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

Optimal. Leaf size=301 \[ -\frac {10 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac {20 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac {20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {c}{a^4 x}+\frac {d \log (x)}{a^4}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

[Out]

-c/a^4/x+1/9*x*(-b*d*x^2-b*c*x+a*e)/a^2/(b*x^3+a)^3+1/54*x*(-15*b*d*x^2-16*b*c*x+8*a*e)/a^3/(b*x^3+a)^2+1/162*
x*(-99*b*d*x^2-118*b*c*x+40*a*e)/a^4/(b*x^3+a)+d*ln(x)/a^4+20/243*(7*b^(2/3)*c+2*a^(2/3)*e)*ln(a^(1/3)+b^(1/3)
*x)/a^(13/3)/b^(1/3)-10/243*(7*b^(2/3)*c+2*a^(2/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)/b^(1/
3)-1/3*d*ln(b*x^3+a)/a^4+20/243*(7*b^(2/3)*c-2*a^(2/3)*e)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^
(13/3)/b^(1/3)*3^(1/2)

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Rubi [A]  time = 0.60, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ -\frac {10 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac {20 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac {20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {c}{a^4 x}+\frac {d \log (x)}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^4*x)) + (x*(a*e - b*c*x - b*d*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*e - 16*b*c*x - 15*b*d*x^2))/(54*a^3
*(a + b*x^3)^2) + (x*(40*a*e - 118*b*c*x - 99*b*d*x^2))/(162*a^4*(a + b*x^3)) + (20*(7*b^(2/3)*c - 2*a^(2/3)*e
)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(13/3)*b^(1/3)) + (d*Log[x])/a^4 + (20*(7*b
^(2/3)*c + 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(13/3)*b^(1/3)) - (10*(7*b^(2/3)*c + 2*a^(2/3)*e)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(13/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^4)

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 )^4} \, dx &=\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac {\int \frac {-9 b c-9 b d x-8 b e x^2+\frac {7 b^2 c x^3}{a}+\frac {6 b^2 d x^4}{a}}{x^2 \left (a+b x^3\right )^3} \, dx}{9 a b}\\ &=\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {\int \frac {54 b^3 c+54 b^3 d x+40 b^3 e x^2-\frac {64 b^4 c x^3}{a}-\frac {45 b^4 d x^4}{a}}{x^2 \left (a+b x^3\right )^2} \, dx}{54 a^2 b^3}\\ &=\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac {\int \frac {-162 b^5 c-162 b^5 d x-80 b^5 e x^2+\frac {118 b^6 c x^3}{a}}{x^2 \left (a+b x^3\right )} \, dx}{162 a^3 b^5}\\ &=\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac {\int \left (-\frac {162 b^5 c}{a x^2}-\frac {162 b^5 d}{a x}-\frac {2 b^5 \left (40 a e-140 b c x-81 b d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{162 a^3 b^5}\\ &=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {\int \frac {40 a e-140 b c x-81 b d x^2}{a+b x^3} \, dx}{81 a^4}\\ &=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {\int \frac {40 a e-140 b c x}{a+b x^3} \, dx}{81 a^4}-\frac {(b d) \int \frac {x^2}{a+b x^3} \, dx}{a^4}\\ &=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}-\frac {d \log \left (a+b x^3\right )}{3 a^4}+\frac {\int \frac {\sqrt [3]{a} \left (-140 \sqrt [3]{a} b c+80 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-140 \sqrt [3]{a} b c-40 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{14/3} \sqrt [3]{b}}+\frac {\left (20 \left (7 b^{2/3} c+2 a^{2/3} e\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{13/3}}\\ &=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {\left (10 \left (7 b^{2/3} c-2 a^{2/3} e\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^4}-\frac {\left (10 \left (7 b^{2/3} c+2 a^{2/3} 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}{243 a^{13/3} \sqrt [3]{b}}\\ &=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {10 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {\left (20 \left (7 b^{2/3} c-2 a^{2/3} 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 )}{81 a^{13/3} \sqrt [3]{b}}\\ &=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {10 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 279, normalized size = 0.93 \[ \frac {-\frac {20 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac {40 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {40 \sqrt {3} a^{2/3} \left (2 a^{2/3} e-7 b^{2/3} c\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {54 a^3 \left (a (d+e x)-b c x^2\right )}{\left (a+b x^3\right )^3}+\frac {9 a^2 \left (9 a d+8 a e x-16 b c x^2\right )}{\left (a+b x^3\right )^2}+\frac {6 a \left (27 a d+20 a e x-59 b c x^2\right )}{a+b x^3}-162 a d \log \left (a+b x^3\right )-\frac {486 a c}{x}+486 a d \log (x)}{486 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-486*a*c)/x + (9*a^2*(9*a*d + 8*a*e*x - 16*b*c*x^2))/(a + b*x^3)^2 + (6*a*(27*a*d + 20*a*e*x - 59*b*c*x^2))/
(a + b*x^3) + (54*a^3*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3)^3 - (40*Sqrt[3]*a^(2/3)*(-7*b^(2/3)*c + 2*a^(2/3
)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 486*a*d*Log[x] + (40*(7*a^(2/3)*b^(2/3)*c + 2*a^(4
/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (20*(7*a^(2/3)*b^(2/3)*c + 2*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/b^(1/3) - 162*a*d*Log[a + b*x^3])/(486*a^5)

