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

Optimal. Leaf size=310 \[ -\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )} \]

[Out]

-(c + d*x + e*x^2 + f*x^3)/(4*b*(a + b*x^4)) + (e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*Sqrt[a]*b^(3/2)) - ((Sqrt[
b]*d + 3*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(3/4)*b^(7/4)) + ((Sqrt[b]*d + 3*Sqr
t[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(3/4)*b^(7/4)) - ((Sqrt[b]*d - 3*Sqrt[a]*f)*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(3/4)*b^(7/4)) + ((Sqrt[b]*d - 3*Sqrt[a]*f)*
Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(3/4)*b^(7/4))

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Rubi [A]  time = 0.274175, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1823, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c + d*x + e*x^2 + f*x^3)/(4*b*(a + b*x^4)) + (e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*Sqrt[a]*b^(3/2)) - ((Sqrt[
b]*d + 3*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(3/4)*b^(7/4)) + ((Sqrt[b]*d + 3*Sqr
t[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(3/4)*b^(7/4)) - ((Sqrt[b]*d - 3*Sqrt[a]*f)*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(3/4)*b^(7/4)) + ((Sqrt[b]*d - 3*Sqrt[a]*f)*
Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(3/4)*b^(7/4))

Rule 1823

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Pq*(a + b*x^n)^(p + 1))/(b*n*(p + 1)),
x] - Dist[1/(b*n*(p + 1)), Int[D[Pq, x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x]
&& EqQ[m - n + 1, 0] && LtQ[p, -1]

Rule 1876

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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 )}{\left (a+b x^4\right )^2} \, dx &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{\int \frac{d+2 e x+3 f x^2}{a+b x^4} \, dx}{4 b}\\ &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{\int \left (\frac{2 e x}{a+b x^4}+\frac{d+3 f x^2}{a+b x^4}\right ) \, dx}{4 b}\\ &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{\int \frac{d+3 f x^2}{a+b x^4} \, dx}{4 b}+\frac{e \int \frac{x}{a+b x^4} \, dx}{2 b}\\ &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{e \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{4 b}+\frac{\left (\frac{\sqrt{b} d}{\sqrt{a}}-3 f\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{8 b^2}+\frac{\left (\frac{\sqrt{b} d}{\sqrt{a}}+3 f\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{8 b^2}\\ &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}+\frac{\left (\frac{\sqrt{b} d}{\sqrt{a}}+3 f\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 b^2}+\frac{\left (\frac{\sqrt{b} d}{\sqrt{a}}+3 f\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 b^2}-\frac{\left (\sqrt{b} d-3 \sqrt{a} f\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}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (\sqrt{b} d-3 \sqrt{a} f\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}{16 \sqrt{2} a^{3/4} b^{7/4}}\\ &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d+3 \sqrt{a} f\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (\sqrt{b} d+3 \sqrt{a} f\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}\\ &=-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{\left (\sqrt{b} d+3 \sqrt{a} f\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d+3 \sqrt{a} f\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}\\ \end{align*}

Mathematica [A]  time = 0.269116, size = 294, normalized size = 0.95 \[ \frac{-\frac{2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )}{a^{3/4}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )}{a^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt{a} f-\sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}+\frac{\sqrt{2} \left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}-\frac{8 b^{3/4} (c+x (d+x (e+f x)))}{a+b x^4}}{32 b^{7/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4) - (2*(Sqrt[2]*Sqrt[b]*d + 4*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*
Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (2*(Sqrt[2]*Sqrt[b]*d - 4*a^(1/4)*b^(1/4)*e + 3*
Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (Sqrt[2]*(-(Sqrt[b]*d) + 3*Sqrt[a]*f)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (Sqrt[2]*(Sqrt[b]*d - 3*Sqrt[a]*f)*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4))/(32*b^(7/4))

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Maple [A]  time = 0.009, size = 334, normalized size = 1.1 \begin{align*}{\frac{1}{b{x}^{4}+a} \left ( -{\frac{f{x}^{3}}{4\,b}}-{\frac{e{x}^{2}}{4\,b}}-{\frac{dx}{4\,b}}-{\frac{c}{4\,b}} \right ) }+{\frac{d\sqrt{2}}{32\,ab}\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}}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{4\,b}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,f\sqrt{2}}{32\,{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{3\,f\sqrt{2}}{16\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{16\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{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)^2,x)

[Out]

(-1/4*f*x^3/b-1/4*e*x^2/b-1/4*d*x/b-1/4*c/b)/(b*x^4+a)+1/32/b*d*(1/b*a)^(1/4)/a*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)))+1/16/b*d*(1/b*a)^(1/4)/a*2^(1/2)*arctan(
2^(1/2)/(1/b*a)^(1/4)*x+1)+1/16/b*d*(1/b*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)+1/4/b*e/(a*b)^(1
/2)*arctan(x^2*(b/a)^(1/2))+3/32/b^2*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)))+3/16/b^2*f/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+
3/16/b^2*f/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)

