3.5.91 \(\int \frac {c+d x+e x^2+f x^3}{(a+b x^4)^3} \, dx\) [491]

Optimal. Leaf size=351 \[ \frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}} \]

[Out]

1/32*x*(5*e*x^2+6*d*x+7*c)/a^2/(b*x^4+a)+1/8*(-a*f+b*x*(e*x^2+d*x+c))/a/b/(b*x^4+a)^2+3/16*d*arctan(x^2*b^(1/2
)/a^(1/2))/a^(5/2)/b^(1/2)-1/256*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-5*e*a^(1/2)+21*c*b^(1/2)
)/a^(11/4)/b^(3/4)*2^(1/2)+1/256*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-5*e*a^(1/2)+21*c*b^(1/2))
/a^(11/4)/b^(3/4)*2^(1/2)+1/128*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*(5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/
4)*2^(1/2)+1/128*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)*2^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1868, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {3 d \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*(a + b*x^4)) - (a*f - b*x*(c + d*x + e*x^2))/(8*a*b*(a + b*x^4)^2) + (3*d*
ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^
(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(3/4)) - ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(3/4)) + ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(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 1868

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 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*} \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx &=-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}-\frac {\int \frac {-7 c-6 d x-5 e x^2}{\left (a+b x^4\right )^2} \, dx}{8 a}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {\int \frac {21 c+12 d x+5 e x^2}{a+b x^4} \, dx}{32 a^2}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {\int \left (\frac {12 d x}{a+b x^4}+\frac {21 c+5 e x^2}{a+b x^4}\right ) \, dx}{32 a^2}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {\int \frac {21 c+5 e x^2}{a+b x^4} \, dx}{32 a^2}+\frac {(3 d) \int \frac {x}{a+b x^4} \, dx}{8 a^2}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {(3 d) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{16 a^2}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a^2 b}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a^2 b}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\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}{128 \sqrt {2} a^{9/4} b^{3/4}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\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}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}\\ &=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 347, normalized size = 0.99 \begin {gather*} \frac {\frac {8 a x (7 c+x (6 d+5 e x))}{a+b x^4}-\frac {32 a^2 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^2}-\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {b} c+24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {b} c-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {\sqrt {2} \left (-21 \sqrt [4]{a} \sqrt {b} c+5 a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {\sqrt {2} \left (21 \sqrt [4]{a} \sqrt {b} c-5 a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{256 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a*x*(7*c + x*(6*d + 5*e*x)))/(a + b*x^4) - (32*a^2*(a*f - b*x*(c + x*(d + e*x))))/(b*(a + b*x^4)^2) - (2*a
^(1/4)*(21*Sqrt[2]*Sqrt[b]*c + 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1
/4)])/b^(3/4) + (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c - 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqr
t[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (Sqrt[2]*(-21*a^(1/4)*Sqrt[b]*c + 5*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (Sqrt[2]*(21*a^(1/4)*Sqrt[b]*c - 5*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(
1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(256*a^3)

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Maple [A]
time = 0.36, size = 391, normalized size = 1.11

method	result	size
risch	\(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {7 b c \,x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {11 c x}{32 a}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d +21 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\)	\(117\)
default	\(c \left (\frac {x}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (b \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+d \left (\frac {x^{2}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (b \,x^{4}+a \right )}+\frac {3 \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{16 a \sqrt {a b}}}{a}\right )+e \left (\frac {x^{3}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (b \,x^{4}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\right )+f \left (\frac {x^{4}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {x^{4}}{8 a^{2} \left (b \,x^{4}+a \right )}\right )\)	\(391\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(1/8*x/a/(b*x^4+a)^2+7/8/a*(1/4*x/a/(b*x^4+a)+3/32/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a
/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(
1/4)*x-1))))+d*(1/8*x^2/a/(b*x^4+a)^2+3/4/a*(1/4*x^2/a/(b*x^4+a)+1/4/a/(a*b)^(1/2)*arctan(x^2*(b/a)^(1/2))))+e
*(1/8*x^3/a/(b*x^4+a)^2+5/8/a*(1/4*x^3/a/(b*x^4+a)+1/32/a/b/(a/b)^(1/4)*2^(1/2)*(ln((x^2-(a/b)^(1/4)*x*2^(1/2)
+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b
)^(1/4)*x-1))))+f*(1/8*x^4/a/(b*x^4+a)^2+1/8/a^2*x^4/(b*x^4+a))

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Maxima [A]
time = 0.50, size = 361, normalized size = 1.03 \begin {gather*} \frac {5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + 9 \, a b x^{3} e + 10 \, a b d x^{2} + 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\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 (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\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 (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 24 \, \sqrt {a} \sqrt {b} d\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 (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 24 \, \sqrt {a} \sqrt {b} d\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}}}}{256 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(5*b^2*x^7*e + 6*b^2*d*x^6 + 7*b^2*c*x^5 + 9*a*b*x^3*e + 10*a*b*d*x^2 + 11*a*b*c*x - 4*a^2*f)/(a^2*b^3*x^
8 + 2*a^3*b^2*x^4 + a^4*b) + 1/256*(sqrt(2)*(21*sqrt(b)*c - 5*sqrt(a)*e)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(
1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) - sqrt(2)*(21*sqrt(b)*c - 5*sqrt(a)*e)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b
^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(21*sqrt(2)*a^(1/4)*b^(3/4)*c + 5*sqrt(2)*a^(3/4)*b^(1/4)*e - 24*sqr
t(a)*sqrt(b)*d)*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)*sqr
t(sqrt(a)*sqrt(b))*b^(3/4)) + 2*(21*sqrt(2)*a^(1/4)*b^(3/4)*c + 5*sqrt(2)*a^(3/4)*b^(1/4)*e + 24*sqrt(a)*sqrt(
b)*d)*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

