3.357 \(\int \frac{c+d x+e x^2}{x^4 (a+b x^3)^3} \, dx\)
Optimal. Leaf size=298 \[ -\frac{x \left (-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x} \]
[Out]
-c/(3*a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(6*a^2*(a + b*x^3)^2) - (x*(11*
b*d + 10*b*e*x - (15*b^2*c*x^2)/a))/(18*a^3*(a + b*x^3)) + (2*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(
1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)) - (3*b*c*Log[x])/a^4 - (2*b^(1/3)*(10*b^(1/3)*d -
7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)) + (b^(1/3)*(10*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])/(27*a^(11/3)) + (b*c*Log[a + b*x^3])/a^4
________________________________________________________________________________________
Rubi [A] time = 0.589456, antiderivative size = 298, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used
= {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{x \left (-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]
[Out]
-c/(3*a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(6*a^2*(a + b*x^3)^2) - (x*(11*
b*d + 10*b*e*x - (15*b^2*c*x^2)/a))/(18*a^3*(a + b*x^3)) + (2*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(
1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)) - (3*b*c*Log[x])/a^4 - (2*b^(1/3)*(10*b^(1/3)*d -
7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)) + (b^(1/3)*(10*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])/(27*a^(11/3)) + (b*c*Log[a + b*x^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^4 \left (a+b x^3\right )^3} \, dx &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b c-6 b d x-6 b e x^2+\frac{6 b^2 c x^3}{a}+\frac{5 b^2 d x^4}{a}+\frac{4 b^2 e x^5}{a}-\frac{3 b^3 c x^6}{a^2}}{x^4 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \frac{18 b^3 c+18 b^3 d x+18 b^3 e x^2-\frac{36 b^4 c x^3}{a}-\frac{22 b^4 d x^4}{a}-\frac{10 b^4 e x^5}{a}}{x^4 \left (a+b x^3\right )} \, dx}{18 a^2 b^3}\\ &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^3 c}{a x^4}+\frac{18 b^3 d}{a x^3}+\frac{18 b^3 e}{a x^2}-\frac{54 b^4 c}{a^2 x}-\frac{2 b^4 \left (20 a d+14 a e x-27 b c x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac{3 b c \log (x)}{a^4}-\frac{b \int \frac{20 a d+14 a e x-27 b c x^2}{a+b x^3} \, dx}{9 a^4}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac{3 b c \log (x)}{a^4}-\frac{b \int \frac{20 a d+14 a e x}{a+b x^3} \, dx}{9 a^4}+\frac{\left (3 b^2 c\right ) \int \frac{x^2}{a+b x^3} \, dx}{a^4}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac{3 b c \log (x)}{a^4}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{b^{2/3} \int \frac{\sqrt [3]{a} \left (40 a \sqrt [3]{b} d+14 a^{4/3} e\right )+\sqrt [3]{b} \left (-20 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}{27 a^{14/3}}-\frac{\left (2 b \left (10 d-\frac{7 \sqrt [3]{a} e}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac{3 b c \log (x)}{a^4}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}+\frac{\left (\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \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}{27 a^{11/3}}-\frac{\left (b^{2/3} \left (10 \sqrt [3]{b} d+7 \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}{9 a^{10/3}}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac{3 b c \log (x)}{a^4}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{\sqrt [3]{b} \left (10 \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 )}{27 a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{\left (2 \sqrt [3]{b} \left (10 \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 )}{9 a^{11/3}}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d+10 b e x-\frac{15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac{2 \sqrt [3]{b} \left (10 \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 )}{9 \sqrt{3} a^{11/3}}-\frac{3 b c \log (x)}{a^4}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{\sqrt [3]{b} \left (10 \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 )}{27 a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}\\ \end{align*}
Mathematica [A] time = 0.370094, size = 255, normalized size = 0.86 \[ -\frac{-2 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{9 a^2 b (c+x (d+e x))}{\left (a+b x^3\right )^2}+4 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a b (12 c+x (11 d+10 e x))}{a+b x^3}-54 b c \log \left (a+b x^3\right )-4 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{18 a c}{x^3}+\frac{27 a d}{x^2}+\frac{54 a e}{x}+162 b c \log (x)}{54 a^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]
[Out]
-((18*a*c)/x^3 + (27*a*d)/x^2 + (54*a*e)/x + (9*a^2*b*(c + x*(d + e*x)))/(a + b*x^3)^2 + (3*a*b*(12*c + x*(11*
d + 10*e*x)))/(a + b*x^3) - 4*Sqrt[3]*a^(1/3)*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a
^(1/3))/Sqrt[3]] + 162*b*c*Log[x] + 4*b^(1/3)*(10*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] -
2*b^(1/3)*(10*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] - 54*b*c*Log[a +
b*x^3])/(54*a^4)
