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

Optimal. Leaf size=416 \[ -\frac {\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (a^3 e^2+3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{\sqrt {3} d^{2/3} e^{7/3}}-\frac {x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac {3 c x^2 \left (a c+b^2\right )}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e} \]

[Out]

-(-6*a*b*c*e-b^3*e+c^3*d)*x/e^2+3/2*c*(a*c+b^2)*x^2/e+b*c^2*x^3/e+1/4*c^3*x^4/e+1/3*(c^3*d^2-6*a*b*c*d*e-e*(-a
^3*e+b^3*d)+3*d^(1/3)*e^(2/3)*(-a^2*b*e+a*c^2*d+b^2*c*d))*ln(d^(1/3)+e^(1/3)*x)/d^(2/3)/e^(7/3)-1/6*(c^3*d^2-6
*a*b*c*d*e-e*(-a^3*e+b^3*d)+3*d^(1/3)*e^(2/3)*(-a^2*b*e+a*c^2*d+b^2*c*d))*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)
*x^2)/d^(2/3)/e^(7/3)-(-a^2*c*e-a*b^2*e+b*c^2*d)*ln(e*x^3+d)/e^2-1/3*(c^3*d^2-3*b^2*c*d^(4/3)*e^(2/3)-3*a*c^2*
d^(4/3)*e^(2/3)-b^3*d*e-6*a*b*c*d*e+3*a^2*b*d^(1/3)*e^(5/3)+a^3*e^2)*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)/d^(1/3)*
3^(1/2))/d^(2/3)/e^(7/3)*3^(1/2)

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Rubi [A]  time = 0.70, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-e \left (b^3 d-a^3 e\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-e \left (b^3 d-a^3 e\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-3 b^2 c d^{4/3} e^{2/3}-b^3 d e+c^3 d^2\right )}{\sqrt {3} d^{2/3} e^{7/3}}-\frac {\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac {x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac {3 c x^2 \left (a c+b^2\right )}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c^3*d - b^3*e - 6*a*b*c*e)*x)/e^2) + (3*c*(b^2 + a*c)*x^2)/(2*e) + (b*c^2*x^3)/e + (c^3*x^4)/(4*e) - ((c^3
*d^2 - 3*b^2*c*d^(4/3)*e^(2/3) - 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e + 3*a^2*b*d^(1/3)*e^(5/3) + a
^3*e^2)*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(7/3)) + ((c^3*d^2 - 6*a*b*c*d*e
- e*(b^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2*c*d + a*c^2*d - a^2*b*e))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e
^(7/3)) - ((c^3*d^2 - 6*a*b*c*d*e - e*(b^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2*c*d + a*c^2*d - a^2*b*e))*Log[d
^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(7/3)) - ((b*c^2*d - a*b^2*e - a^2*c*e)*Log[d + e*x^3]
)/e^2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx &=\int \left (-\frac {c^3 d-b^3 e-6 a b c e}{e^2}+\frac {3 c \left (b^2+a c\right ) x}{e}+\frac {3 b c^2 x^2}{e}+\frac {c^3 x^3}{e}+\frac {c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )-3 e \left (b^2 c d+a c^2 d-a^2 b e\right ) x-3 e \left (b c^2 d-a b^2 e-a^2 c e\right ) x^2}{e^2 \left (d+e x^3\right )}\right ) \, dx\\ &=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}+\frac {\int \frac {c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )-3 e \left (b^2 c d+a c^2 d-a^2 b e\right ) x-3 e \left (b c^2 d-a b^2 e-a^2 c e\right ) x^2}{d+e x^3} \, dx}{e^2}\\ &=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}+\frac {\int \frac {c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )-3 e \left (b^2 c d+a c^2 d-a^2 b e\right ) x}{d+e x^3} \, dx}{e^2}-\frac {\left (3 \left (b c^2 d-a b^2 e-a^2 c e\right )\right ) \int \frac {x^2}{d+e x^3} \, dx}{e}\\ &=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}-\frac {\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}+\frac {\int \frac {\sqrt [3]{d} \left (-3 \sqrt [3]{d} e \left (b^2 c d+a c^2 d-a^2 b e\right )+2 \sqrt [3]{e} \left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )\right )\right )+\sqrt [3]{e} \left (-3 \sqrt [3]{d} e \left (b^2 c d+a c^2 d-a^2 b e\right )-\sqrt [3]{e} \left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )\right )\right ) x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3} e^{7/3}}+\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3} e^2}\\ &=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}+\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}+\frac {\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d} e^2}-\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} e^{7/3}}\\ &=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}+\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}-\frac {\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}+\frac {\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{7/3}}\\ &=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}-\frac {\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{7/3}}+\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}-\frac {\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 439, normalized size = 1.06 \[ \frac {12 \sqrt [3]{e} \log \left (d+e x^3\right ) \left (a^2 c e+a b^2 e-b c^2 d\right )-\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (e \left (a^3 e+3 a^2 b \sqrt [3]{d} e^{2/3}-b^3 d\right )-3 c \left (2 a b d e+b^2 d^{4/3} e^{2/3}\right )-3 a c^2 d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}-\frac {2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^3 e^2-3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}-b^3 d e+3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}+\frac {4 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a^3 e^2-3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}-b^3 d e+3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}+12 \sqrt [3]{e} x \left (6 a b c e+b^3 e-c^3 d\right )+18 c e^{4/3} x^2 \left (a c+b^2\right )+12 b c^2 e^{4/3} x^3+3 c^3 e^{4/3} x^4}{12 e^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*e^(1/3)*(-(c^3*d) + b^3*e + 6*a*b*c*e)*x + 18*c*(b^2 + a*c)*e^(4/3)*x^2 + 12*b*c^2*e^(4/3)*x^3 + 3*c^3*e^(
4/3)*x^4 - (4*Sqrt[3]*(c^3*d^2 - 3*a*c^2*d^(4/3)*e^(2/3) + e*(-(b^3*d) + 3*a^2*b*d^(1/3)*e^(2/3) + a^3*e) - 3*
c*(b^2*d^(4/3)*e^(2/3) + 2*a*b*d*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (4*(c^3*d^2 + 3*b^
2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e - 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log
[d^(1/3) + e^(1/3)*x])/d^(2/3) - (2*(c^3*d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6
*a*b*c*d*e - 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + 12*e
^(1/3)*(-(b*c^2*d) + a*b^2*e + a^2*c*e)*Log[d + e*x^3])/(12*e^(7/3))

