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

Optimal. Leaf size=275 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.921478, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*e - a*h)*x)/b^2 + (f*x^2)/(2*b) + (g*x^3)/(3*b) + (h*x^4)/(4*b) - ((b^(5/3)*c - a^(2/3)*b*e - a*b^(2/3)*f
+ a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*b^(7/3)) - ((b^(2/3)*(b*c - a
*f) + a^(2/3)*(b*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*b^(7/3)) + ((b^(2/3)*(b*c - a*f) + a^(2/3)*(b*
e - a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(1/3)*b^(7/3)) + ((b*d - a*g)*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 \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^4}{4 b}+\frac{\int \frac{x \left (4 b c+4 b d x+4 (b e-a h) x^2+4 b f x^3+4 b g x^4\right )}{a+b x^3} \, dx}{4 b}\\ &=\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{\int \frac{x \left (12 b^2 c+12 b (b d-a g) x+12 b (b e-a h) x^2+12 b^2 f x^3\right )}{a+b x^3} \, dx}{12 b^2}\\ &=\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{\int \frac{x \left (24 b^2 (b c-a f)+24 b^2 (b d-a g) x+24 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{24 b^3}\\ &=\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{\int \left (24 b (b e-a h)-\frac{24 \left (a b (b e-a h)-b^2 (b c-a f) x-b^2 (b d-a g) x^2\right )}{a+b x^3}\right ) \, dx}{24 b^3}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\int \frac{a b (b e-a h)-b^2 (b c-a f) x-b^2 (b d-a g) x^2}{a+b x^3} \, dx}{b^3}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\int \frac{a b (b e-a h)-b^2 (b c-a f) x}{a+b x^3} \, dx}{b^3}+\frac{(b d-a g) \int \frac{x^2}{a+b x^3} \, dx}{b}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}-\frac{\int \frac{\sqrt [3]{a} \left (-\sqrt [3]{a} b^2 (b c-a f)+2 a b^{4/3} (b e-a h)\right )+\sqrt [3]{b} \left (-\sqrt [3]{a} b^2 (b c-a f)-a b^{4/3} (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 (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} b^2}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^2}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\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 \sqrt [3]{a} b^{7/3}}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{7/3}}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}-\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.372748, size = 272, normalized size = 0.99 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{\sqrt [3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+4 \sqrt [3]{b} (b d-a g) \log \left (a+b x^3\right )+12 \sqrt [3]{b} x (b e-a h)+6 b^{4/3} f x^2+4 b^{4/3} g x^3+3 b^{4/3} h x^4}{12 b^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 455, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*h*x^4/b+1/3*g*x^3/b+1/2*f*x^2/b-1/b^2*a*h*x+e*x/b+1/3/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*a^2*h-1/3/b^2*
a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*e-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*h+1/6/b^
2*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e+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*h-1/3/b^2*a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+1/3/b^2*
a/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*f-1/3/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c-1/6/b^2*a/(1/b*a)^(1/3)*ln(x^2
-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f+1/6/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c-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))*f+1/3/b*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/
(1/b*a)^(1/3)*x-1))*c-1/3/b^2*ln(b*x^3+a)*a*g+1/3*d*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*(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*(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 = 33.0156, size = 811, normalized size = 2.95 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a b^{7} + t^{2} \left (27 a^{2} b^{5} g - 27 a b^{6} d\right ) + t \left (- 9 a^{3} b^{3} f h + 9 a^{3} b^{3} g^{2} + 9 a^{2} b^{4} c h - 18 a^{2} b^{4} d g + 9 a^{2} b^{4} e f - 9 a b^{5} c e + 9 a b^{5} d^{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^{2} b^{5} f + 9 t^{2} a b^{6} c + 3 t a^{4} b^{2} h^{2} - 6 t a^{3} b^{3} e h - 6 t a^{3} b^{3} f g + 6 t a^{2} b^{4} c g + 6 t a^{2} b^{4} d f + 3 t a^{2} b^{4} e^{2} - 6 t a b^{5} c d + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h + 2 a^{4} b f^{2} h - a^{4} b f g^{2} - 4 a^{3} b^{2} c f h + a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h + 2 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g - 2 a^{3} b^{2} e f^{2} + 2 a^{2} b^{3} c^{2} h - 2 a^{2} b^{3} c d g + 4 a^{2} b^{3} c e f - a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} - 2 a b^{4} c^{2} e + a b^{4} c d^{2}}{a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{3} b^{2} e^{2} h - a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c f^{2} - a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{2}}{2 b} + \frac{g x^{3}}{3 b} + \frac{h x^{4}}{4 b} - \frac{x \left (a h - b e\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]  time = 1.08277, size = 428, normalized size = 1.56 \begin{align*} \frac{{\left (b d - a g\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}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\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 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} + \frac{3 \, b^{3} h x^{4} + 4 \, b^{3} g x^{3} + 6 \, b^{3} f x^{2} - 12 \, a b^{2} h x + 12 \, b^{3} x e}{12 \, b^{4}} - \frac{{\left (b^{9} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{8} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{7} h - a b^{8} e\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^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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*d - a*g)*log(abs(b*x^3 + a))/b^2 + 1/3*sqrt(3)*((-a*b^2)^(1/3)*a^2*h - (-a*b^2)^(1/3)*a*b*e - (-a*b^2)^
(2/3)*b*c + (-a*b^2)^(2/3)*a*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3) + 1/6*((-a*b^2)^
(1/3)*a^2*h - (-a*b^2)^(1/3)*a*b*e + (-a*b^2)^(2/3)*b*c - (-a*b^2)^(2/3)*a*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b
)^(2/3))/(a*b^3) + 1/12*(3*b^3*h*x^4 + 4*b^3*g*x^3 + 6*b^3*f*x^2 - 12*a*b^2*h*x + 12*b^3*x*e)/b^4 - 1/3*(b^9*c
*(-a/b)^(1/3) - a*b^8*f*(-a/b)^(1/3) + a^2*b^7*h - a*b^8*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^9)