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

Optimal. Leaf size=222 \[ -\frac {\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac {\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac {\left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{2/3}}+\frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {c \log \left (a+b x^3\right )}{3 a^2}+\frac {c \log (x)}{a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.31, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac {\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac {\left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{2/3}}+\frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {c \log \left (a+b x^3\right )}{3 a^2}+\frac {c \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 199, normalized size = 0.90 \[ \frac {\frac {\left (a^{2/3} e-2 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {2 \left (2 \sqrt [3]{a} \sqrt [3]{b} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {2 \sqrt {3} \sqrt [3]{a} \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {6 a (c+x (d+e x))}{a+b x^3}-6 c \log \left (a+b x^3\right )+18 c \log (x)}{18 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [C]  time = 3.15, size = 5018, normalized size = 22.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(108*a*e*x^2 + 108*a*d*x - 2*(a^2*b*x^3 + a^3)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b
))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*
c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2
+ 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e
)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)*log(1/324*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/
27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a
^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*
d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/
(a^6*b^2))^(1/3) + 54*c/a^2)^2*a^4*b*e + 12*b*c*d^2 + 9*b*c^2*e + 4*a*d*e^2 - 1/9*(2*a^2*b*d^2 + 3*a^2*b*c*e)*
((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*
b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(
1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5
*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2) + (8*b*d^3 + a*e^
3)*x) + 108*a*c - (162*b*c*x^3 - (a^2*b*x^3 + a^3)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))
/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^
3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 +
2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*
a*b)/(a^6*b^2))^(1/3) + 54*c/a^2) + 162*a*c - 3*sqrt(1/3)*(a^2*b*x^3 + a^3)*sqrt(-(((-I*sqrt(3) + 1)*(9*c^2/a^
4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^
3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(
-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3
+ a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)^2*a^4*b - 108*((-I*sqrt(3) + 1)*(9*c^2/a^4 -
(9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/
(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/
27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a
^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)*a^2*b*c + 2916*b*c^2 + 2592*a*d*e)/(a^4*b)))*lo
g(-1/324*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e
)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^
6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*
e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)^2*a^4*b*
e - 12*b*c*d^2 - 9*b*c^2*e - 4*a*d*e^2 + 1/9*(2*a^2*b*d^2 + 3*a^2*b*c*e)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c
^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^
2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/
a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3
- 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2) + 2*(8*b*d^3 + a*e^3)*x + 1/108*sqrt(1/3)*(((-I*sqrt(3
) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458
*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(
I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1
458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)*a^4*b*e + 72*a^2*b*d^2 - 54*
a^2*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2
+ 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)
*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b
*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)
^2*a^4*b - 108*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2
*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*
b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^
3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)*a^
2*b*c + 2916*b*c^2 + 2592*a*d*e)/(a^4*b))) - (162*b*c*x^3 - (a^2*b*x^3 + a^3)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (
9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a
^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27
*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2
*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2) + 162*a*c + 3*sqrt(1/3)*(a^2*b*x^3 + a^3)*sqrt(-(
((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*
b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(
1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5
*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)^2*a^4*b - 108*((-
I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b)
+ 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3
) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^
2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)*a^2*b*c + 2916*b*c^2
+ 2592*a*d*e)/(a^4*b)))*log(-1/324*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6
+ 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2
*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^
6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))
^(1/3) + 54*c/a^2)^2*a^4*b*e - 12*b*c*d^2 - 9*b*c^2*e - 4*a*d*e^2 + 1/9*(2*a^2*b*d^2 + 3*a^2*b*c*e)*((-I*sqrt(
3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/145
8*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*
(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/
1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2) + 2*(8*b*d^3 + a*e^3)*x - 1
/108*sqrt(1/3)*(((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c^3/a^6 + 1/162*(9*b*c^2 +
2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a
*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d
^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 54*c/a^2)*a
^4*b*e + 72*a^2*b*d^2 - 54*a^2*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/2
7*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^
2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d
*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(
a^6*b^2))^(1/3) + 54*c/a^2)^2*a^4*b - 108*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b*c^2 + 2*a*d*e)/(a^4*b))/(-1/27*c
^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e
^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b*c^2 + 2*a*d*e)
*c/(a^6*b) + 1/1458*(8*b*d^3 + a*e^3)/(a^5*b^2) - 1/1458*(27*b^2*c^3 + a^2*e^3 - 2*(4*d^3 - 9*c*d*e)*a*b)/(a^6
*b^2))^(1/3) + 54*c/a^2)*a^2*b*c + 2916*b*c^2 + 2592*a*d*e)/(a^4*b))) + 324*(b*c*x^3 + a*c)*log(x))/(a^2*b*x^3
+ a^3)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 217, normalized size = 0.98 \[ -\frac {\sqrt {3} {\left (2 \, b d - \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (2 \, b d + \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {a x^{2} e + a d x + a c}{3 \, {\left (b x^{3} + a\right )} a^{2}} - \frac {{\left (a^{3} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + 2 \, a^{3} b d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(3)*(2*b*d - (-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) - 1/18*(2*b*d + (-a*b^2)^(1/3)*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a) - 1/3*c*log(a
bs(b*x^3 + a))/a^2 + c*log(abs(x))/a^2 + 1/3*(a*x^2*e + a*d*x + a*c)/((b*x^3 + a)*a^2) - 1/9*(a^3*b*(-a/b)^(1/
3)*e + 2*a^3*b*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*b)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 274, normalized size = 1.23 \[ \frac {e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}+\frac {d x}{3 \left (b \,x^{3}+a \right ) a}+\frac {2 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {2 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}-\frac {d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {\sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {c}{3 \left (b \,x^{3}+a \right ) a}+\frac {c \ln \relax (x )}{a^{2}}-\frac {c \ln \left (b \,x^{3}+a \right )}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.94, size = 203, normalized size = 0.91 \[ \frac {e x^{2} + d x + c}{3 \, {\left (a b x^{3} + a^{2}\right )}} + \frac {c \log \relax (x)}{a^{2}} + \frac {\sqrt {3} {\left (a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {{\left (6 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(e*x^2 + d*x + c)/(a*b*x^3 + a^2) + c*log(x)/a^2 + 1/9*sqrt(3)*(a*e*(a/b)^(2/3) + 2*a*d*(a/b)^(1/3))*arcta
n(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^3 - 1/18*(6*b*c*(a/b)^(2/3) - a*e*(a/b)^(1/3) + 2*a*d)*log(x^
2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) - 1/9*(3*b*c*(a/b)^(2/3) + a*e*(a/b)^(1/3) - 2*a*d)*log(x
+ (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))

