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

Optimal. Leaf size=380 \[ -\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2} \]

[Out]

-(c + d*x + e*x^2 + f*x^3)/(12*b*(a + b*x^4)^3) + (x*(d + 2*e*x + 3*f*x^2))/(96*a*b*(a + b*x^4)^2) + (x*(7*d +
12*e*x + 15*f*x^2))/(384*a^2*b*(a + b*x^4)) + (e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(5/2)*b^(3/2)) - ((7*Sq
rt[b]*d + 5*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7*Sqrt[b]*d
+ 5*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(11/4)*b^(7/4)) - ((7*Sqrt[b]*d - 5*Sq
rt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7*Sqrt[b]*
d - 5*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(11/4)*b^(7/4))

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Rubi [A]  time = 0.402178, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {1823, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c + d*x + e*x^2 + f*x^3)/(12*b*(a + b*x^4)^3) + (x*(d + 2*e*x + 3*f*x^2))/(96*a*b*(a + b*x^4)^2) + (x*(7*d +
12*e*x + 15*f*x^2))/(384*a^2*b*(a + b*x^4)) + (e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(5/2)*b^(3/2)) - ((7*Sq
rt[b]*d + 5*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7*Sqrt[b]*d
+ 5*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(11/4)*b^(7/4)) - ((7*Sqrt[b]*d - 5*Sq
rt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7*Sqrt[b]*
d - 5*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(11/4)*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 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 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 )^4} \, dx &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{\int \frac{d+2 e x+3 f x^2}{\left (a+b x^4\right )^3} \, dx}{12 b}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}-\frac{\int \frac{-7 d-12 e x-15 f x^2}{\left (a+b x^4\right )^2} \, dx}{96 a b}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{\int \frac{21 d+24 e x+15 f x^2}{a+b x^4} \, dx}{384 a^2 b}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{\int \left (\frac{24 e x}{a+b x^4}+\frac{21 d+15 f x^2}{a+b x^4}\right ) \, dx}{384 a^2 b}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{\int \frac{21 d+15 f x^2}{a+b x^4} \, dx}{384 a^2 b}+\frac{e \int \frac{x}{a+b x^4} \, dx}{16 a^2 b}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{e \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{32 a^2 b}+\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 f\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{256 a^2 b^2}+\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}+5 f\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{256 a^2 b^2}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}-\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 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}{512 \sqrt{2} a^{9/4} b^{7/4}}-\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 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}{512 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}+5 f\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^2 b^2}+\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}+5 f\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^2 b^2}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}-\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (7 \sqrt{b} d+5 \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 )}{256 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left (7 \sqrt{b} d+5 \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 )}{256 \sqrt{2} a^{11/4} b^{7/4}}\\ &=-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}-\frac{\left (7 \sqrt{b} d+5 \sqrt{a} f\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (7 \sqrt{b} d+5 \sqrt{a} f\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (\frac{7 \sqrt{b} d}{\sqrt{a}}-5 f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{9/4} b^{7/4}}\\ \end{align*}

Mathematica [A]  time = 0.421341, size = 366, normalized size = 0.96 \[ \frac{\frac{8 b^{3/4} x (7 d+3 x (4 e+5 f x))}{a^2 \left (a+b x^4\right )}-\frac{6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt{2} \sqrt{a} f+7 \sqrt{2} \sqrt{b} d\right )}{a^{11/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt{2} \sqrt{a} f+7 \sqrt{2} \sqrt{b} d\right )}{a^{11/4}}+\frac{3 \sqrt{2} \left (5 \sqrt{a} f-7 \sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{11/4}}+\frac{3 \sqrt{2} \left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{11/4}}-\frac{256 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^3}+\frac{32 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )^2}}{3072 b^{7/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)^2) + (8*b^(3/4)*x*(7*d + 3*x*(4*e + 5*f*x)))/(a^2*(a + b*
x^4)) - (256*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4)^3 - (6*(7*Sqrt[2]*Sqrt[b]*d + 16*a^(1/4)*b^(1/4)*e
+ 5*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*(7*Sqrt[2]*Sqrt[b]*d - 16*a^(1/
4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (3*Sqrt[2]*(-7*Sqrt[b]
*d + 5*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4) + (3*Sqrt[2]*(7*Sqrt[b]*d -
5*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4))/(3072*b^(7/4))

