\(\int \frac {x^3 (c+d x+e x^2+f x^3)}{a+b x^4} \, dx\) [488]

Optimal result

Integrand size = 28, antiderivative size = 321 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}-\frac {\sqrt {a} e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{7/4}}+\frac {c \log \left (a+b x^4\right )}{4 b} \]

[Out]

d*x/b+1/2*e*x^2/b+1/3*f*x^3/b+1/4*c*ln(b*x^4+a)/b-1/2*e*arctan(x^2*b^(1/2)/a^(1/2))*a^(1/2)/b^(3/2)+1/8*a^(1/4
)*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-f*a^(1/2)+d*b^(1/2))/b^(7/4)*2^(1/2)-1/8*a^(1/4)*ln(a^(
1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-f*a^(1/2)+d*b^(1/2))/b^(7/4)*2^(1/2)-1/4*a^(1/4)*arctan(-1+b^(1/
4)*x*2^(1/2)/a^(1/4))*(f*a^(1/2)+d*b^(1/2))/b^(7/4)*2^(1/2)-1/4*a^(1/4)*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(f
*a^(1/2)+d*b^(1/2))/b^(7/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {1845, 1266, 788, 649, 211, 266, 1294, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\frac {\sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt {a} e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{7/4}}+\frac {c \log \left (a+b x^4\right )}{4 b}+\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b} \]

[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 210

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

Rule 211

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

Rule 266

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 631

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 642

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 649

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 788

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 1176

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 1179

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 1182

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 1266

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 1294

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 1845

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

Rubi steps \begin{align*} \text {integral}& = \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} \text {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 {\text {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 \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )-\frac {(a e) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.18 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\frac {24 b^{3/4} d x+12 b^{3/4} e x^2+8 b^{3/4} f x^3+6 \sqrt [4]{a} \left (\sqrt {2} \sqrt {b} d+2 \sqrt [4]{a} \sqrt [4]{b} e+\sqrt {2} \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-6 \sqrt [4]{a} \left (\sqrt {2} \sqrt {b} d-2 \sqrt [4]{a} \sqrt [4]{b} e+\sqrt {2} \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-3 \sqrt {2} \left (-\sqrt [4]{a} \sqrt {b} d+a^{3/4} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+3 \sqrt {2} \left (-\sqrt [4]{a} \sqrt {b} d+a^{3/4} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+6 b^{3/4} c \log \left (a+b x^4\right )}{24 b^{7/4}} \]

[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))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.55 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {f \,x^{3}}{3 b}+\frac {e \,x^{2}}{2 b}+\frac {d x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} b c -\textit {\_R}^{2} a f -\textit {\_R} a e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\)	\(75\)
default	\(\frac {\frac {1}{3} f \,x^{3}+\frac {1}{2} e \,x^{2}+d x}{b}+\frac {-\frac {d \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8}-\frac {a e \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}-\frac {a f \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {c \ln \left (b \,x^{4}+a \right )}{4}}{b}\)	\(261\)

[In]

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

[Out]

1/3*f*x^3/b+1/2*e*x^2/b+d*x/b+1/4/b^2*sum((_R^3*b*c-_R^2*a*f-_R*a*e-a*d)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.10 (sec) , antiderivative size = 219615, normalized size of antiderivative = 684.16 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\frac {2 \, f x^{3} + 3 \, e x^{2} + 6 \, d x}{6 \, b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c - a b d + a^{\frac {3}{2}} \sqrt {b} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c + a b d - a^{\frac {3}{2}} \sqrt {b} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} d + \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} f - 2 \, a^{\frac {3}{2}} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} d + \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} f + 2 \, a^{\frac {3}{2}} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \]

[In]

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

[Out]

1/6*(2*f*x^3 + 3*e*x^2 + 6*d*x)/b + 1/8*(sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*c - a*b*d + a^(3/2)*sqrt(b)*f)*log(s
qrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4)) + sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*c + a*b
*d - a^(3/2)*sqrt(b)*f)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4)) - 2*(sqrt(2)*
a^(5/4)*b^(5/4)*d + sqrt(2)*a^(7/4)*b^(3/4)*f - 2*a^(3/2)*b*e)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/
4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(5/4)) - 2*(sqrt(2)*a^(5/4)*b^(5/4)*d + sq
rt(2)*a^(7/4)*b^(3/4)*f + 2*a^(3/2)*b*e)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(
a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(5/4)))/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\frac {c \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} e - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, b^{4}} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} e - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, b^{4}} + \frac {2 \, b^{2} f x^{3} + 3 \, b^{2} e x^{2} + 6 \, b^{2} d x}{6 \, b^{3}} \]

[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*e
*x^2 + 6*b^2*d*x)/b^3

Mupad [B] (verification not implemented)

