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

Optimal. Leaf size=645 \[ -\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (\sqrt [3]{e} \left (-12 a^2 b c d e+a^4 e^2-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{6 d^{2/3} e^{8/3}}+\frac{\log \left (d+e x^3\right ) \left (-4 c e \left (b^3 d-a^3 e\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{3 e^3}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (-12 a^2 b c d e+a^4 e^2-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{3 d^{2/3} e^{8/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-e \left (-3 a^2 b \sqrt [3]{d} e^{2/3}+a^3 (-e)+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e+4 c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{8/3}}-\frac{x^2 \left (-6 a^2 c^2 e-12 a b^2 c e+b^4 (-e)+4 b c^3 d\right )}{2 e^2}-\frac{2 x \left (-6 a^2 b c e-2 a b^3 e+2 a c^3 d+3 b^2 c^2 d\right )}{e^2}-\frac{c x^3 \left (-12 a b c e-4 b^3 e+c^3 d\right )}{3 e^2}+\frac{c^2 x^4 \left (2 a c+3 b^2\right )}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e} \]

[Out]

(-2*(3*b^2*c^2*d + 2*a*c^3*d - 2*a*b^3*e - 6*a^2*b*c*e)*x)/e^2 - ((4*b*c^3*d - b^4*e - 12*a*b^2*c*e - 6*a^2*c^
2*e)*x^2)/(2*e^2) - (c*(c^3*d - 4*b^3*e - 12*a*b*c*e)*x^3)/(3*e^2) + (c^2*(3*b^2 + 2*a*c)*x^4)/(2*e) + (4*b*c^
3*x^5)/(5*e) + (c^4*x^6)/(6*e) - ((b*d^(1/3) + a*e^(1/3))*(4*c^3*d^2 + 6*c^2*(b*d^(5/3)*e^(1/3) - a*d^(4/3)*e^
(2/3)) - 12*a*b*c*d*e - e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^(2/3) - a^3*e))*ArcTan[(d^(1/3)
- 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(8/3)) + ((e^(1/3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b
^3*d*e - 12*a^2*b*c*d*e + a^4*e^2) + d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^
2)))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(8/3)) - ((e^(1/3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 12
*a^2*b*c*d*e + a^4*e^2) + d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^2)))*Log[d^
(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(8/3)) + ((c^4*d^2 - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 - 4
*c*e*(b^3*d - a^3*e))*Log[d + e*x^3])/(3*e^3)

________________________________________________________________________________________

Rubi [A]  time = 1.09664, antiderivative size = 643, normalized size of antiderivative = 1., 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} \[ -\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (\frac{\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )}{\sqrt [3]{e}}-12 a^2 b c d e+a^4 e^2-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (d+e x^3\right ) \left (-4 c e \left (b^3 d-a^3 e\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{3 e^3}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (-12 a^2 b c d e+a^4 e^2-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{3 d^{2/3} e^{8/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-e \left (-3 a^2 b \sqrt [3]{d} e^{2/3}+a^3 (-e)+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e+4 c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{8/3}}-\frac{x^2 \left (-6 a^2 c^2 e-12 a b^2 c e+b^4 (-e)+4 b c^3 d\right )}{2 e^2}-\frac{2 x \left (-6 a^2 b c e-2 a b^3 e+2 a c^3 d+3 b^2 c^2 d\right )}{e^2}-\frac{c x^3 \left (-12 a b c e-4 b^3 e+c^3 d\right )}{3 e^2}+\frac{c^2 x^4 \left (2 a c+3 b^2\right )}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*b^2*c^2*d + 2*a*c^3*d - 2*a*b^3*e - 6*a^2*b*c*e)*x)/e^2 - ((4*b*c^3*d - b^4*e - 12*a*b^2*c*e - 6*a^2*c^
2*e)*x^2)/(2*e^2) - (c*(c^3*d - 4*b^3*e - 12*a*b*c*e)*x^3)/(3*e^2) + (c^2*(3*b^2 + 2*a*c)*x^4)/(2*e) + (4*b*c^
3*x^5)/(5*e) + (c^4*x^6)/(6*e) - ((b*d^(1/3) + a*e^(1/3))*(4*c^3*d^2 + 6*c^2*(b*d^(5/3)*e^(1/3) - a*d^(4/3)*e^
(2/3)) - 12*a*b*c*d*e - e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^(2/3) - a^3*e))*ArcTan[(d^(1/3)
- 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(8/3)) + ((e^(1/3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b
^3*d*e - 12*a^2*b*c*d*e + a^4*e^2) + d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^
2)))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(8/3)) - ((6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 12*a^2*b*c*
d*e + a^4*e^2 + (d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^2)))/e^(1/3))*Log[d^
(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(7/3)) + ((c^4*d^2 - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 - 4
*c*e*(b^3*d - a^3*e))*Log[d + e*x^3])/(3*e^3)