________________________________________________________________________________________

mupad [B]  time = 0.38, size = 490, normalized size = 2.21 \[ \frac {\frac {c}{3\,a}+\frac {e\,x^2}{3\,a}+\frac {d\,x}{3\,a}}{b\,x^3+a}+\left (\sum _{k=1}^3\ln \left (\frac {4\,b^2\,c\,d^2-3\,b^2\,c^2\,e}{9\,a^3}-\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\,\left (\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\,\left (-a\,b^2\,e+24\,b^3\,c\,x+\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\,a^2\,b^3\,x\,36\right )+\frac {4\,a^2\,b^2\,d^2+6\,c\,e\,a^2\,b^2}{9\,a^3}+\frac {x\,\left (60\,d\,e\,a^2\,b^2+108\,a\,b^3\,c^2\right )}{27\,a^3}\right )-\frac {x\,\left (-8\,b^2\,d^3+12\,c\,b^2\,d\,e+a\,b\,e^3\right )}{27\,a^3}\right )\,\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\right )+\frac {c\,\ln \relax (x)}{a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c/(3*a) + (e*x^2)/(3*a) + (d*x)/(3*a))/(a + b*x^3) + symsum(log((4*b^2*c*d^2 - 3*b^2*c^2*e)/(9*a^3) - root(72
9*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54*a^3*b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3
+ a^2*e^3, z, k)*(root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54*a^3*b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*
e - 8*a*b*d^3 + 27*b^2*c^3 + a^2*e^3, z, k)*(24*b^3*c*x - a*b^2*e + 36*root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^
2 + 54*a^3*b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3 + a^2*e^3, z, k)*a^2*b^3*x) + (
4*a^2*b^2*d^2 + 6*a^2*b^2*c*e)/(9*a^3) + (x*(108*a*b^3*c^2 + 60*a^2*b^2*d*e))/(27*a^3)) - (x*(a*b*e^3 - 8*b^2*
d^3 + 12*b^2*c*d*e))/(27*a^3))*root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54*a^3*b*d*e*z + 243*a^2*b^2*c^2*z +
18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3 + a^2*e^3, z, k), k, 1, 3) + (c*log(x))/a^2

________________________________________________________________________________________

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)**2,x)

[Out]

Timed out

________________________________________________________________________________________