\(\int \frac {(d+e x)^2}{(a+c x^4)^3} \, dx\) [409]

Optimal result

Integrand size = 17, antiderivative size = 360 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {3 d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}-\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}} \]

[Out]

1/8*x*(e*x+d)^2/a/(c*x^4+a)^2+1/32*x*(5*e^2*x^2+12*d*e*x+7*d^2)/a^2/(c*x^4+a)+3/8*d*e*arctan(x^2*c^(1/2)/a^(1/
2))/a^(5/2)/c^(1/2)-1/256*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-5*e^2*a^(1/2)+21*d^2*c^(1/2))/a
^(11/4)/c^(3/4)*2^(1/2)+1/256*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-5*e^2*a^(1/2)+21*d^2*c^(1/2)
)/a^(11/4)/c^(3/4)*2^(1/2)+1/128*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(5*e^2*a^(1/2)+21*d^2*c^(1/2))/a^(11/4)/
c^(3/4)*2^(1/2)+1/128*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(5*e^2*a^(1/2)+21*d^2*c^(1/2))/a^(11/4)/c^(3/4)*2^(1
/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e^2+21 \sqrt {c} d^2\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} e^2+21 \sqrt {c} d^2\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}-\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2} \]

[In]

Int[(d + e*x)^2/(a + c*x^4)^3,x]

[Out]

(x*(d + e*x)^2)/(8*a*(a + c*x^4)^2) + (x*(7*d^2 + 12*d*e*x + 5*e^2*x^2))/(32*a^2*(a + c*x^4)) + (3*d*e*ArcTan[
(Sqrt[c]*x^2)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c]) - ((21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)
/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(3/4)) + ((21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^
(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(3/4)) - ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(3/4)) + ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(3/4))

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 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 {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}-\frac {\int \frac {-7 d^2-12 d e x-5 e^2 x^2}{\left (a+c x^4\right )^2} \, dx}{8 a} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {\int \frac {21 d^2+24 d e x+5 e^2 x^2}{a+c x^4} \, dx}{32 a^2} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {\int \left (\frac {24 d e x}{a+c x^4}+\frac {21 d^2+5 e^2 x^2}{a+c x^4}\right ) \, dx}{32 a^2} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {\int \frac {21 d^2+5 e^2 x^2}{a+c x^4} \, dx}{32 a^2}+\frac {(3 d e) \int \frac {x}{a+c x^4} \, dx}{4 a^2} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {(3 d e) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{8 a^2}+\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{64 a^2 c}+\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{64 a^2 c} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {3 d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}-\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{9/4} c^{3/4}}-\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c}+\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {3 d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}-\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}} \\ & = \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {3 d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}-\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {\left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {32 a^2 x (d+e x)^2}{\left (a+c x^4\right )^2}+\frac {8 a x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{a+c x^4}-\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {c} d^2+48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {c} d^2-48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {\sqrt {2} \left (-21 \sqrt [4]{a} \sqrt {c} d^2+5 a^{3/4} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}+\frac {\sqrt {2} \left (21 \sqrt [4]{a} \sqrt {c} d^2-5 a^{3/4} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}}{256 a^3} \]

[In]

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

[Out]

((32*a^2*x*(d + e*x)^2)/(a + c*x^4)^2 + (8*a*x*(7*d^2 + 12*d*e*x + 5*e^2*x^2))/(a + c*x^4) - (2*a^(1/4)*(21*Sq
rt[2]*Sqrt[c]*d^2 + 48*a^(1/4)*c^(1/4)*d*e + 5*Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c
^(3/4) + (2*a^(1/4)*(21*Sqrt[2]*Sqrt[c]*d^2 - 48*a^(1/4)*c^(1/4)*d*e + 5*Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt
[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (Sqrt[2]*(-21*a^(1/4)*Sqrt[c]*d^2 + 5*a^(3/4)*e^2)*Log[Sqrt[a] - Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4) + (Sqrt[2]*(21*a^(1/4)*Sqrt[c]*d^2 - 5*a^(3/4)*e^2)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4))/(256*a^3)

Maple [C] (verified)

