3.408 \(\int \frac {(d+e x)^3}{(a+c x^4)^3} \, dx\)

Optimal. Leaf size=394 \[ -\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}-\frac {3 d \left (5 \sqrt {a} e^2+7 \sqrt {c} d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (5 \sqrt {a} e^2+7 \sqrt {c} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}+\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2} \]

[Out]

1/32*x*(15*d*e^2*x^2+18*d^2*e*x+7*d^3)/a^2/(c*x^4+a)+1/8*(-a*e^3+c*x*(3*d*e^2*x^2+3*d^2*e*x+d^3))/a/c/(c*x^4+a
)^2+9/16*d^2*e*arctan(x^2*c^(1/2)/a^(1/2))/a^(5/2)/c^(1/2)-3/256*d*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)+7*d^2*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2)+3/256*d*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)+7*d^2*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2)+3/128*d*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4
))*(5*e^2*a^(1/2)+7*d^2*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2)+3/128*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(5*e^2*a
^(1/2)+7*d^2*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2)

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Rubi [A]  time = 0.35, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}-\frac {3 d \left (5 \sqrt {a} e^2+7 \sqrt {c} d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (5 \sqrt {a} e^2+7 \sqrt {c} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {x \left (18 d^2 e x+7 d^3+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {a e^3-c x \left (3 d^2 e x+d^3+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + c*x^4)^3,x]

[Out]

(x*(7*d^3 + 18*d^2*e*x + 15*d*e^2*x^2))/(32*a^2*(a + c*x^4)) - (a*e^3 - c*x*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2))/(
8*a*c*(a + c*x^4)^2) + (9*d^2*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[c]) - (3*d*(7*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)) + (3*d*(7*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)) - (3*d*(7*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)) + (3*d*(7*
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 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 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 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 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 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]

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 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 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 1854

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 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

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}-\frac {\int \frac {-7 d^3-18 d^2 e x-15 d e^2 x^2}{\left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\int \frac {21 d^3+36 d^2 e x+15 d e^2 x^2}{a+c x^4} \, dx}{32 a^2}\\ &=\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\int \left (\frac {36 d^2 e x}{a+c x^4}+\frac {21 d^3+15 d e^2 x^2}{a+c x^4}\right ) \, dx}{32 a^2}\\ &=\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\int \frac {21 d^3+15 d e^2 x^2}{a+c x^4} \, dx}{32 a^2}+\frac {\left (9 d^2 e\right ) \int \frac {x}{a+c x^4} \, dx}{8 a^2}\\ &=\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\left (9 d^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2}+\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{64 a^2 c}+\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{64 a^2 c}\\ &=\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\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 (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\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 (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\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 (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\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 \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {3 d \left (\frac {7 \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 {3 d \left (\frac {7 \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 (3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right )\right ) \operatorname {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 (3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right )\right ) \operatorname {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 \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {3 d \left (7 \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 {3 d \left (7 \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 {3 d \left (\frac {7 \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 {3 d \left (\frac {7 \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*}