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fricas [C]  time = 3.51, size = 5250, normalized size = 17.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/236196*(408240*b^3*c*x^9 - 58320*a*b^2*e*x^8 - 78732*a*b^2*d*x^7 + 1122660*a*b^2*c*x^6 - 151632*a^2*b*e*x^5
- 196830*a^2*b*d*x^4 + 976860*a^2*b*c*x^3 - 119556*a^3*e*x^2 - 144342*a^3*d*x + 236196*a^3*c + 2*(a^4*b^3*x^1
0 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x)*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d
^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3
- 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*
(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2
187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)*l
og(-7/236196*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2
- 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b)
- 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(
6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(
a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^9*b*c - 45927*a*b*c*d^2 +
78400*a*b*c^2*e + 6480*a^2*d*e^2 + 1/243*(567*a^5*b*c*d - 40*a^6*e^2)*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561
*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 +
64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^
(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^
2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a
^13*b))^(1/3) + 39366*d/a^4) - 400*(343*b^2*c^3 - 8*a^2*e^3)*x) + (118098*b^3*d*x^10 + 354294*a*b^2*d*x^7 + 35
4294*a^2*b*d*x^4 + 118098*a^3*d*x - (a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x)*((-I*sqrt(3) + 1)*(65
61*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(
2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a
^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28
697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c
^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4) + 3*sqrt(1/3)*(a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7
*x)*sqrt(-(((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 -
5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) -
4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(65
61*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^
13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^8 - 78732*((-I*sqrt(3) + 1)
*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/286978
14*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 -
8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 +
1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b
^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)*a^4*d + 1549681956*d^2 - 5290790400*c*e)/a^8))*log(7/236196
*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)
*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348
907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5
600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 40
00/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^9*b*c + 45927*a*b*c*d^2 - 78400*a*b*c
^2*e - 6480*a^2*d*e^2 - 1/243*(567*a^5*b*c*d - 40*a^6*e^2)*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*
c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^
3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 5904
9*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 6400
0*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3
) + 39366*d/a^4) - 800*(343*b^2*c^3 - 8*a^2*e^3)*x + 1/78732*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6
561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3
+ 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b
))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000
*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)
/(a^13*b))^(1/3) + 39366*d/a^4)*a^9*b*c - 275562*a^5*b*c*d - 38880*a^6*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(6561*d^2
/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(274400
0*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3
)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814
*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8
*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^8 - 78732*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)
/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 -
243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I
*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^
2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) +
39366*d/a^4)*a^4*d + 1549681956*d^2 - 5290790400*c*e)/a^8)) + (118098*b^3*d*x^10 + 354294*a*b^2*d*x^7 + 354294
*a^2*b*d*x^4 + 118098*a^3*d*x - (a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x)*((-I*sqrt(3) + 1)*(6561*d
^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744
000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e
^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/286978
14*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 -
8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4) - 3*sqrt(1/3)*(a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x)*
sqrt(-(((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 560
0*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000
/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d
^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b
) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^8 - 78732*((-I*sqrt(3) + 1)*(65
61*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(
2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a
^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28
697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c
^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)*a^4*d + 1549681956*d^2 - 5290790400*c*e)/a^8))*log(7/236196*((-
I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a
^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*
(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*
c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/1
4348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^9*b*c + 45927*a*b*c*d^2 - 78400*a*b*c^2*e
- 6480*a^2*d*e^2 - 1/243*(567*a^5*b*c*d - 40*a^6*e^2)*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)
/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 -
243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I
*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^
2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) +
39366*d/a^4) - 800*(343*b^2*c^3 - 8*a^2*e^3)*x - 1/78732*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*
d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 6
4000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(
1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2
*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^
13*b))^(1/3) + 39366*d/a^4)*a^9*b*c - 275562*a^5*b*c*d - 38880*a^6*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(6561*d^2/a^8
- (6561*d^2 - 5600*c*e)/a^8)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^
2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a
^13*b))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(27
44000*b^2*c^3 + 64000*a^2*e^3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2
*e^3)/(a^13*b))^(1/3) + 39366*d/a^4)^2*a^8 - 78732*((-I*sqrt(3) + 1)*(6561*d^2/a^8 - (6561*d^2 - 5600*c*e)/a^8
)/(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^3 - 243*
(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 59049*(I*sqr
t(3) + 1)*(-1/27*d^3/a^12 + 1/118098*(6561*d^2 - 5600*c*e)*d/a^12 + 1/28697814*(2744000*b^2*c^3 + 64000*a^2*e^
3 - 243*(2187*d^3 - 5600*c*d*e)*a*b)/(a^13*b) - 4000/14348907*(343*b^2*c^3 - 8*a^2*e^3)/(a^13*b))^(1/3) + 3936
6*d/a^4)*a^4*d + 1549681956*d^2 - 5290790400*c*e)/a^8)) - 236196*(b^3*d*x^10 + 3*a*b^2*d*x^7 + 3*a^2*b*d*x^4 +
a^3*d*x)*log(x))/(a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x)