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]

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 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 31

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

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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx &=\int \left (-\frac{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right )}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x}{e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^2}{e^2}+\frac{2 c^2 \left (3 b^2+2 a c\right ) x^3}{e}+\frac{4 b c^3 x^4}{e}+\frac{c^4 x^5}{e}+\frac{6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2-\left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right ) x+\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) x^2}{e^2 \left (d+e x^3\right )}\right ) \, dx\\ &=-\frac{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac{c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}+\frac{\int \frac{6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2-\left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right ) x+\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) x^2}{d+e x^3} \, dx}{e^2}\\ &=-\frac{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac{c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}+\frac{\int \frac{6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\left (-b^4 d e-12 a b^2 c d e-6 a^2 c^2 d e+4 b \left (c^3 d^2+a^3 e^2\right )\right ) x}{d+e x^3} \, dx}{e^2}+\frac{\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \int \frac{x^2}{d+e x^3} \, dx}{e^2}\\ &=-\frac{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac{c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}+\frac{\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}+\frac{\int \frac{\sqrt [3]{d} \left (2 \sqrt [3]{e} \left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2\right )+\sqrt [3]{d} \left (-b^4 d e-12 a b^2 c d e-6 a^2 c^2 d e+4 b \left (c^3 d^2+a^3 e^2\right )\right )\right )+\sqrt [3]{e} \left (-\sqrt [3]{e} \left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2\right )+\sqrt [3]{d} \left (-b^4 d e-12 a b^2 c d e-6 a^2 c^2 d e+4 b \left (c^3 d^2+a^3 e^2\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 (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3} e^2}\\ &=-\frac{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac{c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}+\frac{\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}+\frac{\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}+\frac{\left (\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right )\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^{7/3}}-\frac{\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\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{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac{c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}+\frac{\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\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 (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}+\frac{\left (\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right )\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^{8/3}}\\ &=-\frac{2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac{\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac{c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac{c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}-\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\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^{8/3}}+\frac{\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac{\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\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 (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.348562, size = 678, normalized size = 1.05 \[ \frac{-\frac{5 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (-4 b \left (3 a^2 c d e^{4/3}+a^3 \sqrt [3]{d} e^2+c^3 d^{7/3}\right )+6 a^2 c^2 d^{4/3} e+a^4 e^{7/3}+6 b^2 \left (2 a c d^{4/3} e+c^2 d^2 \sqrt [3]{e}\right )-4 a b^3 d e^{4/3}+4 a c^3 d^2 \sqrt [3]{e}+b^4 d^{4/3} e\right )}{d^{2/3}}+\frac{10 \log \left (d+e x^3\right ) \left (4 c e \left (a^3 e-b^3 d\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{\sqrt [3]{e}}+\frac{10 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (-4 b \left (3 a^2 c d e^{4/3}+a^3 \sqrt [3]{d} e^2+c^3 d^{7/3}\right )+6 a^2 c^2 d^{4/3} e+a^4 e^{7/3}+6 b^2 \left (2 a c d^{4/3} e+c^2 d^2 \sqrt [3]{e}\right )-4 a b^3 d e^{4/3}+4 a c^3 d^2 \sqrt [3]{e}+b^4 d^{4/3} e\right )}{d^{2/3}}+\frac{10 \sqrt{3} \left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e \left (-3 a^2 b \sqrt [3]{d} e^{2/3}+a^3 (-e)+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+c^2 \left (6 a d^{4/3} e^{2/3}-6 b d^{5/3} \sqrt [3]{e}\right )+12 a b c d e-4 c^3 d^2\right )}{d^{2/3}}+15 e^{2/3} x^2 \left (6 a^2 c^2 e+12 a b^2 c e+b^4 e-4 b c^3 d\right )+60 e^{2/3} x \left (6 a^2 b c e+2 a b^3 e-2 a c^3 d-3 b^2 c^2 d\right )+10 c e^{2/3} x^3 \left (12 a b c e+4 b^3 e-c^3 d\right )+15 c^2 e^{5/3} x^4 \left (2 a c+3 b^2\right )+24 b c^3 e^{5/3} x^5+5 c^4 e^{5/3} x^6}{30 e^{8/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*e^(2/3)*(-3*b^2*c^2*d - 2*a*c^3*d + 2*a*b^3*e + 6*a^2*b*c*e)*x + 15*e^(2/3)*(-4*b*c^3*d + b^4*e + 12*a*b^2
*c*e + 6*a^2*c^2*e)*x^2 + 10*c*e^(2/3)*(-(c^3*d) + 4*b^3*e + 12*a*b*c*e)*x^3 + 15*c^2*(3*b^2 + 2*a*c)*e^(5/3)*
x^4 + 24*b*c^3*e^(5/3)*x^5 + 5*c^4*e^(5/3)*x^6 + (10*Sqrt[3]*(b*d^(1/3) + a*e^(1/3))*(-4*c^3*d^2 + c^2*(-6*b*d
^(5/3)*e^(1/3) + 6*a*d^(4/3)*e^(2/3)) + 12*a*b*c*d*e + e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^
(2/3) - a^3*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (10*(4*a*c^3*d^2*e^(1/3) + b^4*d^(4/3)*
e + 6*a^2*c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4/3)*e) - 4*b*(c
^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (5*(4*a*c^3*d^2*e^(1/3)
+ b^4*d^(4/3)*e + 6*a^2*c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4
/3)*e) - 4*b*(c^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^
2])/d^(2/3) + (10*(c^4*d^2 - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 + 4*c*e*(-(b^3*d) + a^3*e))*Log[d + e*x^3])/e^(1/3
))/(30*e^(8/3))

