3} f^{4} + 4 a^{2} b c e f^{2} + 2 a^{2} b d^{2} f^{2} - 4 a^{2} b d e^{2} f + a^{2} b e^{4} + 4 a b^{2} c^{2} d f + 2 a b^{2} c^{2} e^{2} - 4 a b^{2} c d^{2} e + a b^{2} d^{4} + b^{3} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a b^{5} f^{3} + 64 t^{3} b^{6} d^{2} f - 128 t^{3} b^{6} d e^{2} + 48 t^{2} a b^{4} c f^{3} + 48 t^{2} a b^{4} d e f^{2} - 32 t^{2} a b^{4} e^{3} f - 48 t^{2} b^{5} c d^{2} f + 96 t^{2} b^{5} c d e^{2} + 16 t^{2} b^{5} d^{3} e - 12 t a^{2} b^{2} d f^{4} - 12 t a^{2} b^{2} e^{2} f^{3} - 12 t a b^{3} c^{2} f^{3} - 24 t a b^{3} c d e f^{2} + 16 t a b^{3} c e^{3} f + 16 t a b^{3} d^{3} f^{2} - 36 t a b^{3} d^{2} e^{2} f - 8 t a b^{3} d e^{4} + 12 t b^{4} c^{2} d^{2} f - 24 t b^{4} c^{2} d e^{2} - 8 t b^{4} c d^{3} e - 4 t b^{4} d^{5} + 3 a^{3} e f^{5} + 3 a^{2} b c d f^{4} + 3 a^{2} b c e^{2} f^{3} + 5 a^{2} b d e^{3} f^{2} - 2 a^{2} b e^{5} f + a b^{2} c^{3} f^{3} + 3 a b^{2} c^{2} d e f^{2} - 2 a b^{2} c^{2} e^{3} f - 4 a b^{2} c d^{3} f^{2} + 9 a b^{2} c d^{2} e^{2} f + 2 a b^{2} c d e^{4} + 5 a b^{2} d^{4} e f - 5 a b^{2} d^{3} e^{3} - b^{3} c^{3} d^{2} f + 2 b^{3} c^{3} d e^{2} + b^{3} c^{2} d^{3} e + b^{3} c d^{5}}{a^{3} f^{6} - a^{2} b d^{2} f^{4} + 8 a^{2} b d e^{2} f^{3} - 4 a^{2} b e^{4} f^{2} - a b^{2} d^{4} f^{2} + 8 a b^{2} d^{3} e^{2} f - 4 a b^{2} d^{2} e^{4} + b^{3} d^{6}} \right )} \right )\right )} + \frac{d x}{b} + \frac{e x^{2}}{2 b} + \frac{f x^{3}}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(f*x**3+e*x**2+d*x+c)/(b*x**4+a),x)

[Out]

RootSum(256*_t**4*b**7 - 256*_t**3*b**6*c + _t**2*(64*a*b**4*d*f + 32*a*b**4*e**2 + 96*b**5*c**2) + _t*(-16*a*
*2*b**2*e*f**2 - 32*a*b**3*c*d*f - 16*a*b**3*c*e**2 + 16*a*b**3*d**2*e - 16*b**4*c**3) + a**3*f**4 + 4*a**2*b*
c*e*f**2 + 2*a**2*b*d**2*f**2 - 4*a**2*b*d*e**2*f + a**2*b*e**4 + 4*a*b**2*c**2*d*f + 2*a*b**2*c**2*e**2 - 4*a
*b**2*c*d**2*e + a*b**2*d**4 + b**3*c**4, Lambda(_t, _t*log(x + (-64*_t**3*a*b**5*f**3 + 64*_t**3*b**6*d**2*f
- 128*_t**3*b**6*d*e**2 + 48*_t**2*a*b**4*c*f**3 + 48*_t**2*a*b**4*d*e*f**2 - 32*_t**2*a*b**4*e**3*f - 48*_t**
2*b**5*c*d**2*f + 96*_t**2*b**5*c*d*e**2 + 16*_t**2*b**5*d**3*e - 12*_t*a**2*b**2*d*f**4 - 12*_t*a**2*b**2*e**
2*f**3 - 12*_t*a*b**3*c**2*f**3 - 24*_t*a*b**3*c*d*e*f**2 + 16*_t*a*b**3*c*e**3*f + 16*_t*a*b**3*d**3*f**2 - 3
6*_t*a*b**3*d**2*e**2*f - 8*_t*a*b**3*d*e**4 + 12*_t*b**4*c**2*d**2*f - 24*_t*b**4*c**2*d*e**2 - 8*_t*b**4*c*d
**3*e - 4*_t*b**4*d**5 + 3*a**3*e*f**5 + 3*a**2*b*c*d*f**4 + 3*a**2*b*c*e**2*f**3 + 5*a**2*b*d*e**3*f**2 - 2*a
**2*b*e**5*f + a*b**2*c**3*f**3 + 3*a*b**2*c**2*d*e*f**2 - 2*a*b**2*c**2*e**3*f - 4*a*b**2*c*d**3*f**2 + 9*a*b
**2*c*d**2*e**2*f + 2*a*b**2*c*d*e**4 + 5*a*b**2*d**4*e*f - 5*a*b**2*d**3*e**3 - b**3*c**3*d**2*f + 2*b**3*c**
3*d*e**2 + b**3*c**2*d**3*e + b**3*c*d**5)/(a**3*f**6 - a**2*b*d**2*f**4 + 8*a**2*b*d*e**2*f**3 - 4*a**2*b*e**
4*f**2 - a*b**2*d**4*f**2 + 8*a*b**2*d**3*e**2*f - 4*a*b**2*d**2*e**4 + b**3*d**6)))) + d*x/b + e*x**2/(2*b) +
f*x**3/(3*b)

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Giac [A]  time = 1.10117, size = 416, normalized size = 1.3 \begin{align*} \frac{c \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a b} b^{2} e - \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \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 )}{4 \, b^{4}} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a b} b^{2} e - \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \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 )}{4 \, b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, b^{4}} + \frac{2 \, b^{2} f x^{3} + 3 \, b^{2} x^{2} e + 6 \, b^{2} d x}{6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a),x, algorithm="giac")

[Out]

1/4*c*log(abs(b*x^4 + a))/b + 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^2*e - (a*b^3)^(1/4)*b^2*d - (a*b^3)^(3/4)*f)*ar
ctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/b^4 + 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^2*e - (a*b^3)
^(1/4)*b^2*d - (a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/b^4 - 1/8*sqrt(2)*
((a*b^3)^(1/4)*b^2*d - (a*b^3)^(3/4)*f)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/b^4 + 1/8*sqrt(2)*((a*b^3
)^(1/4)*b^2*d - (a*b^3)^(3/4)*f)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/b^4 + 1/6*(2*b^2*f*x^3 + 3*b^2*x
^2*e + 6*b^2*d*x)/b^3