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Maple [A]  time = 0.012, size = 403, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ({\frac{5\,bf{x}^{11}}{128\,{a}^{2}}}+{\frac{be{x}^{10}}{32\,{a}^{2}}}+{\frac{7\,bd{x}^{9}}{384\,{a}^{2}}}+{\frac{7\,f{x}^{7}}{64\,a}}+{\frac{e{x}^{6}}{12\,a}}+{\frac{3\,d{x}^{5}}{64\,a}}-{\frac{5\,f{x}^{3}}{384\,b}}-{\frac{e{x}^{2}}{32\,b}}-{\frac{7\,dx}{128\,b}}-{\frac{c}{12\,b}} \right ) }+{\frac{7\,d\sqrt{2}}{1024\,b{a}^{3}}\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{7\,d\sqrt{2}}{512\,b{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{7\,d\sqrt{2}}{512\,b{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{32\,b{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,f\sqrt{2}}{1024\,{b}^{2}{a}^{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{5\,f\sqrt{2}}{512\,{b}^{2}{a}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,f\sqrt{2}}{512\,{b}^{2}{a}^{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)^4,x)

[Out]

(5/128*f*b/a^2*x^11+1/32/a^2*b*e*x^10+7/384/a^2*d*b*x^9+7/64*f/a*x^7+1/12/a*e*x^6+3/64*d/a*x^5-5/384*f*x^3/b-1
/32*e*x^2/b-7/128*d*x/b-1/12*c/b)/(b*x^4+a)^3+7/1024/b/a^3*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)))+7/512/b/a^3*d*(1/b*a)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(1/b*a)^(1/4)*x+1)+7/512/b/a^3*d*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)+1/32/b/a^2*e/(a
*b)^(1/2)*arctan(x^2*(b/a)^(1/2))+5/1024/b^2/a^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)))+5/512/b^2/a^2*f/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1
/b*a)^(1/4)*x+1)+5/512/b^2/a^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)^4,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)^4,x, algorithm="fricas")

[Out]

Timed out

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Sympy [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)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.09185, size = 513, normalized size = 1.35 \begin{align*} \frac{\sqrt{2}{\left (8 \, \sqrt{2} \sqrt{a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 5 \, \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 )}{512 \, a^{3} b^{4}} + \frac{\sqrt{2}{\left (8 \, \sqrt{2} \sqrt{a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 5 \, \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 )}{512 \, a^{3} b^{4}} + \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 5 \, \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 )}{1024 \, a^{3} b^{4}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 5 \, \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 )}{1024 \, a^{3} b^{4}} + \frac{15 \, b^{2} f x^{11} + 12 \, b^{2} x^{10} e + 7 \, b^{2} d x^{9} + 42 \, a b f x^{7} + 32 \, a b x^{6} e + 18 \, a b d x^{5} - 5 \, a^{2} f x^{3} - 12 \, a^{2} x^{2} e - 21 \, a^{2} d x - 32 \, a^{2} c}{384 \,{\left (b x^{4} + a\right )}^{3} a^{2} 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)^4,x, algorithm="giac")

[Out]

1/512*sqrt(2)*(8*sqrt(2)*sqrt(a*b)*b^2*e + 7*(a*b^3)^(1/4)*b^2*d + 5*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x
+ sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) + 1/512*sqrt(2)*(8*sqrt(2)*sqrt(a*b)*b^2*e + 7*(a*b^3)^(1/4)*b^2
*d + 5*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) + 1/1024*sqrt(2)
*(7*(a*b^3)^(1/4)*b^2*d - 5*(a*b^3)^(3/4)*f)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4) - 1/1024*s
qrt(2)*(7*(a*b^3)^(1/4)*b^2*d - 5*(a*b^3)^(3/4)*f)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4) + 1/
384*(15*b^2*f*x^11 + 12*b^2*x^10*e + 7*b^2*d*x^9 + 42*a*b*f*x^7 + 32*a*b*x^6*e + 18*a*b*d*x^5 - 5*a^2*f*x^3 -
12*a^2*x^2*e - 21*a^2*d*x - 32*a^2*c)/((b*x^4 + a)^3*a^2*b)