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giac [A]  time = 0.18, size = 310, normalized size = 1.03 \[ -\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {20 \, \sqrt {3} {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5} b} + \frac {10 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} b} - \frac {280 \, b^{3} c x^{9} - 40 \, a b^{2} x^{8} e - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b x^{5} e - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} x^{2} e - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \, {\left (b x^{3} + a\right )}^{3} a^{4} x} + \frac {20 \, {\left (7 \, a^{4} b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{5} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*d*log(abs(b*x^3 + a))/a^4 + d*log(abs(x))/a^4 + 20/243*sqrt(3)*(2*(-a*b^2)^(1/3)*a*e + 7*(-a*b^2)^(2/3)*c
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) + 10/243*(2*(-a*b^2)^(1/3)*a*e - 7*(-a*b^2)^(2
/3)*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/162*(280*b^3*c*x^9 - 40*a*b^2*x^8*e - 54*a*b^2*d*x
^7 + 770*a*b^2*c*x^6 - 104*a^2*b*x^5*e - 135*a^2*b*d*x^4 + 670*a^2*b*c*x^3 - 82*a^3*x^2*e - 99*a^3*d*x + 162*a
^3*c)/((b*x^3 + a)^3*a^4*x) + 20/243*(7*a^4*b^2*c*(-a/b)^(1/3) - 2*a^5*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1
/3)))/(a^9*b)

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maple [A]  time = 0.07, size = 397, normalized size = 1.32 \[ -\frac {59 b^{3} c \,x^{8}}{81 \left (b \,x^{3}+a \right )^{3} a^{4}}+\frac {20 b^{2} e \,x^{7}}{81 \left (b \,x^{3}+a \right )^{3} a^{3}}+\frac {b^{2} d \,x^{6}}{3 \left (b \,x^{3}+a \right )^{3} a^{3}}-\frac {142 b^{2} c \,x^{5}}{81 \left (b \,x^{3}+a \right )^{3} a^{3}}+\frac {52 b e \,x^{4}}{81 \left (b \,x^{3}+a \right )^{3} a^{2}}+\frac {5 b d \,x^{3}}{6 \left (b \,x^{3}+a \right )^{3} a^{2}}-\frac {92 b c \,x^{2}}{81 \left (b \,x^{3}+a \right )^{3} a^{2}}+\frac {41 e x}{81 \left (b \,x^{3}+a \right )^{3} a}+\frac {11 d}{18 \left (b \,x^{3}+a \right )^{3} a}+\frac {40 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b}+\frac {40 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b}-\frac {20 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b}-\frac {140 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {140 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {70 c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {d \ln \relax (x )}{a^{4}}-\frac {d \ln \left (b \,x^{3}+a \right )}{3 a^{4}}-\frac {c}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-59/81/a^4/(b*x^3+a)^3*b^3*c*x^8+20/81/a^3/(b*x^3+a)^3*b^2*e*x^7+1/3/a^3/(b*x^3+a)^3*b^2*d*x^6-142/81/a^3/(b*x
^3+a)^3*b^2*c*x^5+52/81/a^2/(b*x^3+a)^3*b*e*x^4+5/6/a^2/(b*x^3+a)^3*b*d*x^3-92/81/a^2/(b*x^3+a)^3*b*c*x^2+41/8
1/a/(b*x^3+a)^3*e*x+11/18/a/(b*x^3+a)^3*d+40/243/a^3*e/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-20/243/a^3*e/b/(a/b)^(2
/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+40/243/a^3*e/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))+140/243/a^4*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-70/243/a^4*c/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-140/
243/a^4*c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3*d*ln(b*x^3+a)/a^4-c/a^4/x+d*ln(x)/a^
4

