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

Optimal. Leaf size=267 \[ -\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {c}{a^3 x}+\frac {d \log (x)}{a^3}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

-c/a^3/x+1/6*x*(-b*d*x^2-b*c*x+a*e)/a^2/(b*x^3+a)^2+1/18*x*(-9*b*d*x^2-10*b*c*x+5*a*e)/a^3/(b*x^3+a)+d*ln(x)/a
^3+1/27*(14*b^(2/3)*c+5*a^(2/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^(1/3)-1/54*(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^(1/3)-1/3*d*ln(b*x^3+a)/a^3+1/27*(14*b^(2/3)*c-5*a^(2/3)*e)*a
rctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(10/3)/b^(1/3)*3^(1/2)

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Rubi [A]  time = 0.46, antiderivative size = 267, 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} \[ -\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {c}{a^3 x}+\frac {d \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^3*x)) + (x*(a*e - b*c*x - b*d*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*e - 10*b*c*x - 9*b*d*x^2))/(18*a^3*
(a + b*x^3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(
10/3)*b^(1/3)) + (d*Log[x])/a^3 + ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*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])/(54*a^(10/3)*b^(1/3)) - (d*L
og[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^2 \left (a+b x^3\right )^3} \, dx &=\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b c-6 b d x-5 b e x^2+\frac {4 b^2 c x^3}{a}+\frac {3 b^2 d x^4}{a}}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^3 c+18 b^3 d x+10 b^3 e x^2-\frac {10 b^4 c x^3}{a}}{x^2 \left (a+b x^3\right )} \, dx}{18 a^2 b^3}\\ &=\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^3 c}{a x^2}+\frac {18 b^3 d}{a x}+\frac {2 b^3 \left (5 a e-14 b c x-9 b d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {5 a e-14 b c x-9 b d x^2}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {5 a e-14 b c x}{a+b x^3} \, dx}{9 a^3}-\frac {(b d) \int \frac {x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}-\frac {d \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (-14 \sqrt [3]{a} b c+10 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-14 \sqrt [3]{a} b c-5 a \sqrt [3]{b} 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 (14 b^{2/3} c+5 a^{2/3} e\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3}}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^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}{18 a^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}{54 a^{10/3} \sqrt [3]{b}}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\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 )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^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 )}{9 a^{10/3} \sqrt [3]{b}}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\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 )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\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 )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 248, normalized size = 0.93 \[ \frac {-\frac {\left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac {2 \left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {2 \sqrt {3} a^{2/3} \left (5 a^{2/3} e-14 b^{2/3} c\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {9 a^2 \left (a (d+e x)-b c x^2\right )}{\left (a+b x^3\right )^2}+\frac {3 a \left (6 a d+5 a e x-10 b c x^2\right )}{a+b x^3}-18 a d \log \left (a+b x^3\right )-\frac {54 a c}{x}+54 a d \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-54*a*c)/x + (3*a*(6*a*d + 5*a*e*x - 10*b*c*x^2))/(a + b*x^3) + (9*a^2*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^
3)^2 - (2*Sqrt[3]*a^(2/3)*(-14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) +
54*a*d*Log[x] + (2*(14*a^(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - ((14*a^(2/3)*b^(2
/3)*c + 5*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 18*a*d*Log[a + b*x^3])/(54*a^4)

