3.355 \(\int \frac{c+d x+e x^2}{x^2 (a+b x^3)^3} \, dx\)
Optimal. Leaf size=267 \[ -\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt [3]{b}}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3} \]
[Out]
-(c/(a^3*x)) + (x*(a*e - b*c*x - b*d*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*e - 10*b*c*x - 9*b*d*x^2))/(18*a^3*
(a + b*x^3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(
10/3)*b^(1/3)) + (d*Log[x])/a^3 + ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(1/3)
) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*b^(1/3)) - (d*L
og[a + b*x^3])/(3*a^3)
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Rubi [A] time = 0.461799, antiderivative size = 267, 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} \[ -\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt [3]{b}}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^3),x]
[Out]
-(c/(a^3*x)) + (x*(a*e - b*c*x - b*d*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*e - 10*b*c*x - 9*b*d*x^2))/(18*a^3*
(a + b*x^3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(
10/3)*b^(1/3)) + (d*Log[x])/a^3 + ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(1/3)
) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*b^(1/3)) - (d*L
og[a + b*x^3])/(3*a^3)
Rule 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^2 \left (a+b x^3\right )^3} \, dx &=\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b c-6 b d x-5 b e x^2+\frac{4 b^2 c x^3}{a}+\frac{3 b^2 d x^4}{a}}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \frac{18 b^3 c+18 b^3 d x+10 b^3 e x^2-\frac{10 b^4 c x^3}{a}}{x^2 \left (a+b x^3\right )} \, dx}{18 a^2 b^3}\\ &=\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^3 c}{a x^2}+\frac{18 b^3 d}{a x}+\frac{2 b^3 \left (5 a e-14 b c x-9 b d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\int \frac{5 a e-14 b c x-9 b d x^2}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\int \frac{5 a e-14 b c x}{a+b x^3} \, dx}{9 a^3}-\frac{(b d) \int \frac{x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}-\frac{d \log \left (a+b x^3\right )}{3 a^3}+\frac{\int \frac{\sqrt [3]{a} \left (-14 \sqrt [3]{a} b c+10 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-14 \sqrt [3]{a} b c-5 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} \sqrt [3]{b}}+\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3}}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^3}-\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{10/3} \sqrt [3]{b}}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{10/3} \sqrt [3]{b}}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt [3]{b}}+\frac{d \log (x)}{a^3}+\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac{\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac{d \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.221198, size = 248, normalized size = 0.93 \[ \frac{-\frac{\left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{2 \left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} a^{2/3} \left (5 a^{2/3} e-14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}+\frac{9 a^2 \left (a (d+e x)-b c x^2\right )}{\left (a+b x^3\right )^2}+\frac{3 a \left (6 a d+5 a e x-10 b c x^2\right )}{a+b x^3}-18 a d \log \left (a+b x^3\right )-\frac{54 a c}{x}+54 a d \log (x)}{54 a^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^3),x]
[Out]
((-54*a*c)/x + (3*a*(6*a*d + 5*a*e*x - 10*b*c*x^2))/(a + b*x^3) + (9*a^2*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^
3)^2 - (2*Sqrt[3]*a^(2/3)*(-14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) +
54*a*d*Log[x] + (2*(14*a^(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - ((14*a^(2/3)*b^(2
/3)*c + 5*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 18*a*d*Log[a + b*x^3])/(54*a^4)
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Maple [A] time = 0.017, size = 334, normalized size = 1.3 \begin{align*} -{\frac{5\,{b}^{2}c{x}^{5}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,be{x}^{4}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{bd{x}^{3}}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,bc{x}^{2}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{4\,ex}{9\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{d}{2\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,e}{27\,b{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,e}{54\,b{a}^{2}}\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{5\,\sqrt{3}e}{27\,b{a}^{2}}\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\,c}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,c}{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\,c\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{d\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{d\ln \left ( x \right ) }{{a}^{3}}}-{\frac{c}{{a}^{3}x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d*x+c)/x^2/(b*x^3+a)^3,x)
[Out]
-5/9/a^3/(b*x^3+a)^2*b^2*x^5*c+5/18/a^2/(b*x^3+a)^2*x^4*b*e+1/3/a^2/(b*x^3+a)^2*b*d*x^3-13/18/a^2/(b*x^3+a)^2*
b*x^2*c+4/9/a/(b*x^3+a)^2*x*e+1/2/a/(b*x^3+a)^2*d+5/27/a^2/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*e-5/54/a^2/b/(1
/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e+5/27/a^2/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/
b*a)^(1/3)*x-1))*e+14/27/a^3/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c-7/27/a^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x
+(1/b*a)^(2/3))*c-14/27/a^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-1/3*d*ln(b*x^3+a
)/a^3+d*ln(x)/a^3-c/a^3/x
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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^2/(b*x^3+a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [C] time = 10.0854, size = 13300, normalized size = 49.81 \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^2/(b*x^3+a)^3,x, algorithm="fricas")
[Out]
-1/2916*(4536*b^2*c*x^6 - 810*a*b*e*x^5 - 972*a*b*d*x^4 + 7938*a*b*c*x^3 - 1296*a^2*e*x^2 - 1458*a^2*d*x + 291
6*a^2*c + 2*(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(
81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*
(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*log(-7/1458*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2
- 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*
d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1
/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*
b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^7*b*c - 1134*a*b*c*d^2 + 1