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

Time = 0.84 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.35

method	result	size
risch	\(\frac {\frac {5 c \,e^{2} x^{7}}{32 a^{2}}+\frac {3 d c e \,x^{6}}{8 a^{2}}+\frac {7 c \,d^{2} x^{5}}{32 a^{2}}+\frac {9 e^{2} x^{3}}{32 a}+\frac {5 e d \,x^{2}}{8 a}+\frac {11 d^{2} x}{32 a}}{\left (c \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (5 e^{2} \textit {\_R}^{2}+24 e d \textit {\_R} +21 d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} c}\)	\(126\)
default	\(d^{2} \left (\frac {x}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (c \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+2 e d \left (\frac {x^{2}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (c \,x^{4}+a \right )}+\frac {3 \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{16 a \sqrt {a c}}}{a}\right )+e^{2} \left (\frac {x^{3}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (c \,x^{4}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}\right )\)	\(360\)

[In]

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

[Out]

(5/32*c*e^2/a^2*x^7+3/8*d*c*e/a^2*x^6+7/32*c*d^2/a^2*x^5+9/32*e^2/a*x^3+5/8*e*d/a*x^2+11/32/a*d^2*x)/(c*x^4+a)
^2+1/128/a^2/c*sum((5*_R^2*e^2+24*_R*d*e+21*d^2)/_R^3*ln(x-_R),_R=RootOf(_Z^4*c+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.15 (sec) , antiderivative size = 91420, normalized size of antiderivative = 253.94 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [A] (verification not implemented)

Time = 14.65 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\operatorname {RootSum} {\left (268435456 t^{4} a^{11} c^{3} + 25755648 t^{2} a^{6} c^{2} d^{2} e^{2} + t \left (307200 a^{4} c d e^{5} - 5419008 a^{3} c^{2} d^{5} e\right ) + 625 a^{2} e^{8} + 111906 a c d^{4} e^{4} + 194481 c^{2} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {262144000 t^{3} a^{10} c^{2} e^{6} + 46110081024 t^{3} a^{9} c^{3} d^{4} e^{2} - 1645608960 t^{2} a^{7} c^{2} d^{3} e^{5} + 3641573376 t^{2} a^{6} c^{3} d^{7} e + 32688000 t a^{5} c d^{2} e^{8} + 3128219136 t a^{4} c^{2} d^{6} e^{4} + 522764928 t a^{3} c^{3} d^{10} + 225000 a^{3} d e^{11} - 43338240 a^{2} c d^{5} e^{7} - 523431720 a c^{2} d^{9} e^{3}}{15625 a^{3} e^{12} - 21357225 a^{2} c d^{4} e^{8} - 376741449 a c^{2} d^{8} e^{4} + 85766121 c^{3} d^{12}} \right )} \right )\right )} + \frac {11 a d^{2} x + 20 a d e x^{2} + 9 a e^{2} x^{3} + 7 c d^{2} x^{5} + 12 c d e x^{6} + 5 c e^{2} x^{7}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \]

[In]

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

[Out]

RootSum(268435456*_t**4*a**11*c**3 + 25755648*_t**2*a**6*c**2*d**2*e**2 + _t*(307200*a**4*c*d*e**5 - 5419008*a
**3*c**2*d**5*e) + 625*a**2*e**8 + 111906*a*c*d**4*e**4 + 194481*c**2*d**8, Lambda(_t, _t*log(x + (262144000*_
t**3*a**10*c**2*e**6 + 46110081024*_t**3*a**9*c**3*d**4*e**2 - 1645608960*_t**2*a**7*c**2*d**3*e**5 + 36415733
76*_t**2*a**6*c**3*d**7*e + 32688000*_t*a**5*c*d**2*e**8 + 3128219136*_t*a**4*c**2*d**6*e**4 + 522764928*_t*a*
*3*c**3*d**10 + 225000*a**3*d*e**11 - 43338240*a**2*c*d**5*e**7 - 523431720*a*c**2*d**9*e**3)/(15625*a**3*e**1
2 - 21357225*a**2*c*d**4*e**8 - 376741449*a*c**2*d**8*e**4 + 85766121*c**3*d**12)))) + (11*a*d**2*x + 20*a*d*e
*x**2 + 9*a*e**2*x**3 + 7*c*d**2*x**5 + 12*c*d*e*x**6 + 5*c*e**2*x**7)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**
2*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {5 \, c e^{2} x^{7} + 12 \, c d e x^{6} + 7 \, c d^{2} x^{5} + 9 \, a e^{2} x^{3} + 20 \, a d e x^{2} + 11 \, a d^{2} x}{32 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 48 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 48 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}}}{256 \, a^{2}} \]

[In]

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

[Out]