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fricas [C]  time = 3.33, size = 5112, normalized size = 19.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2916*(4536*b^2*c*x^6 - 810*a*b*e*x^5 - 972*a*b*d*x^4 + 7938*a*b*c*x^3 - 1296*a^2*e*x^2 - 1458*a^2*d*x + 291
6*a^2*c + 2*(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(
81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*
(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*log(-7/1458*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2
- 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*
d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1
/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*
b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^7*b*c - 1134*a*b*c*d^2 + 1
960*a*b*c^2*e + 225*a^2*d*e^2 + 1/54*(252*a^4*b*c*d - 25*a^5*e^2)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70
*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3
- 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3) - (2744*b^2*c^3 - 125*a^2*e^3)*x)
+ (1458*b^2*d*x^7 + 2916*a*b*d*x^4 + 1458*a^2*d*x - (a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*((-I*sqrt(3) + 1)*(81*
d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125
*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(
I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*
d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3) + 3*sqrt(1/3
)*(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3
/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^1
0*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*
d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(27
44*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 7
0*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3
- 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27
*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/
(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/
a^6))*log(7/1458*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c
*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3
- 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/3936
6*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a
^10*b))^(1/3) + 486*d/a^3)^2*a^7*b*c + 1134*a*b*c*d^2 - 1960*a*b*c^2*e - 225*a^2*d*e^2 - 1/54*(252*a^4*b*c*d -
25*a^5*e^2)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*
d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 1
25*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2
744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*
b))^(1/3) + 486*d/a^3) - 2*(2744*b^2*c^3 - 125*a^2*e^3)*x + 1/486*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(81*d^2/a^6 -
(81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3
- 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3)
+ 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*
c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^7*b*c - 3402*a^4*b*
c*d - 675*a^5*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d
^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(274
4*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9
+ 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^
2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(
81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*
(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/a^6)) + (1458*b^2*d
*x^7 + 2916*a*b*d*x^4 + 1458*a^2*d*x - (a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81
*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27
*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1
)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*
e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3) - 3*sqrt(1/3)*(a^3*b^2*x^
7 + 2*a^4*b*x^4 + a^5*x)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458
*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/3936
6*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)
*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 -
125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(
-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*
a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/
1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/
39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/a^6))*log(7/1
458*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1
/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^
3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c
^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3)
+ 486*d/a^3)^2*a^7*b*c + 1134*a*b*c*d^2 - 1960*a*b*c^2*e - 225*a^2*d*e^2 - 1/54*(252*a^4*b*c*d - 25*a^5*e^2)*
((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/393
66*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(
a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 +
125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 4
86*d/a^3) - 2*(2744*b^2*c^3 - 125*a^2*e^3)*x - 1/486*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70
*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3
- 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^7*b*c - 3402*a^4*b*c*d - 675*a^5
*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*
d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 1
25*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2
744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*
b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1
458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/3
9366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c
*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3
- 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/a^6)) - 2916*(b^2*d*x^7 + 2*a*b*
d*x^4 + a^2*d*x)*log(x))/(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)

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giac [A]  time = 0.21, size = 273, normalized size = 1.02 \[ -\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e + 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} + \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b} - \frac {28 \, b^{2} c x^{6} - 5 \, a b x^{5} e - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} x^{2} e - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} x} + \frac {{\left (14 \, a^{3} b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a^{4} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*d*log(abs(b*x^3 + a))/a^3 + d*log(abs(x))/a^3 + 1/27*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) + 1/54*(5*(-a*b^2)^(1/3)*a*e - 14*(-a*b^2)^(2/3
)*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/18*(28*b^2*c*x^6 - 5*a*b*x^5*e - 6*a*b*d*x^4 + 49*a*
b*c*x^3 - 8*a^2*x^2*e - 9*a^2*d*x + 18*a^2*c)/((b*x^3 + a)^2*a^3*x) + 1/27*(14*a^3*b^2*c*(-a/b)^(1/3) - 5*a^4*
b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b)

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maple [A]  time = 0.06, size = 334, normalized size = 1.25 \[ -\frac {5 b^{2} c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {5 b e \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {b d \,x^{3}}{3 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {13 b c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {4 e x}{9 \left (b \,x^{3}+a \right )^{2} a}+\frac {d}{2 \left (b \,x^{3}+a \right )^{2} a}+\frac {5 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {5 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {5 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {14 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {14 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {d \ln \relax (x )}{a^{3}}-\frac {d \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {c}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/9/(b*x^3+a)^2/a^3*b^2*c*x^5+5/18/(b*x^3+a)^2/a^2*b*e*x^4+1/3/a^2/(b*x^3+a)^2*b*d*x^3-13/18/(b*x^3+a)^2/a^2*
b*c*x^2+4/9/(b*x^3+a)^2/a*e*x+1/2/(b*x^3+a)^2/a*d+5/27/(a/b)^(2/3)/a^2/b*e*ln(x+(a/b)^(1/3))-5/54/(a/b)^(2/3)/
a^2/b*e*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+5/27/(a/b)^(2/3)*3^(1/2)/a^2/b*e*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x
-1))+14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-7/27/a^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*c-14/27/
a^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c-1/3/a^3*d*ln(b*x^3+a)-1/a^3*c/x+1/a^3*d*ln(x
)

