\(\int \frac {x (c+d x+e x^2)}{(a+b x^3)^4} \, dx\) [359]

Optimal result

Integrand size = 21, antiderivative size = 270 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=-\frac {x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3}+\frac {x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac {6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{10/3} b^{4/3}}-\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}} \]

[Out]

-1/9*x*(-b*d*x^2-b*c*x+a*e)/a/b/(b*x^3+a)^3+1/162*x*(28*b*c*x+5*a*e)/a^3/b/(b*x^3+a)+1/54*(-6*a*d+x*(7*b*c*x+a
*e))/a^2/b/(b*x^3+a)^2-1/243*(14*b^(2/3)*c-5*a^(2/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^(4/3)+1/486*(14*b^(2/
3)*c-5*a^(2/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/b^(4/3)-1/243*(14*b^(2/3)*c+5*a^(2/3)*e)*
arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(10/3)/b^(4/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1842, 1868, 1869, 1874, 31, 648, 631, 210, 642} \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^{2/3} e+14 b^{2/3} c\right )}{81 \sqrt {3} a^{10/3} b^{4/3}}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{10/3} b^{4/3}}-\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{10/3} b^{4/3}}+\frac {x (5 a e+28 b c x)}{162 a^3 b \left (a+b x^3\right )}-\frac {6 a d-x (a e+7 b c x)}{54 a^2 b \left (a+b x^3\right )^2}-\frac {x \left (a e-b c x-b d x^2\right )}{9 a b \left (a+b x^3\right )^3} \]

[In]

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

[Out]

-1/9*(x*(a*e - b*c*x - b*d*x^2))/(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)/
(Sqrt[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])/(2
43*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^(1
0/3)*b^(4/3))

Rule 31

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

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 642

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 648

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 1842

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 1868

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 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[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 1874

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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] (verified)

Time = 0.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=\frac {\frac {3 a b^{2/3} \left (28 b^3 c x^8-2 a^3 (9 d+5 e x)+a b^2 x^5 \left (77 c+5 e x^2\right )+a^2 b x^2 \left (67 c+13 e x^2\right )\right )}{\left (a+b x^3\right )^3}-2 \sqrt {3} a^{2/3} \sqrt [3]{b} \left (14 b^{2/3} c+5 a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \left (-14 a^{2/3} b c+5 a^{4/3} \sqrt [3]{b} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+a^{2/3} \sqrt [3]{b} \left (14 b^{2/3} c-5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^4 b^{5/3}} \]

[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))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.55 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.44

method	result	size
risch	\(\frac {\frac {14 c \,b^{2} x^{8}}{81 a^{3}}+\frac {5 b e \,x^{7}}{162 a^{2}}+\frac {77 b c \,x^{5}}{162 a^{2}}+\frac {13 e \,x^{4}}{162 a}+\frac {67 c \,x^{2}}{162 a}-\frac {5 e x}{81 b}-\frac {d}{9 b}}{\left (b \,x^{3}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\frac {14 c \textit {\_R}}{a}+\frac {5 e}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{243 a^{2} b}\)	\(119\)
default	\(\frac {\frac {14 c \,b^{2} x^{8}}{81 a^{3}}+\frac {5 b e \,x^{7}}{162 a^{2}}+\frac {77 b c \,x^{5}}{162 a^{2}}+\frac {13 e \,x^{4}}{162 a}+\frac {67 c \,x^{2}}{162 a}-\frac {5 e x}{81 b}-\frac {d}{9 b}}{\left (b \,x^{3}+a \right )^{3}}+\frac {5 a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+14 b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{81 a^{3} b}\)	\(273\)

[In]

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

[Out]

(14/81*c/a^3*b^2*x^8+5/162*b*e/a^2*x^7+77/162*b*c/a^2*x^5+13/162/a*e*x^4+67/162*c/a*x^2-5/81*e*x/b-1/9*d/b)/(b
*x^3+a)^3+1/243/a^2/b*sum((14*c/a*_R+5/b*e)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.13 (sec) , antiderivative size = 2646, normalized size of antiderivative = 9.80 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=\text {Too large to display} \]

[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)

Sympy [A] (verification not implemented)

Time = 6.08 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=\operatorname {RootSum} {\left (14348907 t^{3} a^{10} b^{4} + 51030 t a^{4} b^{2} c e - 125 a^{2} e^{3} + 2744 b^{2} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {826686 t^{2} a^{7} b^{3} c + 6075 t a^{5} b e^{2} + 1960 a b c^{2} e}{125 a^{2} e^{3} + 2744 b^{2} c^{3}} \right )} \right )\right )} + \frac {- 18 a^{3} d - 10 a^{3} e x + 67 a^{2} b c x^{2} + 13 a^{2} b e x^{4} + 77 a b^{2} c x^{5} + 5 a b^{2} e x^{7} + 28 b^{3} c x^{8}}{162 a^{6} b + 486 a^{5} b^{2} x^{3} + 486 a^{4} b^{3} x^{6} + 162 a^{3} b^{4} x^{9}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=\frac {28 \, b^{3} c x^{8} + 5 \, a b^{2} e x^{7} + 77 \, a b^{2} c x^{5} + 13 \, a^{2} b e x^{4} + 67 \, a^{2} b c x^{2} - 10 \, a^{3} e x - 18 \, a^{3} d}{162 \, {\left (a^{3} b^{4} x^{9} + 3 \, a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{3} + a^{6} b\right )}} + \frac {\sqrt {3} {\left (14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{486 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

