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

Optimal. Leaf size=257 \[ -\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac {\left (2 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{2/3}}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {c \log (x)}{a^3}+\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

1/6*x*(-b*c*x^2+a*e*x+a*d)/a^2/(b*x^3+a)^2+1/18*x*(-9*b*c*x^2+4*a*e*x+5*a*d)/a^3/(b*x^3+a)+c*ln(x)/a^3+1/27*(5
*b^(1/3)*d-2*a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^(2/3)-1/54*(5*b^(1/3)*d-2*a^(1/3)*e)*ln(a^(2/3)-a^(1/3
)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)/b^(2/3)-1/3*c*ln(b*x^3+a)/a^3-1/27*(5*b^(1/3)*d+2*a^(1/3)*e)*arctan(1/3*(a^(1
/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(8/3)/b^(2/3)*3^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac {\left (2 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{2/3}}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {c \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*d + 4*a*e*x - 9*b*c*x^2))/(18*a^3*(a + b*x^3)) - (
(5*b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(2/3)) + (
c*Log[x])/a^3 + ((5*b^(1/3)*d - 2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(2/3)) - ((5*b^(1/3)*d -
2*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^3)

Rule 31

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

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 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]

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 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 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 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

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 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]

Rubi steps

\begin {align*} \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^3} \, dx &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b c-5 b d x-4 b e x^2+\frac {3 b^2 c x^3}{a}}{x \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^2 c+10 b^2 d x+4 b^2 e x^2}{x \left (a+b x^3\right )} \, dx}{18 a^2 b^2}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^2 c}{a x}+\frac {2 b^2 \left (5 a d+2 a e x-9 b c x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^2}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\int \frac {5 a d+2 a e x-9 b c x^2}{a+b x^3} \, dx}{9 a^3}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\int \frac {5 a d+2 a e x}{a+b x^3} \, dx}{9 a^3}-\frac {(b c) \int \frac {x^2}{a+b x^3} \, dx}{a^3}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (10 a \sqrt [3]{b} d+2 a^{4/3} e\right )+\sqrt [3]{b} \left (-5 a \sqrt [3]{b} d+2 a^{4/3} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (5 d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3}}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (5 \sqrt [3]{b} d-2 \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}{54 a^{8/3} b^{2/3}}+\frac {\left (5 \sqrt [3]{b} d+2 \sqrt [3]{a} e\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{7/3} \sqrt [3]{b}}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\left (5 \sqrt [3]{b} d+2 \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 )}{9 a^{8/3} b^{2/3}}\\ &=\frac {x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {\left (5 \sqrt [3]{b} d+2 \sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{2/3}}+\frac {c \log (x)}{a^3}+\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac {\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 229, normalized size = 0.89 \[ \frac {\frac {\left (2 a^{2/3} e-5 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {2 \left (5 \sqrt [3]{a} \sqrt [3]{b} d-2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {9 a^2 (c+x (d+e x))}{\left (a+b x^3\right )^2}-\frac {2 \sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {3 a (6 c+x (5 d+4 e x))}{a+b x^3}-18 c \log \left (a+b x^3\right )+54 c \log (x)}{54 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((9*a^2*(c + x*(d + e*x)))/(a + b*x^3)^2 + (3*a*(6*c + x*(5*d + 4*e*x)))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(5*b
^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 54*c*Log[x] + (2*(5*a^(1/3)*b^(
1/3)*d - 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-5*a^(1/3)*b^(1/3)*d + 2*a^(2/3)*e)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 18*c*Log[a + b*x^3])/(54*a^3)

