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

Optimal. Leaf size=184 \[ -\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac {\left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a}+\frac {c \log (x)}{a} \]

[Out]

c*ln(x)/a+1/3*(b^(1/3)*d-a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(2/3)-1/6*(d-a^(1/3)*e/b^(1/3))*ln(a^(2/3)
-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(1/3)-1/3*c*ln(b*x^3+a)/a-1/3*(b^(1/3)*d+a^(1/3)*e)*arctan(1/3*(a^(1
/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(2/3)/b^(2/3)*3^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac {\left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a}+\frac {c \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b^(1/3)*d + a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(2/3))) + (c*
Log[x])/a + ((b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(2/3)) - ((d - (a^(1/3)*e)/b^(1/3)
)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(1/3)) - (c*Log[a + b*x^3])/(3*a)

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 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 \left (a+b x^3\right )} \, dx &=\int \left (\frac {c}{a x}+\frac {a d+a e x-b c x^2}{a \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {c \log (x)}{a}+\frac {\int \frac {a d+a e x-b c x^2}{a+b x^3} \, dx}{a}\\ &=\frac {c \log (x)}{a}+\frac {\int \frac {a d+a e x}{a+b x^3} \, dx}{a}-\frac {(b c) \int \frac {x^2}{a+b x^3} \, dx}{a}\\ &=\frac {c \log (x)}{a}-\frac {c \log \left (a+b x^3\right )}{3 a}+\frac {\int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} d+a^{4/3} e\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} d+a^{4/3} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{5/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=\frac {c \log (x)}{a}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {c \log \left (a+b x^3\right )}{3 a}+\frac {1}{2} \left (\frac {d}{\sqrt [3]{a}}+\frac {e}{\sqrt [3]{b}}\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx-\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\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}{6 a^{2/3} \sqrt [3]{b}}\\ &=\frac {c \log (x)}{a}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {c \log \left (a+b x^3\right )}{3 a}+\frac {\left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{2/3}}\\ &=-\frac {\left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3}}+\frac {c \log (x)}{a}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {c \log \left (a+b x^3\right )}{3 a}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 176, normalized size = 0.96 \[ \frac {\left (a^{2/3} e-\sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \left (\sqrt [3]{a} \sqrt [3]{b} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 b^{2/3} c \log \left (a+b x^3\right )-2 \sqrt {3} \sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+6 b^{2/3} c \log (x)}{6 a b^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3]*a^(1/3)*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 6*b^(2/3)*c*Log[x] +
2*(a^(1/3)*b^(1/3)*d - a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] + (-(a^(1/3)*b^(1/3)*d) + a^(2/3)*e)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*b^(2/3)*c*Log[a + b*x^3])/(6*a*b^(2/3))

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fricas [C]  time = 2.62, size = 4588, normalized size = 24.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(2*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b)
+ 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqr
t(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a
^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)*a*log(1/36*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e
)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 +
a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a
^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c
/a)^2*a^2*b*e + b*c*d^2 + b*c^2*e + 2*a*d*e^2 - 1/6*(a*b*d^2 + 2*a*b*c*e)*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2
+ a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2
*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*
e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3
) + 6*c/a) + (b*d^3 + a*e^3)*x) - (((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18
*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(
a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^
2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)*a + 3*sqrt(1/3)*a*sqrt(-(((-
I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d
^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-
1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^
3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)^2*a^2*b - 12*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))
/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 -
(d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/
54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)*a*b*c
+ 36*b*c^2 + 144*a*d*e)/(a^2*b)) - 18*c)*log(-1/36*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/2
7*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 -
3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*
d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)^2*a^2*b*e -
b*c*d^2 - b*c^2*e - 2*a*d*e^2 + 1/6*(a*b*d^2 + 2*a*b*c*e)*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b)
)/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 -
(d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1
/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a) + 2*(
b*d^3 + a*e^3)*x + 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18
*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(
a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^
2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)*a^2*b*e + 6*a*b*d^2 - 6*a*b*
c*e)*sqrt(-(((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3
*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*
sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3
+ a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)^2*a^2*b - 12*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 +
a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c
^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)
*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3)
+ 6*c/a)*a*b*c + 36*b*c^2 + 144*a*d*e)/(a^2*b))) - (((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/
27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3
- 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b
*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)*a - 3*sqrt(
1/3)*a*sqrt(-(((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a
^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(
I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^
3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)^2*a^2*b - 12*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2
+ a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2
*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*
e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3
) + 6*c/a)*a*b*c + 36*b*c^2 + 144*a*d*e)/(a^2*b)) - 18*c)*log(-1/36*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*
e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 +
a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(
a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*
c/a)^2*a^2*b*e - b*c*d^2 - b*c^2*e - 2*a*d*e^2 + 1/6*(a*b*d^2 + 2*a*b*c*e)*((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2
+ a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^
2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d
*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/
3) + 6*c/a) + 2*(b*d^3 + a*e^3)*x - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/
27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3
- 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b
*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)*a^2*b*e + 6
*a*b*d^2 - 6*a*b*c*e)*sqrt(-(((-I*sqrt(3) + 1)*(c^2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^
2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^
2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2)
- 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 6*c/a)^2*a^2*b - 12*((-I*sqrt(3) + 1)*(c^
2/a^2 - (b*c^2 + a*d*e)/(a^2*b))/(-1/27*c^3/a^3 + 1/18*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b
^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^3*b^2))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*c^3/a^3 + 1/1
8*(b*c^2 + a*d*e)*c/(a^3*b) + 1/54*(b*d^3 + a*e^3)/(a^2*b^2) - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/
(a^3*b^2))^(1/3) + 6*c/a)*a*b*c + 36*b*c^2 + 144*a*d*e)/(a^2*b))) - 36*c*log(x))/a

