3.361 \(\int \frac{c+d x+e x^2}{x (a+b x^3)^4} \, dx\)
Optimal. Leaf size=291 \[ -\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} e+20 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{c \log \left (a+b x^3\right )}{3 a^4}+\frac{c \log (x)}{a^4} \]
[Out]
(x*(a*d + a*e*x - b*c*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*d + 7*a*e*x - 15*b*c*x^2))/(54*a^3*(a + b*x^3)^2)
+ (x*(40*a*d + 28*a*e*x - 99*b*c*x^2))/(162*a^4*(a + b*x^3)) - (2*(20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(11/3)*b^(2/3)) + (c*Log[x])/a^4 + (2*(20*b^(1/3)*d - 7*a^(1
/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) - ((20*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(243*a^(11/3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^4)
________________________________________________________________________________________
Rubi [A] time = 0.516882, antiderivative size = 291, normalized size of antiderivative = 1.,
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} \[ -\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} e+20 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{c \log \left (a+b x^3\right )}{3 a^4}+\frac{c \log (x)}{a^4} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x + e*x^2)/(x*(a + b*x^3)^4),x]
[Out]
(x*(a*d + a*e*x - b*c*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*d + 7*a*e*x - 15*b*c*x^2))/(54*a^3*(a + b*x^3)^2)
+ (x*(40*a*d + 28*a*e*x - 99*b*c*x^2))/(162*a^4*(a + b*x^3)) - (2*(20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(11/3)*b^(2/3)) + (c*Log[x])/a^4 + (2*(20*b^(1/3)*d - 7*a^(1
/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) - ((20*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(243*a^(11/3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^4)
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 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]
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 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
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 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 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 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 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]
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2}{x \left (a+b x^3\right )^4} \, dx &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{\int \frac{-9 b c-8 b d x-7 b e x^2+\frac{6 b^2 c x^3}{a}}{x \left (a+b x^3\right )^3} \, dx}{9 a b}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{\int \frac{54 b^2 c+40 b^2 d x+28 b^2 e x^2-\frac{45 b^3 c x^3}{a}}{x \left (a+b x^3\right )^2} \, dx}{54 a^2 b^2}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{\int \frac{-162 b^3 c-80 b^3 d x-28 b^3 e x^2}{x \left (a+b x^3\right )} \, dx}{162 a^3 b^3}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{162 b^3 c}{a x}-\frac{2 b^3 \left (40 a d+14 a e x-81 b c x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{162 a^3 b^3}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{c \log (x)}{a^4}+\frac{\int \frac{40 a d+14 a e x-81 b c x^2}{a+b x^3} \, dx}{81 a^4}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{c \log (x)}{a^4}+\frac{\int \frac{40 a d+14 a e x}{a+b x^3} \, dx}{81 a^4}-\frac{(b c) \int \frac{x^2}{a+b x^3} \, dx}{a^4}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{c \log (x)}{a^4}-\frac{c \log \left (a+b x^3\right )}{3 a^4}+\frac{\int \frac{\sqrt [3]{a} \left (80 a \sqrt [3]{b} d+14 a^{4/3} e\right )+\sqrt [3]{b} \left (-40 a \sqrt [3]{b} d+14 a^{4/3} 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 (2 \left (20 d-\frac{7 \sqrt [3]{a} e}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{11/3}}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{c \log (x)}{a^4}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{c \log \left (a+b x^3\right )}{3 a^4}-\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{11/3} b^{2/3}}+\frac{\left (20 d+\frac{7 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^{10/3}}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{c \log (x)}{a^4}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}-\frac{c \log \left (a+b x^3\right )}{3 a^4}+\frac{\left (2 \left (20 \sqrt [3]{b} d+7 \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 )}{81 a^{11/3} b^{2/3}}\\ &=\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{2 \left (20 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{c \log (x)}{a^4}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}-\frac{c \log \left (a+b x^3\right )}{3 a^4}\\ \end{align*}
Mathematica [A] time = 0.227502, size = 259, normalized size = 0.89 \[ \frac{\frac{2 \left (7 a^{2/3} e-20 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{4 \left (20 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{54 a^3 (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac{9 a^2 (9 c+x (8 d+7 e x))}{\left (a+b x^3\right )^2}-\frac{4 \sqrt{3} \sqrt [3]{a} \left (7 \sqrt [3]{a} e+20 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 a (27 c+2 x (10 d+7 e x))}{a+b x^3}-162 c \log \left (a+b x^3\right )+486 c \log (x)}{486 a^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)^4),x]
[Out]
((54*a^3*(c + x*(d + e*x)))/(a + b*x^3)^3 + (9*a^2*(9*c + x*(8*d + 7*e*x)))/(a + b*x^3)^2 + (6*a*(27*c + 2*x*(
10*d + 7*e*x)))/(a + b*x^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3
))/Sqrt[3]])/b^(2/3) + 486*c*Log[x] + (4*(20*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3
) + (2*(-20*a^(1/3)*b^(1/3)*d + 7*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 162*c*L
og[a + b*x^3])/(486*a^4)
