\(\int \frac {x^3 (c+d x+e x^2+f x^3)}{(a+b x^4)^4} \, dx\) [494]

Optimal result

Integrand size = 28, antiderivative size = 380 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac {x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac {x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac {e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{5/2} b^{3/2}}-\frac {\left (7 \sqrt {b} d+5 \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (7 \sqrt {b} d+5 \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{11/4} b^{7/4}}-\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{11/4} b^{7/4}} \]

[Out]

1/12*(-f*x^3-e*x^2-d*x-c)/b/(b*x^4+a)^3+1/96*x*(3*f*x^2+2*e*x+d)/a/b/(b*x^4+a)^2+1/384*x*(15*f*x^2+12*e*x+7*d)
/a^2/b/(b*x^4+a)+1/32*e*arctan(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(3/2)-1/1024*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/
2)+x^2*b^(1/2))*(-5*f*a^(1/2)+7*d*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/1024*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2
)+x^2*b^(1/2))*(-5*f*a^(1/2)+7*d*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/512*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*
(5*f*a^(1/2)+7*d*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/512*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(5*f*a^(1/2)+7*d*
b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1837, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} f+7 \sqrt {b} d\right )}{256 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} f+7 \sqrt {b} d\right )}{256 \sqrt {2} a^{11/4} b^{7/4}}+\frac {e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{5/2} b^{3/2}}-\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{11/4} b^{7/4}}+\frac {x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac {x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2} \]

[In]

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

[Out]

-1/12*(c + d*x + e*x^2 + f*x^3)/(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*
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*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)*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 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 281

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 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 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 1837

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 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[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 1890

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

Rubi steps \begin{align*} \text {integral}& = -\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 \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.44 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {32 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )^2}+\frac {8 b^{3/4} x (7 d+3 x (4 e+5 f x))}{a^2 \left (a+b x^4\right )}-\frac {256 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^3}-\frac {6 \left (7 \sqrt {2} \sqrt {b} d+16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {6 \left (7 \sqrt {2} \sqrt {b} d-16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {3 \sqrt {2} \left (-7 \sqrt {b} d+5 \sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{11/4}}+\frac {3 \sqrt {2} \left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{11/4}}}{3072 b^{7/4}} \]

[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))

Maple [C] (verified)

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

Time = 1.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.38

method	result	size
risch	\(\frac {\frac {5 b f \,x^{11}}{128 a^{2}}+\frac {b e \,x^{10}}{32 a^{2}}+\frac {7 b d \,x^{9}}{384 a^{2}}+\frac {7 f \,x^{7}}{64 a}+\frac {e \,x^{6}}{12 a}+\frac {3 d \,x^{5}}{64 a}-\frac {5 f \,x^{3}}{384 b}-\frac {e \,x^{2}}{32 b}-\frac {7 d x}{128 b}-\frac {c}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (5 f \,\textit {\_R}^{2}+8 e \textit {\_R} +7 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{2} b^{2}}\)	\(144\)
default	\(\frac {\frac {5 b f \,x^{11}}{128 a^{2}}+\frac {b e \,x^{10}}{32 a^{2}}+\frac {7 b d \,x^{9}}{384 a^{2}}+\frac {7 f \,x^{7}}{64 a}+\frac {e \,x^{6}}{12 a}+\frac {3 d \,x^{5}}{64 a}-\frac {5 f \,x^{3}}{384 b}-\frac {e \,x^{2}}{32 b}-\frac {7 d x}{128 b}-\frac {c}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\frac {7 d \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {4 e \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {5 f \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{2} b}\)	\(334\)

[In]

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

[Out]

(5/128/a^2*b*f*x^11+1/32*b*e/a^2*x^10+7/384*b*d/a^2*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+1/512/a^2/b^2*sum((5*_R^2*f+8*_R*e+7*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 = 21.92 (sec) , antiderivative size = 125996, normalized size of antiderivative = 331.57 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.04 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {15 \, b^{2} f x^{11} + 12 \, b^{2} e x^{10} + 7 \, b^{2} d x^{9} + 42 \, a b f x^{7} + 32 \, a b e x^{6} + 18 \, a b d x^{5} - 5 \, a^{2} f x^{3} - 12 \, a^{2} e x^{2} - 21 \, a^{2} d x - 32 \, a^{2} c}{384 \, {\left (a^{2} b^{4} x^{12} + 3 \, a^{3} b^{3} x^{8} + 3 \, a^{4} b^{2} x^{4} + a^{5} b\right )}} + \frac {\frac {\sqrt {2} {\left (7 \, \sqrt {b} d - 5 \, \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, \sqrt {b} d - 5 \, \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f - 16 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f + 16 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{2} b} \]

[In]

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

[Out]