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

[Out]

Timed out

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giac [A]  time = 0.23, size = 432, normalized size = 1.04 \[ -{\left (b c^{2} d - a b^{2} e - a^{2} c e\right )} e^{\left (-2\right )} \log \left ({\left | x^{3} e + d \right |}\right ) - \frac {\sqrt {3} {\left (c^{3} d^{2} - b^{3} d e - 6 \, a b c d e + 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} b^{2} c d + 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a c^{2} d - 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{2} b e + a^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}}\right ) e^{\left (-1\right )}}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (c^{3} d^{2} - b^{3} d e - 6 \, a b c d e - 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} b^{2} c d - 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a c^{2} d + 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{2} b e + a^{3} e^{2}\right )} e^{\left (-1\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (c^{3} d^{2} e^{7} - 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} b^{2} c d e^{8} - 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} a c^{2} d e^{8} - b^{3} d e^{8} - 6 \, a b c d e^{8} + 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} a^{2} b e^{9} + a^{3} e^{9}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} e^{\left (-9\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d} + \frac {1}{4} \, {\left (c^{3} x^{4} e^{3} + 4 \, b c^{2} x^{3} e^{3} + 6 \, b^{2} c x^{2} e^{3} + 6 \, a c^{2} x^{2} e^{3} - 4 \, c^{3} d x e^{2} + 4 \, b^{3} x e^{3} + 24 \, a b c x e^{3}\right )} e^{\left (-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*c^2*d - a*b^2*e - a^2*c*e)*e^(-2)*log(abs(x^3*e + d)) - 1/3*sqrt(3)*(c^3*d^2 - b^3*d*e - 6*a*b*c*d*e + 3*(
-d*e^2)^(1/3)*b^2*c*d + 3*(-d*e^2)^(1/3)*a*c^2*d - 3*(-d*e^2)^(1/3)*a^2*b*e + a^3*e^2)*arctan(1/3*sqrt(3)*(2*x
+ (-d*e^(-1))^(1/3))/(-d*e^(-1))^(1/3))*e^(-1)/(-d*e^2)^(2/3) - 1/6*(c^3*d^2 - b^3*d*e - 6*a*b*c*d*e - 3*(-d*
e^2)^(1/3)*b^2*c*d - 3*(-d*e^2)^(1/3)*a*c^2*d + 3*(-d*e^2)^(1/3)*a^2*b*e + a^3*e^2)*e^(-1)*log(x^2 + (-d*e^(-1
))^(1/3)*x + (-d*e^(-1))^(2/3))/(-d*e^2)^(2/3) - 1/3*(c^3*d^2*e^7 - 3*(-d*e^(-1))^(1/3)*b^2*c*d*e^8 - 3*(-d*e^
(-1))^(1/3)*a*c^2*d*e^8 - b^3*d*e^8 - 6*a*b*c*d*e^8 + 3*(-d*e^(-1))^(1/3)*a^2*b*e^9 + a^3*e^9)*(-d*e^(-1))^(1/
3)*e^(-9)*log(abs(x - (-d*e^(-1))^(1/3)))/d + 1/4*(c^3*x^4*e^3 + 4*b*c^2*x^3*e^3 + 6*b^2*c*x^2*e^3 + 6*a*c^2*x
^2*e^3 - 4*c^3*d*x*e^2 + 4*b^3*x*e^3 + 24*a*b*c*x*e^3)*e^(-4)