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Mathematica [A]  time = 0.35, size = 388, normalized size = 0.98 \[ \frac {\frac {3 \sqrt {2} \left (5 a^{3/4} d e^2-7 \sqrt [4]{a} \sqrt {c} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{c^{3/4}}+\frac {3 \sqrt {2} \left (7 \sqrt [4]{a} \sqrt {c} d^3-5 a^{3/4} d e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{c^{3/4}}-\frac {32 a^2 \left (a e^3-c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{c \left (a+c x^4\right )^2}-\frac {6 \sqrt [4]{a} d \left (24 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2+7 \sqrt {2} \sqrt {c} d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {6 \sqrt [4]{a} d \left (-24 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2+7 \sqrt {2} \sqrt {c} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac {8 a d x \left (7 d^2+18 d e x+15 e^2 x^2\right )}{a+c x^4}}{256 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a*d*x*(7*d^2 + 18*d*e*x + 15*e^2*x^2))/(a + c*x^4) - (32*a^2*(a*e^3 - c*d*x*(d^2 + 3*d*e*x + 3*e^2*x^2)))/
(c*(a + c*x^4)^2) - (6*a^(1/4)*d*(7*Sqrt[2]*Sqrt[c]*d^2 + 24*a^(1/4)*c^(1/4)*d*e + 5*Sqrt[2]*Sqrt[a]*e^2)*ArcT
an[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (6*a^(1/4)*d*(7*Sqrt[2]*Sqrt[c]*d^2 - 24*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) + (3*Sqrt[2]*(-7*a^(1/4)*Sqrt[c]*d^3 +
5*a^(3/4)*d*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4) + (3*Sqrt[2]*(7*a^(1/4)*Sqrt
[c]*d^3 - 5*a^(3/4)*d*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4))/(256*a^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.35, size = 389, normalized size = 0.99 \[ \frac {3 \, \sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {3 \, \sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} + \frac {15 \, c^{2} d x^{7} e^{2} + 18 \, c^{2} d^{2} x^{6} e + 7 \, c^{2} d^{3} x^{5} + 27 \, a c d x^{3} e^{2} + 30 \, a c d^{2} x^{2} e + 11 \, a c d^{3} x - 4 \, a^{2} e^{3}}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*sqrt(2)*(12*sqrt(2)*sqrt(a*c)*c^2*d^2*e + 7*(a*c^3)^(1/4)*c^2*d^3 + 5*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sq
rt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 3/128*sqrt(2)*(12*sqrt(2)*sqrt(a*c)*c^2*d^2*e + 7*(
a*c^3)^(1/4)*c^2*d^3 + 5*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3
*c^3) + 3/256*sqrt(2)*(7*(a*c^3)^(1/4)*c^2*d^3 - 5*(a*c^3)^(3/4)*d*e^2)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt
(a/c))/(a^3*c^3) - 3/256*sqrt(2)*(7*(a*c^3)^(1/4)*c^2*d^3 - 5*(a*c^3)^(3/4)*d*e^2)*log(x^2 - sqrt(2)*x*(a/c)^(
1/4) + sqrt(a/c))/(a^3*c^3) + 1/32*(15*c^2*d*x^7*e^2 + 18*c^2*d^2*x^6*e + 7*c^2*d^3*x^5 + 27*a*c*d*x^3*e^2 + 3
0*a*c*d^2*x^2*e + 11*a*c*d^3*x - 4*a^2*e^3)/((c*x^4 + a)^2*a^2*c)

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maple [A]  time = 0.01, size = 470, normalized size = 1.19 \[ \frac {e^{3} x^{4}}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {3 d \,e^{2} x^{3}}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {e^{3} x^{4}}{8 \left (c \,x^{4}+a \right ) a^{2}}+\frac {3 d^{2} e \,x^{2}}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {15 d \,e^{2} x^{3}}{32 \left (c \,x^{4}+a \right ) a^{2}}+\frac {d^{3} x}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {9 d^{2} e \,x^{2}}{16 \left (c \,x^{4}+a \right ) a^{2}}+\frac {7 d^{3} x}{32 \left (c \,x^{4}+a \right ) a^{2}}+\frac {9 d^{2} e \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{16 \sqrt {a c}\, a^{2}}+\frac {15 \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{128 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2} c}+\frac {15 \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{128 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2} c}+\frac {15 \sqrt {2}\, d \,e^{2} \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{256 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2} c}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{256 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^3*x/a/(c*x^4+a)^2+7/32*d^3/a^2*x/(c*x^4+a)+21/256*d^3/a^3*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*2^(1/2
)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))+21/128*d^3/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a
/c)^(1/4)*x+1)+21/128*d^3/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)+3/8*e*d^2*x^2/a/(c*x^4+a)^2+
9/16*e*d^2/a^2*x^2/(c*x^4+a)+9/16*e*d^2/a^2/(a*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)+3/8*d*e^2*x^3/a/(c*x^4+a)^2+
15/32*d*e^2/a^2*x^3/(c*x^4+a)+15/256*d*e^2/a^2/c/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)
)/(x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))+15/128*d*e^2/a^2/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x
+1)+15/128*d*e^2/a^2/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)+1/8*e^3*x^4/a/(c*x^4+a)^2+1/8*e^3/a
^2*x^4/(c*x^4+a)

