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

Optimal. Leaf size=262 \[ \frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac {2 b c \log (x)}{a^3}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{3 a^2 \left (a+b x^3\right )}-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.40, antiderivative size = 262, 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} \[ -\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac {2 b c \log (x)}{a^3}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.30, size = 225, normalized size = 0.86 \[ \frac {\sqrt [3]{b} \left (5 \sqrt [3]{a} \sqrt [3]{b} d-4 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (4 a^{2/3} e-5 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\frac {6 a b (c+x (d+e x))}{a+b x^3}+12 b c \log \left (a+b x^3\right )+2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {6 a c}{x^3}-\frac {9 a d}{x^2}-\frac {18 a e}{x}-36 b c \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*a*c)/x^3 - (9*a*d)/x^2 - (18*a*e)/x - (6*a*b*(c + x*(d + e*x)))/(a + b*x^3) + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(
5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 36*b*c*Log[x] + 2*b^(1/3)*(-5*a^(1/3)
*b^(1/3)*d + 4*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*d - 4*a^(2/3)*e)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 12*b*c*Log[a + b*x^3])/(18*a^3)

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fricas [C]  time = 3.18, size = 5373, normalized size = 20.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(48*a*b*e*x^5 + 30*a*b*d*x^4 + 24*a*b*c*x^3 + 36*a^2*e*x^2 + 18*a^2*d*x + 12*a^2*c + 2*(a^3*b*x^6 + a^4*
x^3)*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d
^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e
)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*
c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)
*log((8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d
^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e
)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*
c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)
^2*a^6*e + 150*b^2*c*d^2 + 144*b^2*c^2*e + 160*a*b*d*e^2 + 1/2*(25*a^3*b*d^2 + 48*a^3*b*c*e)*(8*(1/2)^(2/3)*(-
I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 -
72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) +
(1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a
^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3) + (125*b^2*d^3 + 64*a
*b*e^3)*x) - (36*b^2*c*x^6 + 36*a*b*c*x^3 + (a^3*b*x^6 + a^4*x^3)*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a
^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)
*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)
*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*
b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3) + 3*sqrt(1/3)*(a^3*b*x^6 + a^4*x^3)*sqrt(-((8*(1
/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*
e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a
^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*
b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)^2*a^6 + 2
4*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3
+ 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a
*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2
+ 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)*a^
3*b*c + 144*b^2*c^2 + 320*a*b*d*e)/a^6))*log(-(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*
a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*
c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 +
(125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 -
72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)^2*a^6*e - 150*b^2*c*d^2 - 144*b^2*c^2*e - 160*a*b*d*e^2 - 1/2*(25*a^
3*b*d^2 + 48*a^3*b*c*e)*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3
*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5
*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)
*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^
(1/3) - 12*b*c/a^3) + 2*(125*b^2*d^3 + 64*a*b*e^3)*x + 3/2*sqrt(1/3)*(2*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2
*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*
b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3
) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 6
4*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)*a^6*e - 25*a^3*b*d^2 + 24*a^3*b*c*e)*sqrt(
-((8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3
+ 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a
*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2
+ 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)^2*
a^6 + 24*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125
*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c
*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*
b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/
a^3)*a^3*b*c + 144*b^2*c^2 + 320*a*b*d*e)/a^6)) - (36*b^2*c*x^6 + 36*a*b*c*x^3 + (a^3*b*x^6 + a^4*x^3)*(8*(1/2
)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^
3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9
)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*
d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3) - 3*sqrt(1/
3)*(a^3*b*x^6 + a^4*x^3)*sqrt(-((8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/
(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b
*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 6
4*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^
2)/a^9)^(1/3) - 12*b*c/a^3)^2*a^6 + 24*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e
)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 6
4*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*
d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*
e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)*a^3*b*c + 144*b^2*c^2 + 320*a*b*d*e)/a^6))*log(-(8*(1/2)^(2/3)*(-I*sqrt(3)
+ 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2
*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3
)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*
b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)^2*a^6*e - 150*b^2*c*d^2 - 144*b
^2*c^2*e - 160*a*b*d*e^2 - 1/2*(25*a^3*b*d^2 + 48*a^3*b*c*e)*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 -
(9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/
a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432
*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3
- 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3) + 2*(125*b^2*d^3 + 64*a*b*e^3)*x - 3/2*sqrt(1/3)*(2*(
8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 6
4*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^
2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 +
5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)*a^6*e
- 25*a^3*b*d^2 + 24*a^3*b*c*e)*sqrt(-((8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*a*b*d*e
)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 6
4*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 + (125*b*
d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*
e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)^2*a^6 + 24*(8*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*b^2*c^2/a^6 - (9*b^2*c^2 + 5*
a*b*d*e)/a^6)/(432*b^3*c^3/a^9 + (125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*
c^3 + 64*a^2*b*e^3 - 5*(25*d^3 - 72*c*d*e)*a*b^2)/a^9)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(432*b^3*c^3/a^9 +
(125*b*d^3 + 64*a*e^3)*b/a^8 - 72*(9*b^2*c^2 + 5*a*b*d*e)*b*c/a^9 + (216*b^3*c^3 + 64*a^2*b*e^3 - 5*(25*d^3 -
72*c*d*e)*a*b^2)/a^9)^(1/3) - 12*b*c/a^3)*a^3*b*c + 144*b^2*c^2 + 320*a*b*d*e)/a^6)) + 72*(b^2*c*x^6 + a*b*c*x
^3)*log(x))/(a^3*b*x^6 + a^4*x^3)

