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

Optimal. Leaf size=248 \[ -\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2} \]

[Out]

-(c + d*x + e*x^2)/(9*b*(a + b*x^3)^3) + (x*(d + 2*e*x))/(54*a*b*(a + b*x^3)^2) + (x*(5*d + 8*e*x))/(162*a^2*b
*(a + b*x^3)) - ((5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^
(8/3)*b^(5/3)) + ((5*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(8/3)*b^(5/3)) - ((5*b^(1/3)*d
- 4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^(8/3)*b^(5/3))

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Rubi [A]  time = 0.242314, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {1823, 1855, 1860, 31, 634, 617, 204, 628} \[ -\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c + d*x + e*x^2)/(9*b*(a + b*x^3)^3) + (x*(d + 2*e*x))/(54*a*b*(a + b*x^3)^2) + (x*(5*d + 8*e*x))/(162*a^2*b
*(a + b*x^3)) - ((5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^
(8/3)*b^(5/3)) + ((5*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(8/3)*b^(5/3)) - ((5*b^(1/3)*d
- 4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^(8/3)*b^(5/3))

Rule 1823

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Pq*(a + b*x^n)^(p + 1))/(b*n*(p + 1)),
x] - Dist[1/(b*n*(p + 1)), Int[D[Pq, x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x]
&& EqQ[m - n + 1, 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 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]

Rubi steps

\begin{align*} \int \frac{x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{\int \frac{d+2 e x}{\left (a+b x^3\right )^3} \, dx}{9 b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}-\frac{\int \frac{-5 d-8 e x}{\left (a+b x^3\right )^2} \, dx}{54 a b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\int \frac{10 d+8 e x}{a+b x^3} \, dx}{162 a^2 b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\int \frac{\sqrt [3]{a} \left (20 \sqrt [3]{b} d+8 \sqrt [3]{a} e\right )+\sqrt [3]{b} \left (-10 \sqrt [3]{b} d+8 \sqrt [3]{a} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{8/3} b^{4/3}}+\frac{\left (5 d-\frac{4 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{8/3} b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\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}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{162 a^{7/3} b^{4/3}}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{8/3} b^{5/3}}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{\left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}\\ \end{align*}

Mathematica [A]  time = 0.232971, size = 230, normalized size = 0.93 \[ \frac{\frac{3 b^{2/3} x (5 d+8 e x)}{a^2 \left (a+b x^3\right )}+\frac{\left (4 \sqrt [3]{a} e-5 \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}+\frac{2 \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}-\frac{2 \sqrt{3} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{8/3}}-\frac{54 b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac{9 b^{2/3} x (d+2 e x)}{a \left (a+b x^3\right )^2}}{486 b^{5/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((9*b^(2/3)*x*(d + 2*e*x))/(a*(a + b*x^3)^2) + (3*b^(2/3)*x*(5*d + 8*e*x))/(a^2*(a + b*x^3)) - (54*b^(2/3)*(c
+ x*(d + e*x)))/(a + b*x^3)^3 - (2*Sqrt[3]*(5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt
[3]])/a^(8/3) + (2*(5*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) + ((-5*b^(1/3)*d + 4*a^(1/3)*
e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3))/(486*b^(5/3))