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maxima [A]  time = 2.37, size = 392, normalized size = 0.99 \[ \frac {15 \, c^{2} d e^{2} x^{7} + 18 \, c^{2} d^{2} e x^{6} + 7 \, c^{2} d^{3} x^{5} + 27 \, a c d e^{2} x^{3} + 30 \, a c d^{2} e x^{2} + 11 \, a c d^{3} x - 4 \, a^{2} e^{3}}{32 \, {\left (a^{2} c^{3} x^{8} + 2 \, a^{3} c^{2} x^{4} + a^{4} c\right )}} + \frac {3 \, d {\left (\frac {\sqrt {2} {\left (7 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 24 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 24 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}}\right )}}{256 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(15*c^2*d*e^2*x^7 + 18*c^2*d^2*e*x^6 + 7*c^2*d^3*x^5 + 27*a*c*d*e^2*x^3 + 30*a*c*d^2*e*x^2 + 11*a*c*d^3*x
- 4*a^2*e^3)/(a^2*c^3*x^8 + 2*a^3*c^2*x^4 + a^4*c) + 3/256*d*(sqrt(2)*(7*sqrt(c)*d^2 - 5*sqrt(a)*e^2)*log(sqr
t(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(7*sqrt(c)*d^2 - 5*sqrt(a)*e^2)*lo
g(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) + 2*(7*sqrt(2)*a^(1/4)*c^(3/4)*d^2 + 5*
sqrt(2)*a^(3/4)*c^(1/4)*e^2 - 24*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(sqrt(a)*sqrt(c))*c^(3/4)) + 2*(7*sqrt(2)*a^(1/4)*c^(3/4)*d^2 + 5*sqrt(
2)*a^(3/4)*c^(1/4)*e^2 + 24*sqrt(a)*sqrt(c)*d*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sq
rt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(3/4)))/a^2

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mupad [B]  time = 0.48, size = 721, normalized size = 1.83 \[ \frac {\frac {11\,d^3\,x}{32\,a}-\frac {e^3}{8\,c}+\frac {7\,c\,d^3\,x^5}{32\,a^2}+\frac {15\,d^2\,e\,x^2}{16\,a}+\frac {27\,d\,e^2\,x^3}{32\,a}+\frac {9\,c\,d^2\,e\,x^6}{16\,a^2}+\frac {15\,c\,d\,e^2\,x^7}{32\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\left (\sum _{k=1}^4\ln \left (\frac {c\,d^2\,\left (6867\,c\,d^5\,e^2-1125\,a\,d\,e^6+7992\,c\,d^4\,e^3\,x-{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )}^2\,a^5\,c^2\,d\,114688+\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\,a^3\,c\,e^4\,x\,9600-\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\,a^2\,c^2\,d^4\,x\,18816+{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )}^2\,a^5\,c^2\,e\,x\,196608-\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\,a^3\,c\,d\,e^3\,46080\right )\,3}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(a + c*x^4)^3,x)

[Out]

((11*d^3*x)/(32*a) - e^3/(8*c) + (7*c*d^3*x^5)/(32*a^2) + (15*d^2*e*x^2)/(16*a) + (27*d*e^2*x^3)/(32*a) + (9*c
*d^2*e*x^6)/(16*a^2) + (15*c*d*e^2*x^7)/(32*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) + symsum(log((3*c*d^2*(6867*c*d^
5*e^2 - 1125*a*d*e^6 + 7992*c*d^4*e^3*x - 114688*root(268435456*a^11*c^3*z^4 + 63111168*a^6*c^2*d^4*e^2*z^2 -
8128512*a^3*c^2*d^8*e*z + 4147200*a^4*c*d^4*e^5*z + 245106*a*c*d^8*e^4 + 50625*a^2*d^4*e^8 + 194481*c^2*d^12,
z, k)^2*a^5*c^2*d + 9600*root(268435456*a^11*c^3*z^4 + 63111168*a^6*c^2*d^4*e^2*z^2 - 8128512*a^3*c^2*d^8*e*z
+ 4147200*a^4*c*d^4*e^5*z + 245106*a*c*d^8*e^4 + 50625*a^2*d^4*e^8 + 194481*c^2*d^12, z, k)*a^3*c*e^4*x - 1881
6*root(268435456*a^11*c^3*z^4 + 63111168*a^6*c^2*d^4*e^2*z^2 - 8128512*a^3*c^2*d^8*e*z + 4147200*a^4*c*d^4*e^5
*z + 245106*a*c*d^8*e^4 + 50625*a^2*d^4*e^8 + 194481*c^2*d^12, z, k)*a^2*c^2*d^4*x + 196608*root(268435456*a^1
1*c^3*z^4 + 63111168*a^6*c^2*d^4*e^2*z^2 - 8128512*a^3*c^2*d^8*e*z + 4147200*a^4*c*d^4*e^5*z + 245106*a*c*d^8*
e^4 + 50625*a^2*d^4*e^8 + 194481*c^2*d^12, z, k)^2*a^5*c^2*e*x - 46080*root(268435456*a^11*c^3*z^4 + 63111168*
a^6*c^2*d^4*e^2*z^2 - 8128512*a^3*c^2*d^8*e*z + 4147200*a^4*c*d^4*e^5*z + 245106*a*c*d^8*e^4 + 50625*a^2*d^4*e
^8 + 194481*c^2*d^12, z, k)*a^3*c*d*e^3))/(32768*a^6))*root(268435456*a^11*c^3*z^4 + 63111168*a^6*c^2*d^4*e^2*
z^2 - 8128512*a^3*c^2*d^8*e*z + 4147200*a^4*c*d^4*e^5*z + 245106*a*c*d^8*e^4 + 50625*a^2*d^4*e^8 + 194481*c^2*
d^12, z, k), k, 1, 4)