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Maple [A] time = 0.016, size = 394, normalized size = 1.4 \begin{align*}{\frac{14\,{b}^{2}e{x}^{8}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{20\,{b}^{2}d{x}^{7}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{{b}^{2}c{x}^{6}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{77\,be{x}^{5}}{162\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{52\,bd{x}^{4}}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{5\,bc{x}^{3}}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{67\,e{x}^{2}}{162\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{41\,dx}{81\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{11\,c}{18\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{40\,d}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,d}{243\,{a}^{3}b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{40\,d\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,e}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,e}{243\,{a}^{3}b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,e\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{4}}}+{\frac{c\ln \left ( x \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d*x+c)/x/(b*x^3+a)^4,x)
[Out]
14/81/a^3/(b*x^3+a)^3*b^2*e*x^8+20/81/a^3/(b*x^3+a)^3*b^2*d*x^7+1/3/a^3/(b*x^3+a)^3*c*b^2*x^6+77/162/a^2/(b*x^
3+a)^3*b*e*x^5+52/81/a^2/(b*x^3+a)^3*b*d*x^4+5/6/a^2/(b*x^3+a)^3*b*c*x^3+67/162/a/(b*x^3+a)^3*e*x^2+41/81/a/(b
*x^3+a)^3*d*x+11/18/a/(b*x^3+a)^3*c+40/243/a^3*d/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-20/243/a^3*d/b/(1/b*a)^(2
/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+40/243/a^3*d/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(
1/3)*x-1))-14/243/a^3*e/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+7/243/a^3*e/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x
+(1/b*a)^(2/3))+14/243/a^3*e*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/3*c*ln(b*x^3+
a)/a^4+c*ln(x)/a^4
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [C] time = 7.40697, size = 15555, normalized size = 53.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^4,x, algorithm="fricas")
[Out]
1/236196*(40824*a*b^2*e*x^8 + 58320*a*b^2*d*x^7 + 78732*a*b^2*c*x^6 + 112266*a^2*b*e*x^5 + 151632*a^2*b*d*x^4
+ 196830*a^2*b*c*x^3 + 97686*a^3*e*x^2 + 119556*a^3*d*x + 144342*a^3*c - 2*(a^4*b^3*x^9 + 3*a^5*b^2*x^6 + 3*a^
6*b*x^3 + a^7)*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*
(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*
c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12
+ 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(
531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4)*log(7/236196*(
(-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 56
0*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e
^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561
*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 +
2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4)^2*a^8*b*e + 64800*b*c*d^2 + 45
927*b*c^2*e + 7840*a*d*e^2 - 1/243*(400*a^4*b*d^2 + 567*a^4*b*c*e)*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c
^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d
^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*
b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/1434890
7*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*
a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4) + 4*(8000*b*d^3 + 343*a*e^3)*x) - (118098*b^3*c*x^9 + 354294*a*b^2*c*x^6
+ 354294*a^2*b*c*x^3 + 118098*a^3*c - (a^4*b^3*x^9 + 3*a^5*b^2*x^6 + 3*a^6*b*x^3 + a^7)*((-I*sqrt(3) + 1)*(65
61*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b)
+ 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1
701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*
c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(
800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4) - 3*sqrt(1/3)*(a^4*b^3*x^9 + 3*a^5*b^2*x^6 + 3*a^6
*b*x^3 + a^7)*sqrt(-(((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/1
18098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(53144
1*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3
/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/2869
7814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4)^2*a^8*b
- 78732*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b
*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2
744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118
098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*
b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4)*a^4*b*c + 1549681956*
b*c^2 + 529079040*a*d*e)/(a^8*b)))*log(-7/236196*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a
^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a
^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 5904
9*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 34
3*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(
1/3) + 39366*c/a^4)^2*a^8*b*e - 64800*b*c*d^2 - 45927*b*c^2*e - 7840*a*d*e^2 + 1/243*(400*a^4*b*d^2 + 567*a^4*
b*c*e)*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*
c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 27
44*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/1180
98*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b
^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4) + 8*(8000*b*d^3 + 343*
a*e^3)*x + 1/78732*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3
/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/2869
7814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)
*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^
2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^