960*a*b*c^2*e + 225*a^2*d*e^2 + 1/54*(252*a^4*b*c*d - 25*a^5*e^2)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70
*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3
- 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3) - (2744*b^2*c^3 - 125*a^2*e^3)*x)
+ (1458*b^2*d*x^7 + 2916*a*b*d*x^4 + 1458*a^2*d*x - (a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*((-I*sqrt(3) + 1)*(81*
d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125
*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(
I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*
d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3) + 3*sqrt(1/3
)*(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3
/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^1
0*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*
d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(27
44*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 7
0*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3
- 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27
*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/
(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/
a^6))*log(7/1458*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c
*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3
- 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/3936
6*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a
^10*b))^(1/3) + 486*d/a^3)^2*a^7*b*c + 1134*a*b*c*d^2 - 1960*a*b*c^2*e - 225*a^2*d*e^2 - 1/54*(252*a^4*b*c*d -
25*a^5*e^2)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*
d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 1
25*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2
744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*
b))^(1/3) + 486*d/a^3) - 2*(2744*b^2*c^3 - 125*a^2*e^3)*x + 1/486*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(81*d^2/a^6 -
(81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3
- 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3)
+ 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*
c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^7*b*c - 3402*a^4*b*
c*d - 675*a^5*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d
^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(274
4*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9
+ 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^
2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(
81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*
(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/a^6)) + (1458*b^2*d
*x^7 + 2916*a*b*d*x^4 + 1458*a^2*d*x - (a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81
*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27
*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1
)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*
e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3) - 3*sqrt(1/3)*(a^3*b^2*x^
7 + 2*a^4*b*x^4 + a^5*x)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458
*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/3936
6*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)
*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 -
125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(
-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*
a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/
1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/
39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/a^6))*log(7/1
458*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1
/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^
3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c
^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3)
+ 486*d/a^3)^2*a^7*b*c + 1134*a*b*c*d^2 - 1960*a*b*c^2*e - 225*a^2*d*e^2 - 1/54*(252*a^4*b*c*d - 25*a^5*e^2)*
((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/393
66*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(
a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 +
125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 4
86*d/a^3) - 2*(2744*b^2*c^3 - 125*a^2*e^3)*x - 1/486*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70
*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3
- 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*
d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(
a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^7*b*c - 3402*a^4*b*c*d - 675*a^5
*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*
d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 1
25*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2
744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*
b))^(1/3) + 486*d/a^3)^2*a^6 - 972*((-I*sqrt(3) + 1)*(81*d^2/a^6 - (81*d^2 - 70*c*e)/a^6)/(-1/27*d^3/a^9 + 1/1
458*(81*d^2 - 70*c*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/3
9366*(2744*b^2*c^3 - 125*a^2*e^3)/(a^10*b))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(81*d^2 - 70*c
*e)*d/a^9 + 1/39366*(2744*b^2*c^3 + 125*a^2*e^3 - 27*(27*d^3 - 70*c*d*e)*a*b)/(a^10*b) - 1/39366*(2744*b^2*c^3
- 125*a^2*e^3)/(a^10*b))^(1/3) + 486*d/a^3)*a^3*d + 236196*d^2 - 816480*c*e)/a^6)) - 2916*(b^2*d*x^7 + 2*a*b*
d*x^4 + a^2*d*x)*log(x))/(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)
________________________________________________________________________________________
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**2/(b*x**3+a)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.11718, size = 377, normalized size = 1.41 \begin{align*} -\frac{d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{d \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{3}} - \frac{28 \, b^{2} c x^{6} - 5 \, a b x^{5} e - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} x^{2} e - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} x} + \frac{{\left (14 \, a^{3} b^{2} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a^{4} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d*x+c)/x^2/(b*x^3+a)^3,x, algorithm="giac")
[Out]
-1/3*d*log(abs(b*x^3 + a))/a^3 + d*log(abs(x))/a^3 + 1/54*(5*(-a*b^2)^(1/3)*a*e - 14*(-a*b^2)^(2/3)*c)*log(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) + 1/27*sqrt(3)*(5*(-a*b^2)^(1/3)*a*b^2*e + 14*(-a*b^2)^(2/3)*b^2*c)*
arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^3) - 1/18*(28*b^2*c*x^6 - 5*a*b*x^5*e - 6*a*b*d*x
^4 + 49*a*b*c*x^3 - 8*a^2*x^2*e - 9*a^2*d*x + 18*a^2*c)/((b*x^3 + a)^2*a^3*x) + 1/27*(14*a^3*b^2*c*(-a/b)^(1/3
) - 5*a^4*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b)