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Fricas [C] Result contains complex when optimal does not.
time = 7.94, size = 124838, normalized size = 355.66 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/589824*(92160*b^2*e*x^7 + 110592*b^2*d*x^6 + 129024*b^2*c*x^5 + 165888*a*b*e*x^3 + 184320*a*b*d*x^2 + 202752
*a*b*c*x - 73728*a^2*f + 2*(a^2*b^3*x^8 + 2*a^3*b^2*x^4 + a^4*b)*((-I*sqrt(3) + 1)*((a^3*b*sqrt(-1/(a*b))*sqrt
(-(210*a*b*c*e*sqrt(-1/(a*b)) + 441*b*c^2 - 25*a*e^2)/(a^6*b^2*sqrt(-1/(a*b)))) - 12*d)^2/(a^5*b) - 3*(24*a^3*
b*d*sqrt(-(210*a*b*c*e*sqrt(-1/(a*b)) + 441*b*c^2 - 25*a*e^2)/(a^6*b^2*sqrt(-1/(a*b)))) + 441*b*c^2 - (6*(24*d
^2 + 35*c*e)*b*sqrt(-1/(a*b)) + 25*e^2)*a)/(a^6*b^2*sqrt(-1/(a*b))))/(-1/12582912*(24*a^3*b*d*sqrt(-(210*a*b*c
*e*sqrt(-1/(a*b)) + 441*b*c^2 - 25*a*e^2)/(a^6*b^2*sqrt(-1/(a*b)))) + 441*b*c^2 - (6*(24*d^2 + 35*c*e)*b*sqrt(
-1/(a*b)) + 25*e^2)*a)*(a^3*b*sqrt(-1/(a*b))*sqrt(-(210*a*b*c*e*sqrt(-1/(a*b)) + 441*b*c^2 - 25*a*e^2)/(a^6*b^
2*sqrt(-1/(a*b)))) - 12*d)/(a^8*b^2) + 1/4194304*(a^8*b^2*sqrt(-1/(a*b))*(-(210*a*b*c*e*sqrt(-1/(a*b)) + 441*b
*c^2 - 25*a*e^2)/(a^6*b^2*sqrt(-1/(a*b))))^(3/2) - 12*(12*d^2*sqrt(-(210*a*b*c*e*sqrt(-1/(a*b)) + 441*b*c^2 -
25*a*e^2)/ ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.64, size = 354, normalized size = 1.01 \begin {gather*} \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\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 )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\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 )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac {5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + 9 \, a b x^{3} e + 10 \, a b d x^{2} + 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)*b^2*c + 5*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*
x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^3) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)
*b^2*c + 5*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt
(2)*(21*(a*b^3)^(1/4)*b^2*c - 5*(a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^3) - 1/25
6*sqrt(2)*(21*(a*b^3)^(1/4)*b^2*c - 5*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^3)
+ 1/32*(5*b^2*x^7*e + 6*b^2*d*x^6 + 7*b^2*c*x^5 + 9*a*b*x^3*e + 10*a*b*d*x^2 + 11*a*b*c*x - 4*a^2*f)/((b*x^4 +
a)^2*a^2*b)

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Mupad [B]
time = 5.20, size = 832, normalized size = 2.37 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (125\,a\,e^3-3024\,b\,c\,d^2+2205\,b\,c^2\,e-1728\,b\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,c\,344064-\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^3\,b\,e^2\,x\,3200+2520\,b\,c\,d\,e\,x+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^2\,b^2\,c^2\,x\,56448-{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,d\,x\,196608+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^3\,b\,d\,e\,15360\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\right )+\frac {\frac {5\,d\,x^2}{16\,a}-\frac {f}{8\,b}+\frac {9\,e\,x^3}{32\,a}+\frac {11\,c\,x}{32\,a}+\frac {7\,b\,c\,x^5}{32\,a^2}+\frac {3\,b\,d\,x^6}{16\,a^2}+\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2+2\,a\,b\,x^4+b^2\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-(b*(125*a*e^3 - 3024*b*c*d^2 + 2205*b*c^2*e - 1728*b*d^3*x + 344064*root(268435456*a^11*b^3*z^4 +
6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b
*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)^2*a^5*b^2*c - 3200*root(268
435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4
*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)*a^3*b
*e^2*x + 2520*b*c*d*e*x + 56448*root(268435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^
2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 6
25*a^2*e^4 + 194481*b^2*c^4, z, k)*a^2*b^2*c^2*x - 196608*root(268435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^
2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c
^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)^2*a^5*b^2*d*x + 15360*root(268435456*a^11*b^3*z^4
+ 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*
a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)*a^3*b*d*e))/(32768*a^6))
*root(268435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 1
53600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z,
k), k, 1, 4) + ((5*d*x^2)/(16*a) - f/(8*b) + (9*e*x^3)/(32*a) + (11*c*x)/(32*a) + (7*b*c*x^5)/(32*a^2) + (3*b
*d*x^6)/(16*a^2) + (5*b*e*x^7)/(32*a^2))/(a^2 + b^2*x^8 + 2*a*b*x^4)

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