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giac [A]  time = 0.18, size = 179, normalized size = 0.97 \[ -\frac {\sqrt {3} {\left (b d - \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {{\left (b d + \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac {c \log \left ({\left | x \right |}\right )}{a} - \frac {{\left (a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + a^{2} b d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*(b*d - (-a*b^2)^(1/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(-a*b^2)^(2/3) - 1
/6*(b*d + (-a*b^2)^(1/3)*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(-a*b^2)^(2/3) - 1/3*c*log(abs(b*x^3 + a)
)/a + c*log(abs(x))/a - 1/3*(a^2*b*(-a/b)^(1/3)*e + a^2*b*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b)

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maple [A]  time = 0.05, size = 207, normalized size = 1.12 \[ \frac {c \ln \relax (x )}{a}-\frac {c \ln \left (b \,x^{3}+a \right )}{3 a}+\frac {\sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {\sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}-\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(a/b)^(2/3)/b*d*ln(x+(a/b)^(1/3))-1/6/(a/b)^(2/3)/b*d*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/(a/b)^(2/3)*3^
(1/2)/b*d*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3/(a/b)^(1/3)/b*e*ln(x+(a/b)^(1/3))+1/6/(a/b)^(1/3)/b*e*ln
(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/(a/b)^(1/3)/b*e*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3/a*c*ln
(b*x^3+a)+1/a*c*ln(x)

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maxima [A]  time = 3.02, size = 176, normalized size = 0.96 \[ \frac {c \log \relax (x)}{a} + \frac {\sqrt {3} {\left (a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {{\left (2 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*log(x)/a + 1/3*sqrt(3)*(a*e*(a/b)^(2/3) + a*d*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3
))/a^2 - 1/6*(2*b*c*(a/b)^(2/3) - a*e*(a/b)^(1/3) + a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b*(a/b)^(2/
3)) - 1/3*(b*c*(a/b)^(2/3) + a*e*(a/b)^(1/3) - a*d)*log(x + (a/b)^(1/3))/(a*b*(a/b)^(2/3))

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mupad [B]  time = 5.25, size = 716, normalized size = 3.89 \[ \left (\sum _{k=1}^3\ln \left (b^2\,c\,d^2-b^2\,c^2\,e+b^2\,d^3\,x-{\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )}^3\,a^2\,b^3\,x\,36-a\,b\,e^3\,x-\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a\,b^2\,d^2-\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,b^3\,c^2\,x\,4+{\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )}^2\,a^2\,b^2\,e\,3-{\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )}^2\,a\,b^3\,c\,x\,24-\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a\,b^2\,c\,e\,2-2\,b^2\,c\,d\,e\,x-\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a\,b^2\,d\,e\,x\,10\right )\,\mathrm {root}\left (27\,a^3\,b^2\,z^3+27\,a^2\,b^2\,c\,z^2+9\,a^2\,b\,d\,e\,z+9\,a\,b^2\,c^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\right )+\frac {c\,\ln \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(b^2*c*d^2 - b^2*c^2*e + b^2*d^3*x - 36*root(27*a^3*b^2*z^3 + 27*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a
*b^2*c^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)^3*a^2*b^3*x - a*b*e^3*x - root(27*a^3*b^2*z^3 +
27*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a*b^2*c^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)*a*b^2*d^2
- 4*root(27*a^3*b^2*z^3 + 27*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a*b^2*c^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 +
b^2*c^3, z, k)*b^3*c^2*x + 3*root(27*a^3*b^2*z^3 + 27*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a*b^2*c^2*z + 3*a*b*c
*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)^2*a^2*b^2*e - 24*root(27*a^3*b^2*z^3 + 27*a^2*b^2*c*z^2 + 9*a^2*b*d*
e*z + 9*a*b^2*c^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)^2*a*b^3*c*x - 2*root(27*a^3*b^2*z^3 + 2
7*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a*b^2*c^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)*a*b^2*c*e -
2*b^2*c*d*e*x - 10*root(27*a^3*b^2*z^3 + 27*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a*b^2*c^2*z + 3*a*b*c*d*e - a*b
*d^3 + a^2*e^3 + b^2*c^3, z, k)*a*b^2*d*e*x)*root(27*a^3*b^2*z^3 + 27*a^2*b^2*c*z^2 + 9*a^2*b*d*e*z + 9*a*b^2*
c^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k), k, 1, 3) + (c*log(x))/a

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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/(b*x**3+a),x)

[Out]

Timed out

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