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sympy [A]  time = 8.01, size = 413, normalized size = 1.05 \[ \operatorname {RootSum} {\left (268435456 t^{4} a^{11} c^{3} + 63111168 t^{2} a^{6} c^{2} d^{4} e^{2} + t \left (4147200 a^{4} c d^{4} e^{5} - 8128512 a^{3} c^{2} d^{8} e\right ) + 50625 a^{2} d^{4} e^{8} + 245106 a c d^{8} e^{4} + 194481 c^{2} d^{12}, \left (t \mapsto t \log {\left (x + \frac {262144000 t^{3} a^{10} c^{2} e^{6} + 3714056192 t^{3} a^{9} c^{3} d^{4} e^{2} - 539688960 t^{2} a^{7} c^{2} d^{4} e^{5} + 202309632 t^{2} a^{6} c^{3} d^{8} e + 77328000 t a^{5} c d^{4} e^{8} + 660699648 t a^{4} c^{2} d^{8} e^{4} + 19361664 t a^{3} c^{3} d^{12} + 3037500 a^{3} d^{4} e^{11} - 26360640 a^{2} c d^{8} e^{7} - 60566940 a c^{2} d^{12} e^{3}}{421875 a^{3} d^{3} e^{12} - 29598075 a^{2} c d^{7} e^{8} - 58012227 a c^{2} d^{11} e^{4} + 3176523 c^{3} d^{15}} \right )} \right )\right )} + \frac {- 4 a^{2} e^{3} + 11 a c d^{3} x + 30 a c d^{2} e x^{2} + 27 a c d e^{2} x^{3} + 7 c^{2} d^{3} x^{5} + 18 c^{2} d^{2} e x^{6} + 15 c^{2} d e^{2} x^{7}}{32 a^{4} c + 64 a^{3} c^{2} x^{4} + 32 a^{2} c^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(268435456*_t**4*a**11*c**3 + 63111168*_t**2*a**6*c**2*d**4*e**2 + _t*(4147200*a**4*c*d**4*e**5 - 81285
12*a**3*c**2*d**8*e) + 50625*a**2*d**4*e**8 + 245106*a*c*d**8*e**4 + 194481*c**2*d**12, Lambda(_t, _t*log(x +
(262144000*_t**3*a**10*c**2*e**6 + 3714056192*_t**3*a**9*c**3*d**4*e**2 - 539688960*_t**2*a**7*c**2*d**4*e**5
+ 202309632*_t**2*a**6*c**3*d**8*e + 77328000*_t*a**5*c*d**4*e**8 + 660699648*_t*a**4*c**2*d**8*e**4 + 1936166
4*_t*a**3*c**3*d**12 + 3037500*a**3*d**4*e**11 - 26360640*a**2*c*d**8*e**7 - 60566940*a*c**2*d**12*e**3)/(4218
75*a**3*d**3*e**12 - 29598075*a**2*c*d**7*e**8 - 58012227*a*c**2*d**11*e**4 + 3176523*c**3*d**15)))) + (-4*a**
2*e**3 + 11*a*c*d**3*x + 30*a*c*d**2*e*x**2 + 27*a*c*d*e**2*x**3 + 7*c**2*d**3*x**5 + 18*c**2*d**2*e*x**6 + 15
*c**2*d*e**2*x**7)/(32*a**4*c + 64*a**3*c**2*x**4 + 32*a**2*c**3*x**8)

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