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fricas [C]  time = 2.53, size = 5229, normalized size = 20.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2916*(648*a*b*e*x^5 + 810*a*b*d*x^4 + 972*a*b*c*x^3 + 1134*a^2*e*x^2 + 1296*a^2*d*x + 1458*a^2*c - 2*(a^3*b^
2*x^6 + 2*a^4*b*x^3 + a^5)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1
458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e
^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*
a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54
*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)*log(1/1458*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a
^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/3
9366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a
^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 +
8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b*e + 225*b*c*d^2 + 162*b*c^2*e +
40*a*d*e^2 - 1/54*(25*a^3*b*d^2 + 36*a^3*b*c*e)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))
/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(
729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1
/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2
*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3) + (125*b*d^3 + 8*a*e^3)*x) - (1458*b^2*c*x^6 +
2916*a*b*c*x^3 + 1458*a^2*c - (a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 1
0*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8
*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(
-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(72
9*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3) - 3*sqrt(1/3)*(a^3*b^2*x^6 +
2*a^4*b*x^3 + a^5)*sqrt(-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/14
58*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^
3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a
*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*
c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b - 972*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/
(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1
/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3
/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3
+ 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)*a^3*b*c + 236196*b*c^2 + 116640*a*d*e)
/(a^6*b)))*log(-1/1458*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*
(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 -
5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*
e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d
*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b*e - 225*b*c*d^2 - 162*b*c^2*e - 40*a*d*e^2 + 1/54*(25*a^3*b*d^2
+ 36*a^3*b*c*e)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*
c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25
*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(
a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*
b)/(a^9*b^2))^(1/3) + 486*c/a^3) + 2*(125*b*d^3 + 8*a*e^3)*x + 1/486*sqrt(1/3)*(((-I*sqrt(3) + 1)*(81*c^2/a^6
- (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3
+ 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I
*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2
) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)*a^6*b*e + 675*
a^3*b*d^2 - 486*a^3*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^
9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 +
8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^
2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d
^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b - 972*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10
*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*
b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-
1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729
*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)*a^3*b*c + 236196*b*c^2 + 11664
0*a*d*e)/(a^6*b))) - (1458*b^2*c*x^6 + 2916*a*b*c*x^3 + 1458*a^2*c - (a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*((-I*sq
rt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b
) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a
^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*
d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 48
6*c/a^3) + 3*sqrt(1/3)*(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10
*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*
b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-
1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729
*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b - 972*((-I*sqrt(3) + 1
)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/393
66*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^
(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a
*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)*
a^3*b*c + 236196*b*c^2 + 116640*a*d*e)/(a^6*b)))*log(-1/1458*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*
d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2
) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/2
7*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^
2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b*e - 225*b*c*d^2 - 162*b*c
^2*e - 40*a*d*e^2 + 1/54*(25*a^3*b*d^2 + 36*a^3*b*c*e)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(
a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/
39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/
a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3
+ 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3) + 2*(125*b*d^3 + 8*a*e^3)*x - 1/486*sqr
t(1/3)*(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*
a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54
*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) +
1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b
^2))^(1/3) + 486*c/a^3)*a^6*b*e + 675*a^3*b*d^2 - 486*a^3*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b*
c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^
3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 729*(I*sqrt(3)
+ 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39
366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486*c/a^3)^2*a^6*b - 972*((-I*sqr
t(3) + 1)*(81*c^2/a^6 - (81*b*c^2 + 10*a*d*e)/(a^6*b))/(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b)
+ 1/39366*(125*b*d^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^
9*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b*c^2 + 10*a*d*e)*c/(a^9*b) + 1/39366*(125*b*d
^3 + 8*a*e^3)/(a^8*b^2) - 1/39366*(729*b^2*c^3 + 8*a^2*e^3 - 5*(25*d^3 - 54*c*d*e)*a*b)/(a^9*b^2))^(1/3) + 486
*c/a^3)*a^3*b*c + 236196*b*c^2 + 116640*a*d*e)/(a^6*b))) + 2916*(b^2*c*x^6 + 2*a*b*c*x^3 + a^2*c)*log(x))/(a^3
*b^2*x^6 + 2*a^4*b*x^3 + a^5)