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Maple [B]  time = 0.007, size = 1339, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^2*b*c*d-4/e^2*3^(1/2)/(d/e)^(1/3)*arctan(
1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*b^2*c*d-4/3/e^3/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*b*c^3*d^2-1/e^2/(d/e)^(1/3)*l
n(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a^2*c^2*d-4/3/e^2/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a*b^3*d+4/3/e^3/(d/e)^(2/3)*l
n(x+(d/e)^(1/3))*a*c^3*d^2+2/e^3/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*b^2*c^2*d^2+2/3/e^2/(d/e)^(2/3)*ln(x^2-(d/e)^(1
/3)*x+(d/e)^(2/3))*a*b^3*d-2/3/e^3/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a*c^3*d^2-4/e^2/(d/e)^(2/3)*l
n(x+(d/e)^(1/3))*a^2*b*c*d+2/e^2/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a^2*b*c*d-4/3/e^2/(d/e)^(2/3)*3
^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*b^3*d+4/3/e^3/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)
^(1/3)*x-1))*a*c^3*d^2+2/e^3/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^2*c^2*d^2+4/e^2/(d/
e)^(1/3)*ln(x+(d/e)^(1/3))*a*b^2*c*d-2/e^2/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a*b^2*c*d-2/e^2*3^(1/
2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^2*c^2*d+4/3/e^3*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2
)*(2/(d/e)^(1/3)*x-1))*b*c^3*d^2+4/e*x^3*a*b*c^2+6/e*x^2*a*b^2*c-2/e^2*x^2*b*c^3*d-4/3/e/(d/e)^(1/3)*ln(x+(d/e
)^(1/3))*a^3*b+1/3/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*b^4*d+2/3/e/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))
*a^3*b+1/3/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^4-1/6/e^2/(d/e)^(1/3)*ln(x^2-(d/e)^
(1/3)*x+(d/e)^(2/3))*b^4*d-4/3/e^2*ln(e*x^3+d)*b^3*c*d-6/e^2*b^2*c^2*d*x+1/2/e*x^2*b^4+12/e*a^2*b*c*x-4/e^2*a*
c^3*d*x-1/e^3/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*b^2*c^2*d^2-4/e^2*ln(e*x^3+d)*a*b*c^2*d-1/3/e^2*3^
(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^4*d+4/3/e*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(
2/(d/e)^(1/3)*x-1))*a^3*b+1/6*c^4*x^6/e+1/e*x^4*a*c^3+3/2/e*x^4*b^2*c^2+4/3/e*x^3*b^3*c-1/3/e^2*x^3*c^4*d+3/e*
x^2*a^2*c^2+4/e*a*b^3*x+4/3/e*ln(e*x^3+d)*a^3*c+2/e*ln(e*x^3+d)*a^2*b^2+1/3/e^3*ln(e*x^3+d)*c^4*d^2+1/3/e/(d/e
)^(2/3)*ln(x+(d/e)^(1/3))*a^4-1/6/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a^4+2/3/e^3/(d/e)^(1/3)*ln(x
^2-(d/e)^(1/3)*x+(d/e)^(2/3))*b*c^3*d^2+2/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*a^2*c^2*d+4/5*b*c^3*x^5/e