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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)^2,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)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 31.1828, size = 508, normalized size = 1.64 \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{3} b^{7} + t^{2} \left (3072 a^{2} b^{4} d f + 2048 a^{2} b^{4} e^{2}\right ) + t \left (1152 a^{2} b^{2} e f^{2} - 128 a b^{3} d^{2} e\right ) + 81 a^{2} f^{4} + 18 a b d^{2} f^{2} - 48 a b d e^{2} f + 16 a b e^{4} + b^{2} d^{4}, \left ( t \mapsto t \log{\left (x + \frac{110592 t^{3} a^{4} b^{5} f^{3} - 12288 t^{3} a^{3} b^{6} d^{2} f + 32768 t^{3} a^{3} b^{6} d e^{2} + 13824 t^{2} a^{3} b^{4} d e f^{2} - 12288 t^{2} a^{3} b^{4} e^{3} f + 512 t^{2} a^{2} b^{5} d^{3} e + 3888 t a^{3} b^{2} d f^{4} + 5184 t a^{3} b^{2} e^{2} f^{3} - 576 t a^{2} b^{3} d^{3} f^{2} + 1728 t a^{2} b^{3} d^{2} e^{2} f + 512 t a^{2} b^{3} d e^{4} + 16 t a b^{4} d^{5} + 1458 a^{3} e f^{5} + 360 a^{2} b d e^{3} f^{2} - 192 a^{2} b e^{5} f + 30 a b^{2} d^{4} e f - 40 a b^{2} d^{3} e^{3}}{729 a^{3} f^{6} - 81 a^{2} b d^{2} f^{4} + 864 a^{2} b d e^{2} f^{3} - 576 a^{2} b e^{4} f^{2} - 9 a b^{2} d^{4} f^{2} + 96 a b^{2} d^{3} e^{2} f - 64 a b^{2} d^{2} e^{4} + b^{3} d^{6}} \right )} \right )\right )} - \frac{c + d x + e x^{2} + f x^{3}}{4 a b + 4 b^{2} x^{4}} \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)**2,x)

[Out]

RootSum(65536*_t**4*a**3*b**7 + _t**2*(3072*a**2*b**4*d*f + 2048*a**2*b**4*e**2) + _t*(1152*a**2*b**2*e*f**2 -
128*a*b**3*d**2*e) + 81*a**2*f**4 + 18*a*b*d**2*f**2 - 48*a*b*d*e**2*f + 16*a*b*e**4 + b**2*d**4, Lambda(_t,
_t*log(x + (110592*_t**3*a**4*b**5*f**3 - 12288*_t**3*a**3*b**6*d**2*f + 32768*_t**3*a**3*b**6*d*e**2 + 13824*
_t**2*a**3*b**4*d*e*f**2 - 12288*_t**2*a**3*b**4*e**3*f + 512*_t**2*a**2*b**5*d**3*e + 3888*_t*a**3*b**2*d*f**
4 + 5184*_t*a**3*b**2*e**2*f**3 - 576*_t*a**2*b**3*d**3*f**2 + 1728*_t*a**2*b**3*d**2*e**2*f + 512*_t*a**2*b**
3*d*e**4 + 16*_t*a*b**4*d**5 + 1458*a**3*e*f**5 + 360*a**2*b*d*e**3*f**2 - 192*a**2*b*e**5*f + 30*a*b**2*d**4*
e*f - 40*a*b**2*d**3*e**3)/(729*a**3*f**6 - 81*a**2*b*d**2*f**4 + 864*a**2*b*d*e**2*f**3 - 576*a**2*b*e**4*f**
2 - 9*a*b**2*d**4*f**2 + 96*a*b**2*d**3*e**2*f - 64*a*b**2*d**2*e**4 + b**3*d**6)))) - (c + d*x + e*x**2 + f*x
**3)/(4*a*b + 4*b**2*x**4)

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Giac [A]  time = 1.09309, size = 409, normalized size = 1.32 \begin{align*} -\frac{f x^{3} + x^{2} e + d x + c}{4 \,{\left (b x^{4} + a\right )} b} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} e + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \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 )}{16 \, a b^{4}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} e + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \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 )}{16 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 3 \, \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 )}{32 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 3 \, \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 )}{32 \, a b^{4}} \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)^2,x, algorithm="giac")

[Out]

-1/4*(f*x^3 + x^2*e + d*x + c)/((b*x^4 + a)*b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*e + (a*b^3)^(1/4)*b^2*d
+ 3*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + 1/16*sqrt(2)*(2*sq
rt(2)*sqrt(a*b)*b^2*e + (a*b^3)^(1/4)*b^2*d + 3*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4)
)/(a/b)^(1/4))/(a*b^4) + 1/32*sqrt(2)*((a*b^3)^(1/4)*b^2*d - 3*(a*b^3)^(3/4)*f)*log(x^2 + sqrt(2)*x*(a/b)^(1/4
) + sqrt(a/b))/(a*b^4) - 1/32*sqrt(2)*((a*b^3)^(1/4)*b^2*d - 3*(a*b^3)^(3/4)*f)*log(x^2 - sqrt(2)*x*(a/b)^(1/4
) + sqrt(a/b))/(a*b^4)