4)*a^8*b*e + 388800*a^4*b*d^2 - 275562*a^4*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a
*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a
*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3
) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*
d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12
*b^2))^(1/3) + 39366*c/a^4)^2*a^8*b - 78732*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b)
)/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b
^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*
sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e
^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3)
+ 39366*c/a^4)*a^4*b*c + 1549681956*b*c^2 + 529079040*a*d*e)/(a^8*b))) - (118098*b^3*c*x^9 + 354294*a*b^2*c*x^
6 + 354294*a^2*b*c*x^3 + 118098*a^3*c - (a^4*b^3*x^9 + 3*a^5*b^2*x^6 + 3*a^6*b*x^3 + a^7)*((-I*sqrt(3) + 1)*(6
561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b)
+ 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 -
1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)
*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*
(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4) + 3*sqrt(1/3)*(a^4*b^3*x^9 + 3*a^5*b^2*x^6 + 3*a^
6*b*x^3 + a^7)*sqrt(-(((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/
118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(5314
41*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^
3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/286
97814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4)^2*a^8*b
- 78732*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*
b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 +
2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/11
8098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441
*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4)*a^4*b*c + 1549681956
*b*c^2 + 529079040*a*d*e)/(a^8*b)))*log(-7/236196*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(
a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(
a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 590
49*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 3
43*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^
(1/3) + 39366*c/a^4)^2*a^8*b*e - 64800*b*c*d^2 - 45927*b*c^2*e - 7840*a*d*e^2 + 1/243*(400*a^4*b*d^2 + 567*a^4
*b*c*e)*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b
*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2
744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118
098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*
b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a^4) + 8*(8000*b*d^3 + 343
*a*e^3)*x - 1/78732*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b))/(-1/27*c^
3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b^2) - 1/286
97814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I*sqrt(3) + 1
)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*b
^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 39366*c/a
^4)*a^8*b*e + 388800*a^4*b*d^2 - 275562*a^4*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*
a*d*e)/(a^8*b))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*
a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/
3) + 59049*(I*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b
*d^3 + 343*a*e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^1
2*b^2))^(1/3) + 39366*c/a^4)^2*a^8*b - 78732*((-I*sqrt(3) + 1)*(6561*c^2/a^8 - (6561*b*c^2 + 560*a*d*e)/(a^8*b
))/(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*e^3)/(a^11*
b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3) + 59049*(I
*sqrt(3) + 1)*(-1/27*c^3/a^12 + 1/118098*(6561*b*c^2 + 560*a*d*e)*c/(a^12*b) + 4/14348907*(8000*b*d^3 + 343*a*
e^3)/(a^11*b^2) - 1/28697814*(531441*b^2*c^3 + 2744*a^2*e^3 - 80*(800*d^3 - 1701*c*d*e)*a*b)/(a^12*b^2))^(1/3)
+ 39366*c/a^4)*a^4*b*c + 1549681956*b*c^2 + 529079040*a*d*e)/(a^8*b))) + 236196*(b^3*c*x^9 + 3*a*b^2*c*x^6 +
3*a^2*b*c*x^3 + a^3*c)*log(x))/(a^4*b^3*x^9 + 3*a^5*b^2*x^6 + 3*a^6*b*x^3 + a^7)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**2+d*x+c)/x/(b*x**3+a)**4,x)
[Out]
Timed out
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Giac [A] time = 1.09416, size = 409, normalized size = 1.41 \begin{align*} -\frac{c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac{c \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{2 \, \sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - 7 \, \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 )}{243 \, a^{4} b^{2}} + \frac{28 \, a b^{2} x^{8} e + 40 \, a b^{2} d x^{7} + 54 \, a b^{2} c x^{6} + 77 \, a^{2} b x^{5} e + 104 \, a^{2} b d x^{4} + 135 \, a^{2} b c x^{3} + 67 \, a^{3} x^{2} e + 82 \, a^{3} d x + 99 \, a^{3} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{4}} + \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} 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^{5} b^{4}} - \frac{2 \,{\left (7 \, a^{5} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 20 \, a^{5} b d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{9} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^4,x, algorithm="giac")
[Out]
-1/3*c*log(abs(b*x^3 + a))/a^4 + c*log(abs(x))/a^4 + 2/243*sqrt(3)*(20*(-a*b^2)^(1/3)*b*d - 7*(-a*b^2)^(2/3)*e
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/162*(28*a*b^2*x^8*e + 40*a*b^2*d*x^7 + 5
4*a*b^2*c*x^6 + 77*a^2*b*x^5*e + 104*a^2*b*d*x^4 + 135*a^2*b*c*x^3 + 67*a^3*x^2*e + 82*a^3*d*x + 99*a^3*c)/((b
*x^3 + a)^3*a^4) + 1/243*(20*(-a*b^2)^(1/3)*a*b^3*d + 7*(-a*b^2)^(2/3)*a*b^2*e)*log(x^2 + x*(-a/b)^(1/3) + (-a
/b)^(2/3))/(a^5*b^4) - 2/243*(7*a^5*b*(-a/b)^(1/3)*e + 20*a^5*b*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^
9*b)