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giac [A]  time = 0.18, size = 269, normalized size = 1.03 \[ \frac {2 \, b c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac {2 \, b c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b} + \frac {{\left (4 \, a^{4} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + 5 \, a^{4} b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{7} b} - \frac {8 \, a b x^{5} e + 5 \, a b d x^{4} + 4 \, a b c x^{3} + 6 \, a^{2} x^{2} e + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, {\left (b x^{3} + a\right )} a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 289, normalized size = 1.10 \[ -\frac {b e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {b d x}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {b c}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {5 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {5 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {4 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {4 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 b c \ln \relax (x )}{a^{3}}+\frac {2 b c \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {e}{a^{2} x}-\frac {d}{2 a^{2} x^{2}}-\frac {c}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.07, size = 236, normalized size = 0.90 \[ -\frac {8 \, b e x^{5} + 5 \, b d x^{4} + 4 \, b c x^{3} + 6 \, a e x^{2} + 3 \, a d x + 2 \, a c}{6 \, {\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} - \frac {2 \, b c \log \relax (x)}{a^{3}} - \frac {\sqrt {3} {\left (4 \, a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} + \frac {{\left (12 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (6 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(8*b*e*x^5 + 5*b*d*x^4 + 4*b*c*x^3 + 6*a*e*x^2 + 3*a*d*x + 2*a*c)/(a^2*b*x^6 + a^3*x^3) - 2*b*c*log(x)/a^
3 - 1/9*sqrt(3)*(4*a*e*(a/b)^(2/3) + 5*a*d*(a/b)^(1/3))*b*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/
a^4 + 1/18*(12*b*c*(a/b)^(2/3) - 4*a*e*(a/b)^(1/3) + 5*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^
(2/3)) + 1/9*(6*b*c*(a/b)^(2/3) + 4*a*e*(a/b)^(1/3) - 5*a*d)*log(x + (a/b)^(1/3))/(a^3*(a/b)^(2/3))

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mupad [B]  time = 5.48, size = 537, normalized size = 2.05 \[ \left (\sum _{k=1}^3\ln \left (-\frac {50\,b^5\,c\,d^2-48\,b^5\,c^2\,e}{9\,a^6}-\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\,\left (\frac {25\,a^3\,b^4\,d^2+48\,c\,e\,a^3\,b^4}{9\,a^6}+\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\,\left (4\,b^3\,e+\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\,a^2\,b^3\,x\,36-\frac {48\,b^4\,c\,x}{a}\right )+\frac {x\,\left (600\,d\,e\,a^3\,b^4+432\,a^2\,b^5\,c^2\right )}{27\,a^6}\right )+\frac {x\,\left (-125\,b^5\,d^3+240\,c\,b^5\,d\,e+64\,a\,b^4\,e^3\right )}{27\,a^6}\right )\,\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {d\,x}{2\,a}+\frac {2\,b\,c\,x^3}{3\,a^2}+\frac {5\,b\,d\,x^4}{6\,a^2}+\frac {4\,b\,e\,x^5}{3\,a^2}}{b\,x^6+a\,x^3}-\frac {2\,b\,c\,\ln \relax (x)}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((x*(64*a*b^4*e^3 - 125*b^5*d^3 + 240*b^5*c*d*e))/(27*a^6) - root(729*a^9*z^3 - 1458*a^6*b*c*z^2 + 5
40*a^4*b*d*e*z + 972*a^3*b^2*c^2*z - 360*a*b^2*c*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3*c^3, z, k)*((25*
a^3*b^4*d^2 + 48*a^3*b^4*c*e)/(9*a^6) + root(729*a^9*z^3 - 1458*a^6*b*c*z^2 + 540*a^4*b*d*e*z + 972*a^3*b^2*c^
2*z - 360*a*b^2*c*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3*c^3, z, k)*(4*b^3*e + 36*root(729*a^9*z^3 - 145
8*a^6*b*c*z^2 + 540*a^4*b*d*e*z + 972*a^3*b^2*c^2*z - 360*a*b^2*c*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3
*c^3, z, k)*a^2*b^3*x - (48*b^4*c*x)/a) + (x*(432*a^2*b^5*c^2 + 600*a^3*b^4*d*e))/(27*a^6)) - (50*b^5*c*d^2 -
48*b^5*c^2*e)/(9*a^6))*root(729*a^9*z^3 - 1458*a^6*b*c*z^2 + 540*a^4*b*d*e*z + 972*a^3*b^2*c^2*z - 360*a*b^2*c
*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3*c^3, z, k), k, 1, 3) - (c/(3*a) + (e*x^2)/a + (d*x)/(2*a) + (2*b
*c*x^3)/(3*a^2) + (5*b*d*x^4)/(6*a^2) + (4*b*e*x^5)/(3*a^2))/(a*x^3 + b*x^6) - (2*b*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**4/(b*x**3+a)**2,x)

[Out]

Timed out

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