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.08789, size = 1060, normalized size = 1.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c^4*d^2 - 4*b^3*c*d*e - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 + 4*a^3*c*e^2)*e^(-3)*log(abs(x^3*e + d)) + 1/3*sq
rt(3)*(6*(-d*e^2)^(1/3)*b^2*c^2*d^2*e + 4*(-d*e^2)^(1/3)*a*c^3*d^2*e - 4*(-d*e^2)^(2/3)*b*c^3*d^2 + (-d*e^2)^(
2/3)*b^4*d*e + 12*(-d*e^2)^(2/3)*a*b^2*c*d*e + 6*(-d*e^2)^(2/3)*a^2*c^2*d*e - 4*(-d*e^2)^(1/3)*a*b^3*d*e^2 - 1
2*(-d*e^2)^(1/3)*a^2*b*c*d*e^2 - 4*(-d*e^2)^(2/3)*a^3*b*e^2 + (-d*e^2)^(1/3)*a^4*e^3)*arctan(1/3*sqrt(3)*(2*x
+ (-d*e^(-1))^(1/3))/(-d*e^(-1))^(1/3))*e^(-4)/d - 1/3*(4*(-d*e^(-1))^(1/3)*b*c^3*d^2*e^11 + 6*b^2*c^2*d^2*e^1
1 + 4*a*c^3*d^2*e^11 - (-d*e^(-1))^(1/3)*b^4*d*e^12 - 12*(-d*e^(-1))^(1/3)*a*b^2*c*d*e^12 - 6*(-d*e^(-1))^(1/3
)*a^2*c^2*d*e^12 - 4*a*b^3*d*e^12 - 12*a^2*b*c*d*e^12 + 4*(-d*e^(-1))^(1/3)*a^3*b*e^13 + a^4*e^13)*(-d*e^(-1))
^(1/3)*e^(-13)*log(abs(x - (-d*e^(-1))^(1/3)))/d + 1/30*(5*c^4*x^6*e^5 + 24*b*c^3*x^5*e^5 + 45*b^2*c^2*x^4*e^5
+ 30*a*c^3*x^4*e^5 - 10*c^4*d*x^3*e^4 + 40*b^3*c*x^3*e^5 + 120*a*b*c^2*x^3*e^5 - 60*b*c^3*d*x^2*e^4 + 15*b^4*
x^2*e^5 + 180*a*b^2*c*x^2*e^5 + 90*a^2*c^2*x^2*e^5 - 180*b^2*c^2*d*x*e^4 - 120*a*c^3*d*x*e^4 + 120*a*b^3*x*e^5
+ 360*a^2*b*c*x*e^5)*e^(-6) + 1/6*(6*(-d*e^2)^(1/3)*b^2*c^2*d^2*e + 4*(-d*e^2)^(1/3)*a*c^3*d^2*e + 4*(-d*e^2)
^(2/3)*b*c^3*d^2 - (-d*e^2)^(2/3)*b^4*d*e - 12*(-d*e^2)^(2/3)*a*b^2*c*d*e - 6*(-d*e^2)^(2/3)*a^2*c^2*d*e - 4*(
-d*e^2)^(1/3)*a*b^3*d*e^2 - 12*(-d*e^2)^(1/3)*a^2*b*c*d*e^2 + 4*(-d*e^2)^(2/3)*a^3*b*e^2 + (-d*e^2)^(1/3)*a^4*
e^3)*e^(-4)*log(x^2 + (-d*e^(-1))^(1/3)*x + (-d*e^(-1))^(2/3))/d