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

Optimal. Leaf size=270 \[ \frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}-\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{10/3} b^{4/3}}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3} \]

[Out]

-(x*(a*e - b*c*x - b*d*x^2))/(9*a*b*(a + b*x^3)^3) + (x*(5*a*e + 28*b*c*x))/(162*a^3*b*(a + b*x^3)) - (6*a*d -
x*(a*e + 7*b*c*x))/(54*a^2*b*(a + b*x^3)^2) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(S
qrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(10/3)*b^(4/3)) - ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243
*a^(10/3)*b^(4/3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^(10/
3)*b^(4/3))

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Rubi [A]  time = 0.25355, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1828, 1854, 1855, 1860, 31, 634, 617, 204, 628} \[ \frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}-\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{10/3} b^{4/3}}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(a*e - b*c*x - b*d*x^2))/(9*a*b*(a + b*x^3)^3) + (x*(5*a*e + 28*b*c*x))/(162*a^3*b*(a + b*x^3)) - (6*a*d -
x*(a*e + 7*b*c*x))/(54*a^2*b*(a + b*x^3)^2) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(S
qrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(10/3)*b^(4/3)) - ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243
*a^(10/3)*b^(4/3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^(10/
3)*b^(4/3))

Rule 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q
- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rule 1854

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 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 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}-\frac{\int \frac{-a e-7 b c x-6 b d x^2}{\left (a+b x^3\right )^3} \, dx}{9 a b}\\ &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}+\frac{\int \frac{5 a e+28 b c x}{\left (a+b x^3\right )^2} \, dx}{54 a^2 b}\\ &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{\int \frac{-10 a e-28 b c x}{a+b x^3} \, dx}{162 a^3 b}\\ &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{\int \frac{\sqrt [3]{a} \left (-28 \sqrt [3]{a} b c-20 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-28 \sqrt [3]{a} b c+10 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{11/3} b^{4/3}}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{10/3} b}\\ &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} 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^{10/3} b^{4/3}}+\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{162 a^3 b}\\ &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}+\frac{\left (14 b^{2/3} c+5 a^{2/3} 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^{10/3} b^{4/3}}\\ &=-\frac{x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{10/3} b^{4/3}}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.345652, size = 241, normalized size = 0.89 \[ \frac{\frac{3 a b^{2/3} \left (a^2 b x^2 \left (67 c+13 e x^2\right )-2 a^3 (9 d+5 e x)+a b^2 x^5 \left (77 c+5 e x^2\right )+28 b^3 c x^8\right )}{\left (a+b x^3\right )^3}+a^{2/3} \sqrt [3]{b} \left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt{3} a^{2/3} \sqrt [3]{b} \left (5 a^{2/3} e+14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+2 \left (5 a^{4/3} \sqrt [3]{b} e-14 a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{486 a^4 b^{5/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a*b^(2/3)*(28*b^3*c*x^8 - 2*a^3*(9*d + 5*e*x) + a*b^2*x^5*(77*c + 5*e*x^2) + a^2*b*x^2*(67*c + 13*e*x^2)))
/(a + b*x^3)^3 - 2*Sqrt[3]*a^(2/3)*b^(1/3)*(14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqr
t[3]] + 2*(-14*a^(2/3)*b*c + 5*a^(4/3)*b^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x] + a^(2/3)*b^(1/3)*(14*b^(2/3)*c - 5
*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^4*b^(5/3))

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Maple [A]  time = 0.012, size = 278, normalized size = 1. \begin{align*}{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{3}} \left ({\frac{14\,{b}^{2}c{x}^{8}}{81\,{a}^{3}}}+{\frac{5\,be{x}^{7}}{162\,{a}^{2}}}+{\frac{77\,bc{x}^{5}}{162\,{a}^{2}}}+{\frac{13\,e{x}^{4}}{162\,a}}+{\frac{67\,c{x}^{2}}{162\,a}}-{\frac{5\,ex}{81\,b}}-{\frac{d}{9\,b}} \right ) }+{\frac{5\,e}{243\,{b}^{2}{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,e}{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\,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 ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,c}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,c}{243\,{a}^{3}b}\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{14\,c\sqrt{3}}{243\,{a}^{3}b}\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*(e*x^2+d*x+c)/(b*x^3+a)^4,x)

[Out]