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Maple [A]  time = 0.01, size = 275, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{3}} \left ({\frac{4\,be{x}^{8}}{81\,{a}^{2}}}+{\frac{5\,bd{x}^{7}}{162\,{a}^{2}}}+{\frac{11\,e{x}^{5}}{81\,a}}+{\frac{13\,d{x}^{4}}{162\,a}}-{\frac{2\,e{x}^{2}}{81\,b}}-{\frac{5\,dx}{81\,b}}-{\frac{c}{9\,b}} \right ) }+{\frac{5\,d}{243\,{b}^{2}{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,d}{486\,{b}^{2}{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,d\sqrt{3}}{243\,{b}^{2}{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,e}{243\,{b}^{2}{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,e}{243\,{b}^{2}{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{4\,e\sqrt{3}}{243\,{b}^{2}{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4/81/a^2*b*e*x^8+5/162/a^2*d*b*x^7+11/81/a*e*x^5+13/162*d/a*x^4-2/81*e*x^2/b-5/81*d*x/b-1/9*c/b)/(b*x^3+a)^3+
5/243/b^2/a^2*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-5/486/b^2/a^2*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)
^(2/3))+5/243/b^2/a^2*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-4/243/b^2/a^2*e/(1/b*a
)^(1/3)*ln(x+(1/b*a)^(1/3))+2/243/b^2/a^2*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+4/243/b^2/a^2*
e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 8.78528, size = 5952, normalized size = 24. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/972*(48*b^2*e*x^8 + 30*b^2*d*x^7 + 132*a*b*e*x^5 + 78*a*b*d*x^4 - 24*a^2*e*x^2 - 60*a^2*d*x - 108*a^2*c - 2*
(a^2*b^4*x^9 + 3*a^3*b^3*x^6 + 3*a^4*b^2*x^3 + a^5*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^
8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 +
64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))*log(((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 +
64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*
((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^6*b^3*e - 25/2*((1/2)^(1/3)*
(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d
*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))*a^3
*b^2*d^2 + 160*a*d*e^2 + (125*b*d^3 + 64*a*e^3)*x) + ((a^2*b^4*x^9 + 3*a^3*b^3*x^6 + 3*a^4*b^2*x^3 + a^5*b)*((
1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1
/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^
(1/3))) + 3*sqrt(1/3)*(a^2*b^4*x^9 + 3*a^3*b^3*x^6 + 3*a^4*b^2*x^3 + a^5*b)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1
)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3)
+ 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^5*b^3 + 320*d
*e)/(a^5*b^3)))*log(-((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(
a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3
- 64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^6*b^3*e + 25/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^
5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a
*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))*a^3*b^2*d^2 - 160*a*d*e^2 + 2*(125*b*d^3 + 64*a*e^
3)*x + 3/2*sqrt(1/3)*(2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3
)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d
^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))*a^6*b^3*e + 25*a^3*b^2*d^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3
+ 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3
*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^5*b^3 + 320*d*e)/(a^5*b^3))
) + ((a^2*b^4*x^9 + 3*a^3*b^3*x^6 + 3*a^4*b^2*x^3 + a^5*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3
)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d
^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3))) - 3*sqrt(1/3)*(a^2*b^4*x^9 + 3*a^3*b^3*x^
6 + 3*a^4*b^2*x^3 + a^5*b)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3
- 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5)
+ (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^5*b^3 + 320*d*e)/(a^5*b^3)))*log(-((1/2)^(1/3)*(I*sqrt(3) + 1)
*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3)
+ 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^6*b^3*e + 25/2
*((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40
*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5
))^(1/3)))*a^3*b^2*d^2 - 160*a*d*e^2 + 2*(125*b*d^3 + 64*a*e^3)*x - 3/2*sqrt(1/3)*(2*((1/2)^(1/3)*(I*sqrt(3) +
1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(
3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a^8*b^5))^(1/3)))*a^6*b^3*e + 25*
a^3*b^2*d^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 - 64*a*e^3)/(a
^8*b^5))^(1/3) - 40*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^5*b^3*((125*b*d^3 + 64*a*e^3)/(a^8*b^5) + (125*b*d^3 -
64*a*e^3)/(a^8*b^5))^(1/3)))^2*a^5*b^3 + 320*d*e)/(a^5*b^3))))/(a^2*b^4*x^9 + 3*a^3*b^3*x^6 + 3*a^4*b^2*x^3 +
a^5*b)

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Sympy [A]  time = 13.5946, size = 201, normalized size = 0.81 \begin{align*} \operatorname{RootSum}{\left (14348907 t^{3} a^{8} b^{5} + 14580 t a^{3} b^{2} d e + 64 a e^{3} - 125 b d^{3}, \left ( t \mapsto t \log{\left (x + \frac{236196 t^{2} a^{6} b^{3} e + 6075 t a^{3} b^{2} d^{2} + 160 a d e^{2}}{64 a e^{3} + 125 b d^{3}} \right )} \right )\right )} + \frac{- 18 a^{2} c - 10 a^{2} d x - 4 a^{2} e x^{2} + 13 a b d x^{4} + 22 a b e x^{5} + 5 b^{2} d x^{7} + 8 b^{2} e x^{8}}{162 a^{5} b + 486 a^{4} b^{2} x^{3} + 486 a^{3} b^{3} x^{6} + 162 a^{2} b^{4} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(14348907*_t**3*a**8*b**5 + 14580*_t*a**3*b**2*d*e + 64*a*e**3 - 125*b*d**3, Lambda(_t, _t*log(x + (236
196*_t**2*a**6*b**3*e + 6075*_t*a**3*b**2*d**2 + 160*a*d*e**2)/(64*a*e**3 + 125*b*d**3)))) + (-18*a**2*c - 10*
a**2*d*x - 4*a**2*e*x**2 + 13*a*b*d*x**4 + 22*a*b*e*x**5 + 5*b**2*d*x**7 + 8*b**2*e*x**8)/(162*a**5*b + 486*a*
*4*b**2*x**3 + 486*a**3*b**3*x**6 + 162*a**2*b**4*x**9)

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Giac [A]  time = 1.07474, size = 333, normalized size = 1.34 \begin{align*} -\frac{{\left (4 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{3} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} 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 )}{243 \, a^{3} b^{3}} + \frac{8 \, b^{2} x^{8} e + 5 \, b^{2} d x^{7} + 22 \, a b x^{5} e + 13 \, a b d x^{4} - 4 \, a^{2} x^{2} e - 10 \, a^{2} d x - 18 \, a^{2} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{2} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a 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 )}{486 \, a^{4} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243*(4*(-a/b)^(1/3)*e + 5*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b) + 1/243*sqrt(3)*(5*(-a*b^2)^(1
/3)*b*d - 4*(-a*b^2)^(2/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^3) + 1/162*(8*b^2*x
^8*e + 5*b^2*d*x^7 + 22*a*b*x^5*e + 13*a*b*d*x^4 - 4*a^2*x^2*e - 10*a^2*d*x - 18*a^2*c)/((b*x^3 + a)^3*a^2*b)
+ 1/486*(5*(-a*b^2)^(1/3)*a*b^2*d + 4*(-a*b^2)^(2/3)*a*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^4)