3.405 \(\int \frac{x^2 (c+d x+e x^2+f x^3+g x^4+h x^5)}{a+b x^3} \, dx\)

Optimal. Leaf size=294 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.976263, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]

[Out]

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

Rule 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

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{x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx &=\frac{h x^5}{5 b}+\frac{\int \frac{x^2 \left (5 b c+5 b d x+5 (b e-a h) x^2+5 b f x^3+5 b g x^4\right )}{a+b x^3} \, dx}{5 b}\\ &=\frac{g x^4}{4 b}+\frac{h x^5}{5 b}+\frac{\int \frac{x^2 \left (20 b^2 c+20 b (b d-a g) x+20 b (b e-a h) x^2+20 b^2 f x^3\right )}{a+b x^3} \, dx}{20 b^2}\\ &=\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}+\frac{\int \frac{x^2 \left (60 b^2 (b c-a f)+60 b^2 (b d-a g) x+60 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{60 b^3}\\ &=\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}+\frac{\int \left (60 b (b d-a g)+60 b (b e-a h) x-\frac{60 \left (a b (b d-a g)+a b (b e-a h) x-b^2 (b c-a f) x^2\right )}{a+b x^3}\right ) \, dx}{60 b^3}\\ &=\frac{(b d-a g) x}{b^2}+\frac{(b e-a h) x^2}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}-\frac{\int \frac{a b (b d-a g)+a b (b e-a h) x-b^2 (b c-a f) x^2}{a+b x^3} \, dx}{b^3}\\ &=\frac{(b d-a g) x}{b^2}+\frac{(b e-a h) x^2}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}-\frac{\int \frac{a b (b d-a g)+a b (b e-a h) x}{a+b x^3} \, dx}{b^3}+\frac{(b c-a f) \int \frac{x^2}{a+b x^3} \, dx}{b}\\ &=\frac{(b d-a g) x}{b^2}+\frac{(b e-a h) x^2}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac{\int \frac{\sqrt [3]{a} \left (2 a b^{4/3} (b d-a g)+a^{4/3} b (b e-a h)\right )+\sqrt [3]{b} \left (-a b^{4/3} (b d-a g)+a^{4/3} b (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^{10/3}}-\frac{\left (\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{7/3}}\\ &=\frac{(b d-a g) x}{b^2}+\frac{(b e-a h) x^2}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac{\left (a^{2/3} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{7/3}}+\frac{\left (\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{8/3}}\\ &=\frac{(b d-a g) x}{b^2}+\frac{(b e-a h) x^2}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac{\left (\sqrt [3]{a} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{8/3}}\\ &=\frac{(b d-a g) x}{b^2}+\frac{(b e-a h) x^2}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}+\frac{\sqrt [3]{a} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.26219, size = 290, normalized size = 0.99 \[ \frac{10 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+20 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )-20 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )+20 b^{2/3} (b c-a f) \log \left (a+b x^3\right )+60 b^{2/3} x (b d-a g)+30 b^{2/3} x^2 (b e-a h)+20 b^{5/3} f x^3+15 b^{5/3} g x^4+12 b^{5/3} h x^5}{60 b^{8/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]

[Out]

(60*b^(2/3)*(b*d - a*g)*x + 30*b^(2/3)*(b*e - a*h)*x^2 + 20*b^(5/3)*f*x^3 + 15*b^(5/3)*g*x^4 + 12*b^(5/3)*h*x^
5 - 20*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3
))/Sqrt[3]] + 20*a^(1/3)*(-(b^(4/3)*d) + a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x] + 10*
a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2
0*b^(2/3)*(b*c - a*f)*Log[a + b*x^3])/(60*b^(8/3))

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Maple [B]  time = 0.003, size = 483, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x)

[Out]