1/384*(15*b^2*f*x^11 + 12*b^2*e*x^10 + 7*b^2*d*x^9 + 42*a*b*f*x^7 + 32*a*b*e*x^6 + 18*a*b*d*x^5 - 5*a^2*f*x^3
- 12*a^2*e*x^2 - 21*a^2*d*x - 32*a^2*c)/(a^2*b^4*x^12 + 3*a^3*b^3*x^8 + 3*a^4*b^2*x^4 + a^5*b) + 1/1024*(sqrt(
2)*(7*sqrt(b)*d - 5*sqrt(a)*f)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) - sqrt
(2)*(7*sqrt(b)*d - 5*sqrt(a)*f)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(
7*sqrt(2)*a^(1/4)*b^(3/4)*d + 5*sqrt(2)*a^(3/4)*b^(1/4)*f - 16*sqrt(a)*sqrt(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^(3/4)) + 2*(7*sqrt(2)*a
^(1/4)*b^(3/4)*d + 5*sqrt(2)*a^(3/4)*b^(1/4)*f + 16*sqrt(a)*sqrt(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^(3/4)))/(a^2*b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {\sqrt {2} {\left (8 \, \sqrt {2} \sqrt {a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (8 \, \sqrt {2} \sqrt {a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{3} b^{4}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{3} b^{4}} + \frac {15 \, b^{2} f x^{11} + 12 \, b^{2} e x^{10} + 7 \, b^{2} d x^{9} + 42 \, a b f x^{7} + 32 \, a b e x^{6} + 18 \, a b d x^{5} - 5 \, a^{2} f x^{3} - 12 \, a^{2} e x^{2} - 21 \, a^{2} d x - 32 \, a^{2} c}{384 \, {\left (b x^{4} + a\right )}^{3} a^{2} b} \]

[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*e*x^10 + 7*b^2*d*x^9 + 42*a*b*f*x^7 + 32*a*b*e*x^6 + 18*a*b*d*x^5 - 5*a^2*f*x^3 -
12*a^2*e*x^2 - 21*a^2*d*x - 32*a^2*c)/((b*x^4 + a)^3*a^2*b)

Mupad [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.34 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {3\,d\,x^5}{64\,a}-\frac {c}{12\,b}+\frac {e\,x^6}{12\,a}-\frac {e\,x^2}{32\,b}+\frac {7\,f\,x^7}{64\,a}-\frac {5\,f\,x^3}{384\,b}-\frac {7\,d\,x}{128\,b}+\frac {7\,b\,d\,x^9}{384\,a^2}+\frac {b\,e\,x^{10}}{32\,a^2}+\frac {5\,b\,f\,x^{11}}{128\,a^2}}{a^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^8+b^3\,x^{12}}+\left (\sum _{k=1}^4\ln \left (-\frac {125\,a\,f^3-448\,b\,d\,e^2+245\,b\,d^2\,f-512\,b\,e^3\,x+{\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )}^2\,a^5\,b^4\,d\,1835008+560\,b\,d\,e\,f\,x+\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\,a^2\,b^3\,d^2\,x\,25088-{\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )}^2\,a^5\,b^4\,e\,x\,2097152-\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\,a^3\,b^2\,f^2\,x\,12800+\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\,a^3\,b^2\,e\,f\,40960}{a^6\,b^2\,2097152}\right )\,\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\right ) \]

[In]

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

[Out]

((3*d*x^5)/(64*a) - c/(12*b) + (e*x^6)/(12*a) - (e*x^2)/(32*b) + (7*f*x^7)/(64*a) - (5*f*x^3)/(384*b) - (7*d*x
)/(128*b) + (7*b*d*x^9)/(384*a^2) + (b*e*x^10)/(32*a^2) + (5*b*f*x^11)/(128*a^2))/(a^3 + b^3*x^12 + 3*a^2*b*x^
4 + 3*a*b^2*x^8) + symsum(log(-(125*a*f^3 - 448*b*d*e^2 + 245*b*d^2*f - 512*b*e^3*x + 1835008*root(68719476736
*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*
d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2401*b^2*d^4, z, k)^2*a^5*b^4*d +
560*b*d*e*f*x + 25088*root(68719476736*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 4
09600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^2*
f^4 + 2401*b^2*d^4, z, k)*a^2*b^3*d^2*x - 2097152*root(68719476736*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 3
3554432*a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^
2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2401*b^2*d^4, z, k)^2*a^5*b^4*e*x - 12800*root(68719476736*a^11*b^7*z^4 + 367
00160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a*b*
d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2401*b^2*d^4, z, k)*a^3*b^2*f^2*x + 40960*root(68719
476736*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a^
3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2401*b^2*d^4, z, k)*a^3*b^2
*e*f)/(2097152*a^6*b^2))*root(68719476736*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 +
409600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^
2*f^4 + 2401*b^2*d^4, z, k), k, 1, 4)