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maple [B]  time = 0.05, size = 837, normalized size = 2.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*b*c*d-1/2/e^2/(d/e)^(1/3)*ln(x^2-(d/e)^(1
/3)*x+(d/e)^(2/3))*b^2*c*d+1/e*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^2*b+1/e^2/(d/e)^(
1/3)*ln(x+(d/e)^(1/3))*a*c^2*d+1/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*b^2*c*d-1/2/e^2/(d/e)^(1/3)*ln(x^2-(d/e)^(1
/3)*x+(d/e)^(2/3))*a*c^2*d-1/3/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^3*d+1/3/e^3/(
d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*c^3*d^2-2/e^2/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a*b*c*d
+1/e^2/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a*b*c*d-1/e^2*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(
d/e)^(1/3)*x-1))*a*c^2*d-1/e^2*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^2*c*d-1/6/e/(d/e)
^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a^3+1/e*ln(e*x^3+d)*a^2*c+1/e*ln(e*x^3+d)*a*b^2+1/3/e/(d/e)^(2/3)*ln(
x+(d/e)^(1/3))*a^3+3/2/e*x^2*a*c^2+3/2/e*x^2*b^2*c-1/e^2*c^3*d*x+6/e*a*b*c*x-1/e/(d/e)^(1/3)*ln(x+(d/e)^(1/3))
*a^2*b+1/2/e/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a^2*b-1/e^2*ln(e*x^3+d)*b*c^2*d+1/3/e^3/(d/e)^(2/3)
*ln(x+(d/e)^(1/3))*c^3*d^2+1/6/e^2/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*b^3*d-1/6/e^3/(d/e)^(2/3)*ln(
x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*c^3*d^2+1/3/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^3-1
/3/e^2/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*b^3*d+1/e*b^3*x+1/4*c^3*x^4/e+b*c^2*x^3/e