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giac [A]  time = 0.24, size = 253, normalized size = 0.98 \[ -\frac {\sqrt {3} {\left (5 \, b d - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} - \frac {{\left (5 \, b d + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} - \frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {4 \, a b x^{5} e + 5 \, a b d x^{4} + 6 \, a b c x^{3} + 7 \, a^{2} x^{2} e + 8 \, a^{2} d x + 9 \, a^{2} c}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3}} - \frac {{\left (2 \, a^{4} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + 5 \, a^{4} b d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*sqrt(3)*(5*b*d - 2*(-a*b^2)^(1/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/
3)*a^2) - 1/54*(5*b*d + 2*(-a*b^2)^(1/3)*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a^2) - 1/
3*c*log(abs(b*x^3 + a))/a^3 + c*log(abs(x))/a^3 + 1/18*(4*a*b*x^5*e + 5*a*b*d*x^4 + 6*a*b*c*x^3 + 7*a^2*x^2*e
+ 8*a^2*d*x + 9*a^2*c)/((b*x^3 + a)^2*a^3) - 1/27*(2*a^4*b*(-a/b)^(1/3)*e + 5*a^4*b*d)*(-a/b)^(1/3)*log(abs(x
- (-a/b)^(1/3)))/(a^7*b)

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maple [A]  time = 0.07, size = 331, normalized size = 1.29 \[ \frac {2 b e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {5 b d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {b c \,x^{3}}{3 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {7 e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a}+\frac {4 d x}{9 \left (b \,x^{3}+a \right )^{2} a}+\frac {c}{2 \left (b \,x^{3}+a \right )^{2} a}+\frac {5 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {5 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}-\frac {2 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {c \ln \relax (x )}{a^{3}}-\frac {c \ln \left (b \,x^{3}+a \right )}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9/a^2/(b*x^3+a)^2*b*e*x^5+5/18/(b*x^3+a)^2/a^2*b*d*x^4+1/3/a^2/(b*x^3+a)^2*x^3*c*b+7/18/a/(b*x^3+a)^2*e*x^2+
4/9/(b*x^3+a)^2/a*d*x+1/2/(b*x^3+a)^2/a*c+5/27/(a/b)^(2/3)/a^2/b*d*ln(x+(a/b)^(1/3))-5/54/(a/b)^(2/3)/a^2/b*d*
ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+5/27/(a/b)^(2/3)*3^(1/2)/a^2/b*d*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-2/2
7/a^2*e/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/27/a^2*e/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+2/27/a^2*e*
3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3/a^3*c*ln(b*x^3+a)+1/a^3*c*ln(x)

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maxima [A]  time = 3.00, size = 246, normalized size = 0.96 \[ \frac {4 \, b e x^{5} + 5 \, b d x^{4} + 6 \, b c x^{3} + 7 \, a e x^{2} + 8 \, a d x + 9 \, a c}{18 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} + \frac {c \log \relax (x)}{a^{3}} + \frac {\sqrt {3} {\left (2 \, a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4}} - \frac {{\left (18 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(4*b*e*x^5 + 5*b*d*x^4 + 6*b*c*x^3 + 7*a*e*x^2 + 8*a*d*x + 9*a*c)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) + c*l
og(x)/a^3 + 1/27*sqrt(3)*(2*a*e*(a/b)^(2/3) + 5*a*d*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^
(1/3))/a^4 - 1/54*(18*b*c*(a/b)^(2/3) - 2*a*e*(a/b)^(1/3) + 5*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3
*b*(a/b)^(2/3)) - 1/27*(9*b*c*(a/b)^(2/3) + 2*a*e*(a/b)^(1/3) - 5*a*d)*log(x + (a/b)^(1/3))/(a^3*b*(a/b)^(2/3)
)