1/32*(5*c*e^2*x^7 + 12*c*d*e*x^6 + 7*c*d^2*x^5 + 9*a*e^2*x^3 + 20*a*d*e*x^2 + 11*a*d^2*x)/(a^2*c^2*x^8 + 2*a^3
*c*x^4 + a^4) + 1/256*(sqrt(2)*(21*sqrt(c)*d^2 - 5*sqrt(a)*e^2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x +
sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(21*sqrt(c)*d^2 - 5*sqrt(a)*e^2)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4
)*x + sqrt(a))/(a^(3/4)*c^(3/4)) + 2*(21*sqrt(2)*a^(1/4)*c^(3/4)*d^2 + 5*sqrt(2)*a^(3/4)*c^(1/4)*e^2 - 48*sqrt
(a)*sqrt(c)*d*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sq
rt(sqrt(a)*sqrt(c))*c^(3/4)) + 2*(21*sqrt(2)*a^(1/4)*c^(3/4)*d^2 + 5*sqrt(2)*a^(3/4)*c^(1/4)*e^2 + 48*sqrt(a)*
sqrt(c)*d*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(s
qrt(a)*sqrt(c))*c^(3/4)))/a^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {5 \, c e^{2} x^{7} + 12 \, c d e x^{6} + 7 \, c d^{2} x^{5} + 9 \, a e^{2} x^{3} + 20 \, a d e x^{2} + 11 \, a d^{2} x}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2}} + \frac {\sqrt {2} {\left (24 \, \sqrt {2} \sqrt {a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (24 \, \sqrt {2} \sqrt {a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} \]

[In]

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

[Out]

1/32*(5*c*e^2*x^7 + 12*c*d*e*x^6 + 7*c*d^2*x^5 + 9*a*e^2*x^3 + 20*a*d*e*x^2 + 11*a*d^2*x)/((c*x^4 + a)^2*a^2)
+ 1/128*sqrt(2)*(24*sqrt(2)*sqrt(a*c)*c^2*d*e + 21*(a*c^3)^(1/4)*c^2*d^2 + 5*(a*c^3)^(3/4)*e^2)*arctan(1/2*sqr
t(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 1/128*sqrt(2)*(24*sqrt(2)*sqrt(a*c)*c^2*d*e + 21*(a*
c^3)^(1/4)*c^2*d^2 + 5*(a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3
) + 1/256*sqrt(2)*(21*(a*c^3)^(1/4)*c^2*d^2 - 5*(a*c^3)^(3/4)*e^2)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c)
)/(a^3*c^3) - 1/256*sqrt(2)*(21*(a*c^3)^(1/4)*c^2*d^2 - 5*(a*c^3)^(3/4)*e^2)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) +
sqrt(a/c))/(a^3*c^3)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {11\,d^2\,x}{32\,a}+\frac {9\,e^2\,x^3}{32\,a}+\frac {7\,c\,d^2\,x^5}{32\,a^2}+\frac {5\,c\,e^2\,x^7}{32\,a^2}+\frac {5\,d\,e\,x^2}{8\,a}+\frac {3\,c\,d\,e\,x^6}{8\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\frac {c\,\left (125\,a\,e^6-9891\,c\,d^4\,e^2+{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )}^2\,a^5\,c^2\,d^2\,344064-8784\,c\,d^3\,e^3\,x-\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\,a^3\,c\,e^4\,x\,3200+\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\,a^2\,c^2\,d^4\,x\,56448+\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\,a^3\,c\,d\,e^3\,30720-{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )}^2\,a^5\,c^2\,d\,e\,x\,393216\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\right ) \]

[In]

int((d + e*x)^2/(a + c*x^4)^3,x)

[Out]

((11*d^2*x)/(32*a) + (9*e^2*x^3)/(32*a) + (7*c*d^2*x^5)/(32*a^2) + (5*c*e^2*x^7)/(32*a^2) + (5*d*e*x^2)/(8*a)
+ (3*c*d*e*x^6)/(8*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) + symsum(log(-(c*(125*a*e^6 - 9891*c*d^4*e^2 + 344064*roo
t(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111
906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)^2*a^5*c^2*d^2 - 8784*c*d^3*e^3*x - 3200*root(268435456*a
^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e
^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)*a^3*c*e^4*x + 56448*root(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^
2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8
, z, k)*a^2*c^2*d^4*x + 30720*root(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5
*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)*a^3*c*d*e^3 - 393216*ro
ot(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 11
1906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)^2*a^5*c^2*d*e*x))/(32768*a^6))*root(268435456*a^11*c^3*
z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194
481*c^2*d^8 + 625*a^2*e^8, z, k), k, 1, 4)