(14/81*c/a^3*b^2*x^8+5/162/a^2*b*e*x^7+77/162/a^2*c*b*x^5+13/162/a*e*x^4+67/162*c/a*x^2-5/81*e*x/b-1/9/b*d)/(b
*x^3+a)^3+5/243/b^2/a^2*e/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-5/486/b^2/a^2*e/(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*e/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-14/243/b/a^3
*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+7/243/b/a^3*c/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+14/243/
b/a^3*c*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*(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 = 8.83698, size = 6597, normalized size = 24.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/972*(168*b^3*c*x^8 + 30*a*b^2*e*x^7 + 462*a*b^2*c*x^5 + 78*a^2*b*e*x^4 + 402*a^2*b*c*x^2 - 60*a^3*e*x - 108*
a^3*d - 2*(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 +
125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1
)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))*log(7/2
*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^
4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2
*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))^2*a^7*b^3*c - 25/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a
^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^
6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))*a^5*b*e^2 +
1960*a*b*c^2*e + (2744*b^2*c^3 + 125*a^2*e^3)*x) + ((a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*((1/
2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(
1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3
- 125*a^2*e^3)/(a^10*b^4))^(1/3))) + 3*sqrt(1/3)*(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*sqrt(-(
((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4
))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*
c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))^2*a^6*b^2 + 1120*c*e)/(a^6*b^2)))*log(-7/2*((1/2)^(1/3)*(I*sqrt(3) + 1)
*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c
*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4
))^(1/3)))^2*a^7*b^3*c + 25/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^
2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^
2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))*a^5*b*e^2 - 1960*a*b*c^2*e + 2*(2744*b^2*
c^3 + 125*a^2*e^3)*x + 3/2*sqrt(1/3)*(7*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4)
- (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^
3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))*a^7*b^3*c + 25*a^5*b*e^2)*sqrt(
-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b
^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^
2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))^2*a^6*b^2 + 1120*c*e)/(a^6*b^2))) + ((a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3
*a^5*b^2*x^3 + a^6*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 -
125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(
a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3))) - 3*sqrt(1/3)*(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^
5*b^2*x^3 + a^6*b)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^
3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^
3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))^2*a^6*b^2 + 1120*c*e)/(a^6*b^2)))*log(-7/2*((
1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))
^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^
3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))^2*a^7*b^3*c + 25/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*
e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b
^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))*a^5*b*e^2 - 196
0*a*b*c^2*e + 2*(2744*b^2*c^3 + 125*a^2*e^3)*x - 3/2*sqrt(1/3)*(7*((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3
+ 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) +
1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))*a^7*b
^3*c + 25*a^5*b*e^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((2744*b^2*c^3 + 125*a^2*e^3)/(a^10*b^4) - (2744*b^2*
c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3) - 140*(1/2)^(2/3)*c*e*(-I*sqrt(3) + 1)/(a^6*b^2*((2744*b^2*c^3 + 125*a^2*
e^3)/(a^10*b^4) - (2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b^4))^(1/3)))^2*a^6*b^2 + 1120*c*e)/(a^6*b^2))))/(a^3*b^4
*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)

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Sympy [A]  time = 7.04865, size = 214, normalized size = 0.79 \begin{align*} \operatorname{RootSum}{\left (14348907 t^{3} a^{10} b^{4} + 51030 t a^{4} b^{2} c e - 125 a^{2} e^{3} + 2744 b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{826686 t^{2} a^{7} b^{3} c + 6075 t a^{5} b e^{2} + 1960 a b c^{2} e}{125 a^{2} e^{3} + 2744 b^{2} c^{3}} \right )} \right )\right )} + \frac{- 18 a^{3} d - 10 a^{3} e x + 67 a^{2} b c x^{2} + 13 a^{2} b e x^{4} + 77 a b^{2} c x^{5} + 5 a b^{2} e x^{7} + 28 b^{3} c x^{8}}{162 a^{6} b + 486 a^{5} b^{2} x^{3} + 486 a^{4} b^{3} x^{6} + 162 a^{3} b^{4} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(14348907*_t**3*a**10*b**4 + 51030*_t*a**4*b**2*c*e - 125*a**2*e**3 + 2744*b**2*c**3, Lambda(_t, _t*log
(x + (826686*_t**2*a**7*b**3*c + 6075*_t*a**5*b*e**2 + 1960*a*b*c**2*e)/(125*a**2*e**3 + 2744*b**2*c**3)))) +
(-18*a**3*d - 10*a**3*e*x + 67*a**2*b*c*x**2 + 13*a**2*b*e*x**4 + 77*a*b**2*c*x**5 + 5*a*b**2*e*x**7 + 28*b**3
*c*x**8)/(162*a**6*b + 486*a**5*b**2*x**3 + 486*a**4*b**3*x**6 + 162*a**3*b**4*x**9)

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Giac [A]  time = 1.08782, size = 346, normalized size = 1.28 \begin{align*} -\frac{{\left (14 \, b c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a e\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^{4} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\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^{4} b^{2}} + \frac{28 \, b^{3} c x^{8} + 5 \, a b^{2} x^{7} e + 77 \, a b^{2} c x^{5} + 13 \, a^{2} b x^{4} e + 67 \, a^{2} b c x^{2} - 10 \, a^{3} x e - 18 \, a^{3} d}{162 \,{\left (b x^{3} + a\right )}^{3} a^{3} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\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*(e*x^2+d*x+c)/(b*x^3+a)^4,x, algorithm="giac")

[Out]

-1/243*(14*b*c*(-a/b)^(1/3) + 5*a*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b) + 1/243*sqrt(3)*(5*(-a*b^
2)^(1/3)*a*e - 14*(-a*b^2)^(2/3)*c)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/162*(2
8*b^3*c*x^8 + 5*a*b^2*x^7*e + 77*a*b^2*c*x^5 + 13*a^2*b*x^4*e + 67*a^2*b*c*x^2 - 10*a^3*x*e - 18*a^3*d)/((b*x^
3 + a)^3*a^3*b) + 1/486*(5*(-a*b^2)^(1/3)*a*b^2*e + 14*(-a*b^2)^(2/3)*b^2*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)
^(2/3))/(a^4*b^4)