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Maple [A] time = 0.017, size = 351, normalized size = 1.2 \begin{align*} -{\frac{5\,{x}^{5}e{b}^{2}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{11\,{x}^{4}{b}^{2}d}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{2\,{b}^{2}c{x}^{3}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,be{x}^{2}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,bdx}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,bc}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{20\,d}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,d}{27\,{a}^{3}}\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{20\,d\sqrt{3}}{27\,{a}^{3}}\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}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,e}{27\,{a}^{3}}\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}}{27\,{a}^{3}}\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{bc\ln \left ( b{x}^{3}+a \right ) }{{a}^{4}}}-{\frac{c}{3\,{a}^{3}{x}^{3}}}-{\frac{d}{2\,{a}^{3}{x}^{2}}}-{\frac{e}{{a}^{3}x}}-3\,{\frac{bc\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^4/(b*x^3+a)^3,x)
[Out]
-5/9/a^3/(b*x^3+a)^2*x^5*e*b^2-11/18/a^3/(b*x^3+a)^2*x^4*b^2*d-2/3/a^3*b^2/(b*x^3+a)^2*c*x^3-13/18/a^2/(b*x^3+
a)^2*x^2*b*e-7/9/a^2/(b*x^3+a)^2*b*x*d-5/6/a^2*b/(b*x^3+a)^2*c-20/27/a^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d+1
0/27/a^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d-20/27/a^3/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1
/2)*(2/(1/b*a)^(1/3)*x-1))*d+14/27/a^3*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-7/27/a^3*e/(1/b*a)^(1/3)*ln(x^2-(1/
b*a)^(1/3)*x+(1/b*a)^(2/3))-14/27/a^3*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+b*c*ln
(b*x^3+a)/a^4-1/3*c/a^3/x^3-1/2*d/a^3/x^2-e/a^3/x-3*b*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^4/(b*x^3+a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [C] time = 12.2192, size = 14256, normalized size = 47.84 \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^4/(b*x^3+a)^3,x, algorithm="fricas")
[Out]
-1/108*(168*a*b^2*e*x^8 + 120*a*b^2*d*x^7 + 108*a*b^2*c*x^6 + 294*a^2*b*e*x^5 + 192*a^2*b*d*x^4 + 162*a^2*b*c*
x^3 + 108*a^3*e*x^2 + 54*a^3*d*x + 36*a^3*c + 2*(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)*(2*(1/2)^(2/3)*(-I*sqrt(
3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b
/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a
*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(
729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^
(1/3) - 54*b*c/a^4)*log(7/4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8
)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b
^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3
*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 274
4*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)^2*a^8*e + 5400*b^2*c*d^2 + 5103*b^2*c^
2*e + 3920*a*b*d*e^2 + (100*a^4*b*d^2 + 189*a^4*b*c*e)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729
*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a
*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)
*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/
a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4) + 4*(1000*b
^2*d^3 + 343*a*b*e^3)*x) - (162*b^3*c*x^9 + 324*a*b^2*c*x^6 + 162*a^2*b*c*x^3 + (a^4*b^2*x^9 + 2*a^5*b*x^6 + a
^6*x^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^1
2 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*
e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*
b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(20
0*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4) + 3*sqrt(1/3)*(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)*sqrt(-
((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(
1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 4
0*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 +
343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 -
567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)^2*a^8 + 108*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (7
29*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280
*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/
3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*
c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)*a^4*b*c +
2916*b^2*c^2 + 4480*a*b*d*e)/a^8))*log(-7/4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 +
280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*
c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3
) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19
683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)^2*a^8*e - 5400*b^2*c*
d^2 - 5103*b^2*c^2*e - 3920*a*b*d*e^2 - (100*a^4*b*d^2 + 189*a^4*b*c*e)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b
^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(72
9*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1
/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 +
280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c
/a^4) + 8*(1000*b^2*d^3 + 343*a*b*e^3)*x + 3/4*sqrt(1/3)*(7*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 -
(729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 +
280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^
(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)
*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)*a^8*e