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maxima [A]  time = 3.00, size = 313, normalized size = 1.04 \[ -\frac {280 \, b^{3} c x^{9} - 40 \, a b^{2} e x^{8} - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b e x^{5} - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} e x^{2} - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \, {\left (a^{4} b^{3} x^{10} + 3 \, a^{5} b^{2} x^{7} + 3 \, a^{6} b x^{4} + a^{7} x\right )}} + \frac {d \log \relax (x)}{a^{4}} - \frac {20 \, \sqrt {3} {\left (7 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5}} - \frac {{\left (81 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 70 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (81 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 140 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 40 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/162*(280*b^3*c*x^9 - 40*a*b^2*e*x^8 - 54*a*b^2*d*x^7 + 770*a*b^2*c*x^6 - 104*a^2*b*e*x^5 - 135*a^2*b*d*x^4
+ 670*a^2*b*c*x^3 - 82*a^3*e*x^2 - 99*a^3*d*x + 162*a^3*c)/(a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x
) + d*log(x)/a^4 - 20/243*sqrt(3)*(7*b*c*(a/b)^(2/3) - 2*a*e*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3
))/(a/b)^(1/3))/a^5 - 1/243*(81*b*d*(a/b)^(2/3) + 70*b*c*(a/b)^(1/3) + 20*a*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)
^(2/3))/(a^4*b*(a/b)^(2/3)) - 1/243*(81*b*d*(a/b)^(2/3) - 140*b*c*(a/b)^(1/3) - 40*a*e)*log(x + (a/b)^(1/3))/(
a^4*b*(a/b)^(2/3))

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mupad [B]  time = 5.43, size = 840, normalized size = 2.79 \[ \frac {\frac {41\,e\,x^2}{81\,a}-\frac {c}{a}+\frac {11\,d\,x}{18\,a}-\frac {385\,b^2\,c\,x^6}{81\,a^3}-\frac {140\,b^3\,c\,x^9}{81\,a^4}+\frac {b^2\,d\,x^7}{3\,a^3}+\frac {20\,b^2\,e\,x^8}{81\,a^3}-\frac {335\,b\,c\,x^3}{81\,a^2}+\frac {5\,b\,d\,x^4}{6\,a^2}+\frac {52\,b\,e\,x^5}{81\,a^2}}{a^3\,x+3\,a^2\,b\,x^4+3\,a\,b^2\,x^7+b^3\,x^{10}}+\left (\sum _{k=1}^3\ln \left (\frac {b^2\,\left (-\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^6\,e^2\,32400+32400\,a^2\,d\,e^2+686000\,b^2\,c^3\,x+16000\,a^2\,e^3\,x+229635\,a\,b\,c\,d^2-{\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )}^2\,a^9\,b\,c\,688905-{\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )}^3\,a^{13}\,b\,x\,4782969-\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^5\,b\,d^2\,x\,531441-{\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )}^2\,a^9\,b\,d\,x\,3188646+\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^5\,b\,c\,d\,459270+\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^5\,b\,c\,e\,x\,1134000+226800\,a\,b\,c\,d\,e\,x\right )\,4}{a^{11}\,531441}\right )\,\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\right )+\frac {d\,\ln \relax (x)}{a^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((41*e*x^2)/(81*a) - c/a + (11*d*x)/(18*a) - (385*b^2*c*x^6)/(81*a^3) - (140*b^3*c*x^9)/(81*a^4) + (b^2*d*x^7)
/(3*a^3) + (20*b^2*e*x^8)/(81*a^3) - (335*b*c*x^3)/(81*a^2) + (5*b*d*x^4)/(6*a^2) + (52*b*e*x^5)/(81*a^2))/(a^
3*x + b^3*x^10 + 3*a^2*b*x^4 + 3*a*b^2*x^7) + symsum(log((4*b^2*(32400*a^2*d*e^2 - 32400*root(14348907*a^13*b*
z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 -
64000*a^2*e^3 - 2744000*b^2*c^3, z, k)*a^6*e^2 + 686000*b^2*c^3*x + 16000*a^2*e^3*x + 229635*a*b*c*d^2 - 68890
5*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*
d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)^2*a^9*b*c - 4782969*root(14348907*a^13*b*z^3 + 1
4348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a
^2*e^3 - 2744000*b^2*c^3, z, k)^3*a^13*b*x - 531441*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*
a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k
)*a^5*b*d^2*x - 3188646*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*
d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)^2*a^9*b*d*x + 459270*root(
14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 5
31441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)*a^5*b*c*d + 1134000*root(14348907*a^13*b*z^3 + 14348907
*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3
- 2744000*b^2*c^3, z, k)*a^5*b*c*e*x + 226800*a*b*c*d*e*x))/(531441*a^11))*root(14348907*a^13*b*z^3 + 14348907
*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3
- 2744000*b^2*c^3, z, k), k, 1, 3) + (d*log(x))/a^4

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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)**4,x)

[Out]

Timed out

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