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maxima [A]  time = 3.09, size = 266, normalized size = 1.00 \[ -\frac {28 \, b^{2} c x^{6} - 5 \, a b e x^{5} - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} e x^{2} - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} + \frac {d \log \relax (x)}{a^{3}} - \frac {\sqrt {3} {\left (14 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4}} - \frac {{\left (18 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(28*b^2*c*x^6 - 5*a*b*e*x^5 - 6*a*b*d*x^4 + 49*a*b*c*x^3 - 8*a^2*e*x^2 - 9*a^2*d*x + 18*a^2*c)/(a^3*b^2*
x^7 + 2*a^4*b*x^4 + a^5*x) + d*log(x)/a^3 - 1/27*sqrt(3)*(14*b*c*(a/b)^(2/3) - 5*a*e*(a/b)^(1/3))*arctan(1/3*s
qrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^4 - 1/54*(18*b*d*(a/b)^(2/3) + 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*(a/b)^(2/3)) - 1/27*(9*b*d*(a/b)^(2/3) - 14*b*c*(a/b)^(1/3) - 5*a*e)*log(
x + (a/b)^(1/3))/(a^3*b*(a/b)^(2/3))

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mupad [B]  time = 5.46, size = 793, normalized size = 2.97 \[ \frac {\frac {4\,e\,x^2}{9\,a}-\frac {c}{a}+\frac {d\,x}{2\,a}-\frac {14\,b^2\,c\,x^6}{9\,a^3}-\frac {49\,b\,c\,x^3}{18\,a^2}+\frac {b\,d\,x^4}{3\,a^2}+\frac {5\,b\,e\,x^5}{18\,a^2}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}+\left (\sum _{k=1}^3\ln \left (\frac {b^2\,\left (-\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^5\,e^2\,225+225\,a^2\,d\,e^2+2744\,b^2\,c^3\,x+125\,a^2\,e^3\,x+1134\,a\,b\,c\,d^2-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b\,c\,3402-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^3\,a^{10}\,b\,x\,26244-\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,d^2\,x\,2916-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b\,d\,x\,17496+\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,c\,d\,2268+\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,c\,e\,x\,6300+1260\,a\,b\,c\,d\,e\,x\right )}{a^8\,729}\right )\,\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\right )+\frac {d\,\ln \relax (x)}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((4*e*x^2)/(9*a) - c/a + (d*x)/(2*a) - (14*b^2*c*x^6)/(9*a^3) - (49*b*c*x^3)/(18*a^2) + (b*d*x^4)/(3*a^2) + (5
*b*e*x^5)/(18*a^2))/(a^2*x + b^2*x^7 + 2*a*b*x^4) + symsum(log((b^2*(225*a^2*d*e^2 - 225*root(19683*a^10*b*z^3
+ 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744
*b^2*c^3, z, k)*a^5*e^2 + 2744*b^2*c^3*x + 125*a^2*e^3*x + 1134*a*b*c*d^2 - 3402*root(19683*a^10*b*z^3 + 19683
*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3
, z, k)^2*a^7*b*c - 26244*root(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 18
90*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3, z, k)^3*a^10*b*x - 2916*root(19683*a^10*b*z^3 + 19683
*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3
, z, k)*a^4*b*d^2*x - 17496*root(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z -
1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3, z, k)^2*a^7*b*d*x + 2268*root(19683*a^10*b*z^3 + 19
683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*
c^3, z, k)*a^4*b*c*d + 6300*root(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z -
1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3, z, k)*a^4*b*c*e*x + 1260*a*b*c*d*e*x))/(729*a^8))*r
oot(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3
- 125*a^2*e^3 - 2744*b^2*c^3, z, k), k, 1, 3) + (d*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**2/(b*x**3+a)**3,x)

[Out]

Timed out

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