- 400*a^4*b*d^2 + 378*a^4*b*c*e)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*
a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^1
2 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1
)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b
^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)^2*a^8 + 108*(2*(1/2)^(2/3)
*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 34
3*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 56
7*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a
^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b
^2)/a^12)^(1/3) - 54*b*c/a^4)*a^4*b*c + 2916*b^2*c^2 + 4480*a*b*d*e)/a^8)) - (162*b^3*c*x^9 + 324*a*b^2*c*x^6
+ 162*a^2*b*c*x^3 + (a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (
729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 28
0*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1
/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b
*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4) - 3*sqrt
(1/3)*(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2
*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d
*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*
sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12
+ (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)^2*a^8 + 108*(2*
(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000
*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(2
00*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343
*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567
*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)*a^4*b*c + 2916*b^2*c^2 + 4480*a*b*d*e)/a^8))*log(-7/4*(2*(1/2)^(2/3)*
(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343
*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567
*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^
11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^
2)/a^12)^(1/3) - 54*b*c/a^4)^2*a^8*e - 5400*b^2*c*d^2 - 5103*b^2*c^2*e - 3920*a*b*d*e^2 - (100*a^4*b*d^2 + 189
*a^4*b*c*e)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3
/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^
2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1
000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40
*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4) + 8*(1000*b^2*d^3 + 343*a*b*e^3)*x - 3/4*sqrt(1/3)*(7*
(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1
000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40
*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 +
343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 -
567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)*a^8*e - 400*a^4*b*d^2 + 378*a^4*b*c*e)*sqrt(-((2*(1/2)^(2/3)*(-I*s
qrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^
3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*
e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 -
81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^
12)^(1/3) - 54*b*c/a^4)^2*a^8 + 108*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(729*b^2*c^2/a^8 - (729*b^2*c^2 + 280*a*b*
d*e)/a^8)/(39366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 +
(19683*b^3*c^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(3
9366*b^3*c^3/a^12 + 8*(1000*b*d^3 + 343*a*e^3)*b/a^11 - 81*(729*b^2*c^2 + 280*a*b*d*e)*b*c/a^12 + (19683*b^3*c
^3 + 2744*a^2*b*e^3 - 40*(200*d^3 - 567*c*d*e)*a*b^2)/a^12)^(1/3) - 54*b*c/a^4)*a^4*b*c + 2916*b^2*c^2 + 4480*
a*b*d*e)/a^8)) + 324*(b^3*c*x^9 + 2*a*b^2*c*x^6 + a^2*b*c*x^3)*log(x))/(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)
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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**4/(b*x**3+a)**3,x)
[Out]
Timed out
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Giac [A] time = 1.08487, size = 421, normalized size = 1.41 \begin{align*} \frac{b c \log \left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac{3 \, b c \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d + 7 \, \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 )}{27 \, a^{4} b} - \frac{2 \, \sqrt{3}{\left (10 \, \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 )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b^{3}} + \frac{2 \,{\left (7 \, a^{5} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 10 \, a^{5} 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 )}{27 \, a^{9} b} - \frac{28 \, a b^{2} x^{8} e + 20 \, a b^{2} d x^{7} + 18 \, a b^{2} c x^{6} + 49 \, a^{2} b x^{5} e + 32 \, a^{2} b d x^{4} + 27 \, a^{2} b c x^{3} + 18 \, a^{3} x^{2} e + 9 \, a^{3} d x + 6 \, a^{3} c}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,x, algorithm="giac")
[Out]
b*c*log(abs(b*x^3 + a))/a^4 - 3*b*c*log(abs(x))/a^4 - 1/27*(10*(-a*b^2)^(1/3)*b*d + 7*(-a*b^2)^(2/3)*e)*log(x^
2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 2/27*sqrt(3)*(10*(-a*b^2)^(1/3)*a*b^3*d - 7*(-a*b^2)^(2/3)*a*b^2*
e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b^3) + 2/27*(7*a^5*b^2*(-a/b)^(1/3)*e + 10*a^5*b
^2*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^9*b) - 1/18*(28*a*b^2*x^8*e + 20*a*b^2*d*x^7 + 18*a*b^2*c*x^6
+ 49*a^2*b*x^5*e + 32*a^2*b*d*x^4 + 27*a^2*b*c*x^3 + 18*a^3*x^2*e + 9*a^3*d*x + 6*a^3*c)/((b*x^3 + a)^2*a^4*x
^3)