Time = 8.99 (sec) , antiderivative size = 838, normalized size of antiderivative = 2.61 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx=\left (\sum _{k=1}^4\ln \left (\frac {a^4\,f^3+2\,a^3\,b\,c\,e\,f+a^3\,b\,d^2\,f-a^3\,b\,d\,e^2+a^2\,b^2\,c^2\,d}{b^2}+\mathrm {root}\left (256\,b^7\,z^4-256\,b^6\,c\,z^3+64\,a\,b^4\,d\,f\,z^2+32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z-16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z-16\,b^4\,c^3\,z-4\,a^2\,b\,d\,e^2\,f+4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e+2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a^2\,b\,e^4+a\,b^2\,d^4+a^3\,f^4+b^3\,c^4,z,k\right )\,\left (\mathrm {root}\left (256\,b^7\,z^4-256\,b^6\,c\,z^3+64\,a\,b^4\,d\,f\,z^2+32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z-16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z-16\,b^4\,c^3\,z-4\,a^2\,b\,d\,e^2\,f+4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e+2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a^2\,b\,e^4+a\,b^2\,d^4+a^3\,f^4+b^3\,c^4,z,k\right )\,\left (16\,a^2\,b^2\,d-16\,a^2\,b^2\,e\,x\right )-\frac {8\,e\,f\,a^3\,b^2+8\,c\,d\,a^2\,b^3}{b^2}+\frac {x\,\left (4\,a^3\,b\,f^2-4\,a^2\,b^2\,d^2+8\,c\,e\,a^2\,b^2\right )}{b}\right )-\frac {x\,\left (a^3\,c\,f^2-2\,a^3\,d\,e\,f+a^3\,e^3+b\,a^2\,c^2\,e-b\,a^2\,c\,d^2\right )}{b}\right )\,\mathrm {root}\left (256\,b^7\,z^4-256\,b^6\,c\,z^3+64\,a\,b^4\,d\,f\,z^2+32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z-16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z-16\,b^4\,c^3\,z-4\,a^2\,b\,d\,e^2\,f+4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e+2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a^2\,b\,e^4+a\,b^2\,d^4+a^3\,f^4+b^3\,c^4,z,k\right )\right )+\frac {e\,x^2}{2\,b}+\frac {f\,x^3}{3\,b}+\frac {d\,x}{b} \]

[In]

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

[Out]

symsum(log((a^4*f^3 + a^2*b^2*c^2*d - a^3*b*d*e^2 + a^3*b*d^2*f + 2*a^3*b*c*e*f)/b^2 + root(256*b^7*z^4 - 256*
b^6*c*z^3 + 64*a*b^4*d*f*z^2 + 32*a*b^4*e^2*z^2 + 96*b^5*c^2*z^2 - 32*a*b^3*c*d*f*z - 16*a^2*b^2*e*f^2*z + 16*
a*b^3*d^2*e*z - 16*a*b^3*c*e^2*z - 16*b^4*c^3*z - 4*a^2*b*d*e^2*f + 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^
2*c*d^2*e + 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a^2*b*e^4 + a*b^2*d^4 + a^3*f^4 + b^3*c^4, z, k)*(root(256*b^7
*z^4 - 256*b^6*c*z^3 + 64*a*b^4*d*f*z^2 + 32*a*b^4*e^2*z^2 + 96*b^5*c^2*z^2 - 32*a*b^3*c*d*f*z - 16*a^2*b^2*e*
f^2*z + 16*a*b^3*d^2*e*z - 16*a*b^3*c*e^2*z - 16*b^4*c^3*z - 4*a^2*b*d*e^2*f + 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d
*f - 4*a*b^2*c*d^2*e + 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a^2*b*e^4 + a*b^2*d^4 + a^3*f^4 + b^3*c^4, z, k)*(1
6*a^2*b^2*d - 16*a^2*b^2*e*x) - (8*a^2*b^3*c*d + 8*a^3*b^2*e*f)/b^2 + (x*(4*a^3*b*f^2 - 4*a^2*b^2*d^2 + 8*a^2*
b^2*c*e))/b) - (x*(a^3*e^3 + a^3*c*f^2 - 2*a^3*d*e*f - a^2*b*c*d^2 + a^2*b*c^2*e))/b)*root(256*b^7*z^4 - 256*b
^6*c*z^3 + 64*a*b^4*d*f*z^2 + 32*a*b^4*e^2*z^2 + 96*b^5*c^2*z^2 - 32*a*b^3*c*d*f*z - 16*a^2*b^2*e*f^2*z + 16*a
*b^3*d^2*e*z - 16*a*b^3*c*e^2*z - 16*b^4*c^3*z - 4*a^2*b*d*e^2*f + 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2
*c*d^2*e + 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a^2*b*e^4 + a*b^2*d^4 + a^3*f^4 + b^3*c^4, z, k), k, 1, 4) + (e
*x^2)/(2*b) + (f*x^3)/(3*b) + (d*x)/b