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maxima [A]  time = 3.05, size = 520, normalized size = 1.25 \[ \frac {\sqrt {3} {\left ({\left (c^{3} \left (\frac {d}{e}\right )^{\frac {1}{3}} + 2 \, b c^{2}\right )} d^{2} - {\left (b^{3} \left (\frac {d}{e}\right )^{\frac {1}{3}} + 2 \, a b^{2} + {\left (3 \, a \left (\frac {d}{e}\right )^{\frac {2}{3}} + \frac {2 \, b d}{e}\right )} c^{2} + {\left (3 \, b^{2} \left (\frac {d}{e}\right )^{\frac {2}{3}} + 6 \, a b \left (\frac {d}{e}\right )^{\frac {1}{3}} + 2 \, a^{2}\right )} c\right )} d e + {\left (3 \, a^{2} b \left (\frac {d}{e}\right )^{\frac {2}{3}} + a^{3} \left (\frac {d}{e}\right )^{\frac {1}{3}} + \frac {2 \, a b^{2} d}{e} + \frac {2 \, a^{2} c d}{e}\right )} e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, d e^{2}} + \frac {c^{3} e x^{4} + 4 \, b c^{2} e x^{3} + 6 \, {\left (b^{2} c + a c^{2}\right )} e x^{2} - 4 \, {\left (c^{3} d - {\left (b^{3} + 6 \, a b c\right )} e\right )} x}{4 \, e^{2}} - \frac {{\left (c^{3} d^{2} - {\left (b^{3} - 3 \, {\left (2 \, b \left (\frac {d}{e}\right )^{\frac {2}{3}} + a \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )} c^{2} - 3 \, {\left (b^{2} \left (\frac {d}{e}\right )^{\frac {1}{3}} - 2 \, a b\right )} c\right )} d e - {\left (6 \, a b^{2} \left (\frac {d}{e}\right )^{\frac {2}{3}} + 6 \, a^{2} c \left (\frac {d}{e}\right )^{\frac {2}{3}} + 3 \, a^{2} b \left (\frac {d}{e}\right )^{\frac {1}{3}} - a^{3}\right )} e^{2}\right )} \log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} + \frac {{\left (c^{3} d^{2} - {\left (b^{3} + 3 \, {\left (b \left (\frac {d}{e}\right )^{\frac {2}{3}} - a \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )} c^{2} - 3 \, {\left (b^{2} \left (\frac {d}{e}\right )^{\frac {1}{3}} - 2 \, a b\right )} c\right )} d e + {\left (3 \, a b^{2} \left (\frac {d}{e}\right )^{\frac {2}{3}} + 3 \, a^{2} c \left (\frac {d}{e}\right )^{\frac {2}{3}} - 3 \, a^{2} b \left (\frac {d}{e}\right )^{\frac {1}{3}} + a^{3}\right )} e^{2}\right )} \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 \, e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*((c^3*(d/e)^(1/3) + 2*b*c^2)*d^2 - (b^3*(d/e)^(1/3) + 2*a*b^2 + (3*a*(d/e)^(2/3) + 2*b*d/e)*c^2 +
(3*b^2*(d/e)^(2/3) + 6*a*b*(d/e)^(1/3) + 2*a^2)*c)*d*e + (3*a^2*b*(d/e)^(2/3) + a^3*(d/e)^(1/3) + 2*a*b^2*d/e
+ 2*a^2*c*d/e)*e^2)*arctan(1/3*sqrt(3)*(2*x - (d/e)^(1/3))/(d/e)^(1/3))/(d*e^2) + 1/4*(c^3*e*x^4 + 4*b*c^2*e*x
^3 + 6*(b^2*c + a*c^2)*e*x^2 - 4*(c^3*d - (b^3 + 6*a*b*c)*e)*x)/e^2 - 1/6*(c^3*d^2 - (b^3 - 3*(2*b*(d/e)^(2/3)
+ a*(d/e)^(1/3))*c^2 - 3*(b^2*(d/e)^(1/3) - 2*a*b)*c)*d*e - (6*a*b^2*(d/e)^(2/3) + 6*a^2*c*(d/e)^(2/3) + 3*a^
2*b*(d/e)^(1/3) - a^3)*e^2)*log(x^2 - x*(d/e)^(1/3) + (d/e)^(2/3))/(e^3*(d/e)^(2/3)) + 1/3*(c^3*d^2 - (b^3 + 3
*(b*(d/e)^(2/3) - a*(d/e)^(1/3))*c^2 - 3*(b^2*(d/e)^(1/3) - 2*a*b)*c)*d*e + (3*a*b^2*(d/e)^(2/3) + 3*a^2*c*(d/
e)^(2/3) - 3*a^2*b*(d/e)^(1/3) + a^3)*e^2)*log(x + (d/e)^(1/3))/(e^3*(d/e)^(2/3))