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mupad [B]  time = 5.44, size = 540, normalized size = 2.10 \[ \frac {\frac {c}{2\,a}+\frac {7\,e\,x^2}{18\,a}+\frac {4\,d\,x}{9\,a}+\frac {b\,c\,x^3}{3\,a^2}+\frac {5\,b\,d\,x^4}{18\,a^2}+\frac {2\,b\,e\,x^5}{9\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\left (\sum _{k=1}^3\ln \left (\frac {25\,b^2\,c\,d^2-18\,b^2\,c^2\,e}{81\,a^6}-\mathrm {root}\left (19683\,a^9\,b^2\,z^3+19683\,a^6\,b^2\,c\,z^2+810\,a^4\,b\,d\,e\,z+6561\,a^3\,b^2\,c^2\,z+270\,a\,b\,c\,d\,e-125\,a\,b\,d^3+8\,a^2\,e^3+729\,b^2\,c^3,z,k\right )\,\left (\frac {25\,a^3\,b^2\,d^2+36\,c\,e\,a^3\,b^2}{81\,a^6}+\mathrm {root}\left (19683\,a^9\,b^2\,z^3+19683\,a^6\,b^2\,c\,z^2+810\,a^4\,b\,d\,e\,z+6561\,a^3\,b^2\,c^2\,z+270\,a\,b\,c\,d\,e-125\,a\,b\,d^3+8\,a^2\,e^3+729\,b^2\,c^3,z,k\right )\,\left (-\frac {2\,b^2\,e}{3}+\mathrm {root}\left (19683\,a^9\,b^2\,z^3+19683\,a^6\,b^2\,c\,z^2+810\,a^4\,b\,d\,e\,z+6561\,a^3\,b^2\,c^2\,z+270\,a\,b\,c\,d\,e-125\,a\,b\,d^3+8\,a^2\,e^3+729\,b^2\,c^3,z,k\right )\,a^2\,b^3\,x\,36+\frac {24\,b^3\,c\,x}{a}\right )+\frac {x\,\left (900\,d\,e\,a^3\,b^2+2916\,a^2\,b^3\,c^2\right )}{729\,a^6}\right )-\frac {x\,\left (-125\,b^2\,d^3+180\,c\,b^2\,d\,e+8\,a\,b\,e^3\right )}{729\,a^6}\right )\,\mathrm {root}\left (19683\,a^9\,b^2\,z^3+19683\,a^6\,b^2\,c\,z^2+810\,a^4\,b\,d\,e\,z+6561\,a^3\,b^2\,c^2\,z+270\,a\,b\,c\,d\,e-125\,a\,b\,d^3+8\,a^2\,e^3+729\,b^2\,c^3,z,k\right )\right )+\frac {c\,\ln \relax (x)}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c/(2*a) + (7*e*x^2)/(18*a) + (4*d*x)/(9*a) + (b*c*x^3)/(3*a^2) + (5*b*d*x^4)/(18*a^2) + (2*b*e*x^5)/(9*a^2))/
(a^2 + b^2*x^6 + 2*a*b*x^3) + symsum(log((25*b^2*c*d^2 - 18*b^2*c^2*e)/(81*a^6) - root(19683*a^9*b^2*z^3 + 196
83*a^6*b^2*c*z^2 + 810*a^4*b*d*e*z + 6561*a^3*b^2*c^2*z + 270*a*b*c*d*e - 125*a*b*d^3 + 8*a^2*e^3 + 729*b^2*c^
3, z, k)*((25*a^3*b^2*d^2 + 36*a^3*b^2*c*e)/(81*a^6) + root(19683*a^9*b^2*z^3 + 19683*a^6*b^2*c*z^2 + 810*a^4*
b*d*e*z + 6561*a^3*b^2*c^2*z + 270*a*b*c*d*e - 125*a*b*d^3 + 8*a^2*e^3 + 729*b^2*c^3, z, k)*(36*root(19683*a^9
*b^2*z^3 + 19683*a^6*b^2*c*z^2 + 810*a^4*b*d*e*z + 6561*a^3*b^2*c^2*z + 270*a*b*c*d*e - 125*a*b*d^3 + 8*a^2*e^
3 + 729*b^2*c^3, z, k)*a^2*b^3*x - (2*b^2*e)/3 + (24*b^3*c*x)/a) + (x*(2916*a^2*b^3*c^2 + 900*a^3*b^2*d*e))/(7
29*a^6)) - (x*(8*a*b*e^3 - 125*b^2*d^3 + 180*b^2*c*d*e))/(729*a^6))*root(19683*a^9*b^2*z^3 + 19683*a^6*b^2*c*z
^2 + 810*a^4*b*d*e*z + 6561*a^3*b^2*c^2*z + 270*a*b*c*d*e - 125*a*b*d^3 + 8*a^2*e^3 + 729*b^2*c^3, z, k), k, 1
, 3) + (c*log(x))/a^3

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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