1/162*(28*b^3*c*x^8 + 5*a*b^2*e*x^7 + 77*a*b^2*c*x^5 + 13*a^2*b*e*x^4 + 67*a^2*b*c*x^2 - 10*a^3*e*x - 18*a^3*d
)/(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b) + 1/243*sqrt(3)*(14*b*c*(a/b)^(1/3) + 5*a*e)*arctan(1/
3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b^2*(a/b)^(2/3)) + 1/486*(14*b*c*(a/b)^(1/3) - 5*a*e)*log(x^2
- x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b^2*(a/b)^(2/3)) - 1/243*(14*b*c*(a/b)^(1/3) - 5*a*e)*log(x + (a/b)^(1/3))
/(a^3*b^2*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=-\frac {\sqrt {3} {\left (5 \, a e - 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {{\left (5 \, a e + 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{486 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {{\left (14 \, b c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{4} b} + \frac {28 \, b^{3} c x^{8} + 5 \, a b^{2} e x^{7} + 77 \, a b^{2} c x^{5} + 13 \, a^{2} b e x^{4} + 67 \, a^{2} b c x^{2} - 10 \, a^{3} e x - 18 \, a^{3} d}{162 \, {\left (b x^{3} + a\right )}^{3} a^{3} b} \]

[In]

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

[Out]

-1/243*sqrt(3)*(5*a*e - 14*(-a*b^2)^(1/3)*c)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(
2/3)*a^3) - 1/486*(5*a*e + 14*(-a*b^2)^(1/3)*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a^3)
- 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) + 1/162*(28*b^3*c*x^8 +
5*a*b^2*e*x^7 + 77*a*b^2*c*x^5 + 13*a^2*b*e*x^4 + 67*a^2*b*c*x^2 - 10*a^3*e*x - 18*a^3*d)/((b*x^3 + a)^3*a^3*b
)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.98 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx=\frac {\frac {67\,c\,x^2}{162\,a}-\frac {d}{9\,b}+\frac {13\,e\,x^4}{162\,a}-\frac {5\,e\,x}{81\,b}+\frac {14\,b^2\,c\,x^8}{81\,a^3}+\frac {77\,b\,c\,x^5}{162\,a^2}+\frac {5\,b\,e\,x^7}{162\,a^2}}{a^3+3\,a^2\,b\,x^3+3\,a\,b^2\,x^6+b^3\,x^9}+\left (\sum _{k=1}^3\ln \left (\frac {70\,a\,c\,e+{\mathrm {root}\left (14348907\,a^{10}\,b^4\,z^3+51030\,a^4\,b^2\,c\,e\,z-125\,a^2\,e^3+2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b^2\,59049+196\,b\,c^2\,x+\mathrm {root}\left (14348907\,a^{10}\,b^4\,z^3+51030\,a^4\,b^2\,c\,e\,z-125\,a^2\,e^3+2744\,b^2\,c^3,z,k\right )\,a^4\,b\,e\,x\,1215}{a^6\,6561}\right )\,\mathrm {root}\left (14348907\,a^{10}\,b^4\,z^3+51030\,a^4\,b^2\,c\,e\,z-125\,a^2\,e^3+2744\,b^2\,c^3,z,k\right )\right ) \]

[In]

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

[Out]

((67*c*x^2)/(162*a) - d/(9*b) + (13*e*x^4)/(162*a) - (5*e*x)/(81*b) + (14*b^2*c*x^8)/(81*a^3) + (77*b*c*x^5)/(
162*a^2) + (5*b*e*x^7)/(162*a^2))/(a^3 + b^3*x^9 + 3*a^2*b*x^3 + 3*a*b^2*x^6) + symsum(log((70*a*c*e + 59049*r
oot(14348907*a^10*b^4*z^3 + 51030*a^4*b^2*c*e*z - 125*a^2*e^3 + 2744*b^2*c^3, z, k)^2*a^7*b^2 + 196*b*c^2*x +
1215*root(14348907*a^10*b^4*z^3 + 51030*a^4*b^2*c*e*z - 125*a^2*e^3 + 2744*b^2*c^3, z, k)*a^4*b*e*x)/(6561*a^6
))*root(14348907*a^10*b^4*z^3 + 51030*a^4*b^2*c*e*z - 125*a^2*e^3 + 2744*b^2*c^3, z, k), k, 1, 3)