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mupad [B]  time = 4.91, size = 1700, normalized size = 4.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((b^3 + 6*a*b*c)/e - (c^3*d)/e^2) + symsum(log(root(27*d^2*e^7*z^3 + 81*b*c^2*d^3*e^5*z^2 - 81*a^2*c*d^2*e^6
*z^2 - 81*a*b^2*d^2*e^6*z^2 - 27*a^3*b^2*c*d^2*e^5*z + 27*a^2*b*c^3*d^3*e^4*z + 27*a*b^3*c^2*d^3*e^4*z + 54*b^
2*c^4*d^4*e^3*z + 54*a^4*c^2*d^2*e^5*z + 54*a^2*b^4*d^2*e^5*z + 27*b^5*c*d^3*e^4*z - 27*a*c^5*d^4*e^3*z + 27*a
^5*b*d*e^6*z + 18*a^4*b^4*c*d^2*e^4 - 18*a^4*b*c^4*d^3*e^3 + 18*a*b^4*c^4*d^4*e^2 - 9*a*b^7*c*d^3*e^3 - 27*a^5
*b^2*c^2*d^2*e^4 + 27*a^2*b^5*c^2*d^3*e^3 - 27*a^2*b^2*c^5*d^4*e^2 - 21*a^3*b^3*c^3*d^3*e^3 - 9*a^7*b*c*d*e^5
- 9*a*b*c^7*d^5*e - 3*b^6*c^3*d^4*e^2 - 3*a^6*c^3*d^2*e^4 - 3*a^3*c^6*d^4*e^2 - 3*a^3*b^6*d^2*e^4 + 3*b^3*c^6*
d^5*e + 3*a^6*b^3*d*e^5 + b^9*d^3*e^3 - c^9*d^6 - a^9*e^6, z, k)*((3*x*(a^3*e^4 - b^3*d*e^3 + c^3*d^2*e^2 - 6*
a*b*c*d*e^3))/e^2 - (3*(6*a*b^2*d*e^3 - 6*b*c^2*d^2*e^2 + 6*a^2*c*d*e^3))/e^2 + 9*root(27*d^2*e^7*z^3 + 81*b*c
^2*d^3*e^5*z^2 - 81*a^2*c*d^2*e^6*z^2 - 81*a*b^2*d^2*e^6*z^2 - 27*a^3*b^2*c*d^2*e^5*z + 27*a^2*b*c^3*d^3*e^4*z
+ 27*a*b^3*c^2*d^3*e^4*z + 54*b^2*c^4*d^4*e^3*z + 54*a^4*c^2*d^2*e^5*z + 54*a^2*b^4*d^2*e^5*z + 27*b^5*c*d^3*
e^4*z - 27*a*c^5*d^4*e^3*z + 27*a^5*b*d*e^6*z + 18*a^4*b^4*c*d^2*e^4 - 18*a^4*b*c^4*d^3*e^3 + 18*a*b^4*c^4*d^4
*e^2 - 9*a*b^7*c*d^3*e^3 - 27*a^5*b^2*c^2*d^2*e^4 + 27*a^2*b^5*c^2*d^3*e^3 - 27*a^2*b^2*c^5*d^4*e^2 - 21*a^3*b
^3*c^3*d^3*e^3 - 9*a^7*b*c*d*e^5 - 9*a*b*c^7*d^5*e - 3*b^6*c^3*d^4*e^2 - 3*a^6*c^3*d^2*e^4 - 3*a^3*c^6*d^4*e^2
- 3*a^3*b^6*d^2*e^4 + 3*b^3*c^6*d^5*e + 3*a^6*b^3*d*e^5 + b^9*d^3*e^3 - c^9*d^6 - a^9*e^6, z, k)*d*e^2) + (3*
(a^5*b*e^3 - a*c^5*d^3 + 2*b^2*c^4*d^3 + 2*a^2*b^4*d*e^2 + 2*a^4*c^2*d*e^2 + b^5*c*d^2*e + a*b^3*c^2*d^2*e + a
^2*b*c^3*d^2*e - a^3*b^2*c*d*e^2))/e^2 + (3*x*(b*c^5*d^3 - a^5*c*e^3 + 2*a^4*b^2*e^3 + 2*a^2*c^4*d^2*e + 2*b^4
*c^2*d^2*e + a*b^5*d*e^2 - a*b^2*c^3*d^2*e + a^2*b^3*c*d*e^2 + a^3*b*c^2*d*e^2))/e^2)*root(27*d^2*e^7*z^3 + 81
*b*c^2*d^3*e^5*z^2 - 81*a^2*c*d^2*e^6*z^2 - 81*a*b^2*d^2*e^6*z^2 - 27*a^3*b^2*c*d^2*e^5*z + 27*a^2*b*c^3*d^3*e
^4*z + 27*a*b^3*c^2*d^3*e^4*z + 54*b^2*c^4*d^4*e^3*z + 54*a^4*c^2*d^2*e^5*z + 54*a^2*b^4*d^2*e^5*z + 27*b^5*c*
d^3*e^4*z - 27*a*c^5*d^4*e^3*z + 27*a^5*b*d*e^6*z + 18*a^4*b^4*c*d^2*e^4 - 18*a^4*b*c^4*d^3*e^3 + 18*a*b^4*c^4
*d^4*e^2 - 9*a*b^7*c*d^3*e^3 - 27*a^5*b^2*c^2*d^2*e^4 + 27*a^2*b^5*c^2*d^3*e^3 - 27*a^2*b^2*c^5*d^4*e^2 - 21*a
^3*b^3*c^3*d^3*e^3 - 9*a^7*b*c*d*e^5 - 9*a*b*c^7*d^5*e - 3*b^6*c^3*d^4*e^2 - 3*a^6*c^3*d^2*e^4 - 3*a^3*c^6*d^4
*e^2 - 3*a^3*b^6*d^2*e^4 + 3*b^3*c^6*d^5*e + 3*a^6*b^3*d*e^5 + b^9*d^3*e^3 - c^9*d^6 - a^9*e^6, z, k), k, 1, 3
) + (c^3*x^4)/(4*e) + (b*c^2*x^3)/e + (3*c*x^2*(a*c + b^2))/(2*e)

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sympy [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((c*x**2+b*x+a)**3/(e*x**3+d),x)

[Out]

Timed out

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