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

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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(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),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(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 63.1816, size = 789, normalized size = 2.68 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{8} + t^{2} \left (27 a b^{6} f - 27 b^{7} c\right ) + t \left (9 a^{3} b^{3} g h - 9 a^{2} b^{4} d h - 9 a^{2} b^{4} e g + 9 a^{2} b^{4} f^{2} - 18 a b^{5} c f + 9 a b^{5} d e + 9 b^{6} c^{2}\right ) + a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{4} b f g h - a^{4} b g^{3} - 3 a^{3} b^{2} c g h - 3 a^{3} b^{2} d f h + 3 a^{3} b^{2} d g^{2} + 3 a^{3} b^{2} e^{2} h - 3 a^{3} b^{2} e f g + a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c d h + 3 a^{2} b^{3} c e g - 3 a^{2} b^{3} c f^{2} - 3 a^{2} b^{3} d^{2} g + 3 a^{2} b^{3} d e f - a^{2} b^{3} e^{3} + 3 a b^{4} c^{2} f - 3 a b^{4} c d e + a b^{4} d^{3} - b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a b^{5} h - 9 t^{2} b^{6} e + 6 t a^{2} b^{3} f h + 3 t a^{2} b^{3} g^{2} - 6 t a b^{4} c h - 6 t a b^{4} d g - 6 t a b^{4} e f + 6 t b^{5} c e + 3 t b^{5} d^{2} + 2 a^{4} g h^{2} - 2 a^{3} b d h^{2} - 4 a^{3} b e g h + a^{3} b f^{2} h + a^{3} b f g^{2} - 2 a^{2} b^{2} c f h - a^{2} b^{2} c g^{2} + 4 a^{2} b^{2} d e h - 2 a^{2} b^{2} d f g + 2 a^{2} b^{2} e^{2} g - a^{2} b^{2} e f^{2} + a b^{3} c^{2} h + 2 a b^{3} c d g + 2 a b^{3} c e f + a b^{3} d^{2} f - 2 a b^{3} d e^{2} - b^{4} c^{2} e - b^{4} c d^{2}}{a^{4} h^{3} - 3 a^{3} b e h^{2} + a^{3} b g^{3} - 3 a^{2} b^{2} d g^{2} + 3 a^{2} b^{2} e^{2} h + 3 a b^{3} d^{2} g - a b^{3} e^{3} - b^{4} d^{3}} \right )} \right )\right )} + \frac{f x^{3}}{3 b} + \frac{g x^{4}}{4 b} + \frac{h x^{5}}{5 b} - \frac{x^{2} \left (a h - b e\right )}{2 b^{2}} - \frac{x \left (a g - b d\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**8 + _t**2*(27*a*b**6*f - 27*b**7*c) + _t*(9*a**3*b**3*g*h - 9*a**2*b**4*d*h - 9*a**2*b**4*
e*g + 9*a**2*b**4*f**2 - 18*a*b**5*c*f + 9*a*b**5*d*e + 9*b**6*c**2) + a**5*h**3 - 3*a**4*b*e*h**2 + 3*a**4*b*
f*g*h - a**4*b*g**3 - 3*a**3*b**2*c*g*h - 3*a**3*b**2*d*f*h + 3*a**3*b**2*d*g**2 + 3*a**3*b**2*e**2*h - 3*a**3
*b**2*e*f*g + a**3*b**2*f**3 + 3*a**2*b**3*c*d*h + 3*a**2*b**3*c*e*g - 3*a**2*b**3*c*f**2 - 3*a**2*b**3*d**2*g
+ 3*a**2*b**3*d*e*f - a**2*b**3*e**3 + 3*a*b**4*c**2*f - 3*a*b**4*c*d*e + a*b**4*d**3 - b**5*c**3, Lambda(_t,
_t*log(x + (9*_t**2*a*b**5*h - 9*_t**2*b**6*e + 6*_t*a**2*b**3*f*h + 3*_t*a**2*b**3*g**2 - 6*_t*a*b**4*c*h -
6*_t*a*b**4*d*g - 6*_t*a*b**4*e*f + 6*_t*b**5*c*e + 3*_t*b**5*d**2 + 2*a**4*g*h**2 - 2*a**3*b*d*h**2 - 4*a**3*
b*e*g*h + a**3*b*f**2*h + a**3*b*f*g**2 - 2*a**2*b**2*c*f*h - a**2*b**2*c*g**2 + 4*a**2*b**2*d*e*h - 2*a**2*b*
*2*d*f*g + 2*a**2*b**2*e**2*g - a**2*b**2*e*f**2 + a*b**3*c**2*h + 2*a*b**3*c*d*g + 2*a*b**3*c*e*f + a*b**3*d*
*2*f - 2*a*b**3*d*e**2 - b**4*c**2*e - b**4*c*d**2)/(a**4*h**3 - 3*a**3*b*e*h**2 + a**3*b*g**3 - 3*a**2*b**2*d
*g**2 + 3*a**2*b**2*e**2*h + 3*a*b**3*d**2*g - a*b**3*e**3 - b**4*d**3)))) + f*x**3/(3*b) + g*x**4/(4*b) + h*x
**5/(5*b) - x**2*(a*h - b*e)/(2*b**2) - x*(a*g - b*d)/b**2

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Giac [A]  time = 1.08309, size = 450, normalized size = 1.53 \begin{align*} \frac{{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} + \frac{12 \, b^{4} h x^{5} + 15 \, b^{4} g x^{4} + 20 \, b^{4} f x^{3} - 30 \, a b^{3} h x^{2} + 30 \, b^{4} x^{2} e + 60 \, b^{4} d x - 60 \, a b^{3} g x}{60 \, b^{5}} - \frac{{\left (a^{2} b^{9} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{10} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - a b^{10} d + a^{2} b^{9} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="giac")

[Out]

1/3*(b*c - a*f)*log(abs(b*x^3 + a))/b^2 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^2*d - (-a*b^2)^(1/3)*a*b*g + (-a*b^2)^
(2/3)*a*h - (-a*b^2)^(2/3)*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)^(1/3
)*b^2*d - (-a*b^2)^(1/3)*a*b*g - (-a*b^2)^(2/3)*a*h + (-a*b^2)^(2/3)*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/b^4 + 1/60*(12*b^4*h*x^5 + 15*b^4*g*x^4 + 20*b^4*f*x^3 - 30*a*b^3*h*x^2 + 30*b^4*x^2*e + 60*b^4*d*x - 60*
a*b^3*g*x)/b^5 - 1/3*(a^2*b^9*h*(-a/b)^(1/3) - a*b^10*(-a/b)^(1/3)*e - a*b^10*d + a^2*b^9*g)*(-a/b)^(1/3)*log(
abs(x - (-a/b)^(1/3)))/(a*b^11)