3.5.10 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 (a+b x^3)} \, dx\) [410]

Optimal. Leaf size=260 \[ -\frac {c}{2 a x^2}-\frac {d}{a x}+\frac {\left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} b^{2/3}}+\frac {e \log (x)}{a}-\frac {\left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} b^{2/3}}+\frac {\left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} b^{2/3}}-\frac {(b e-a h) \log \left (a+b x^3\right )}{3 a b} \]

[Out]

-1/2*c/a/x^2-d/a/x+e*ln(x)/a-1/3*(b^(1/3)*(-a*f+b*c)-a^(1/3)*(-a*g+b*d))*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(2/3)
+1/6*(b^(1/3)*(-a*f+b*c)-a^(1/3)*(-a*g+b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(2/3)-1/3*(-a
*h+b*e)*ln(b*x^3+a)/a/b+1/3*(b^(4/3)*c+a^(1/3)*b*d-a*b^(1/3)*f-a^(4/3)*g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(
1/3)*3^(1/2))/a^(5/3)/b^(2/3)*3^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 258, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt {3} a^{5/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}-a f+b c\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{5/3} b^{2/3}}-\frac {(b e-a h) \log \left (a+b x^3\right )}{3 a b}-\frac {c}{2 a x^2}-\frac {d}{a x}+\frac {e \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)),x]

[Out]

-1/2*c/(a*x^2) - d/(a*x) + ((b^(4/3)*c + a^(1/3)*b*d - a*b^(1/3)*f - a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)
/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*b^(2/3)) + (e*Log[x])/a - ((b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*L
og[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)*b^(2/3)) + ((b*c - a*f - (a^(1/3)*(b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1
/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(5/3)*b^(1/3)) - ((b*e - a*h)*Log[a + b*x^3])/(3*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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

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 631

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 642

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 648

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 1848

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 1874

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 1885

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

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Mathematica [A]
time = 0.22, size = 257, normalized size = 0.99 \begin {gather*} \frac {-\frac {3 a^{2/3} c}{x^2}-\frac {6 a^{2/3} d}{x}+\frac {2 \sqrt {3} \left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+6 a^{2/3} e \log (x)-\frac {2 \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {\left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {2 a^{2/3} (-b e+a h) \log \left (a+b x^3\right )}{b}}{6 a^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)),x]

[Out]

((-3*a^(2/3)*c)/x^2 - (6*a^(2/3)*d)/x + (2*Sqrt[3]*(b^(4/3)*c + a^(1/3)*b*d - a*b^(1/3)*f - a^(4/3)*g)*ArcTan[
(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 6*a^(2/3)*e*Log[x] - (2*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f
+ a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(
2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + (2*a^(2/3)*(-(b*e) + a*h)*Log[a + b*x^3])/b)/(6*a^(5/3))

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Maple [A]
time = 0.42, size = 251, normalized size = 0.97

method	result	size
default	\(\frac {\left (a f -b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a g -b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a h -b e \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{a}-\frac {c}{2 a \,x^{2}}-\frac {d}{a x}+\frac {e \ln \left (x \right )}{a}\)	\(251\)
risch	\(\frac {-\frac {x d}{a}-\frac {c}{2 a}}{x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{5} b^{3} \textit {\_Z}^{3}+\left (-3 a^{5} b^{2} h +3 a^{4} b^{3} e \right ) \textit {\_Z}^{2}+\left (3 a^{5} b \,h^{2}-6 a^{4} b^{2} e h +3 a^{4} b^{2} f g -3 a^{3} b^{3} c g -3 a^{3} b^{3} d f +3 a^{3} b^{3} e^{2}+3 a^{2} b^{4} c d \right ) \textit {\_Z} -a^{5} h^{3}+3 a^{4} b e \,h^{2}-3 a^{4} b f g h +a^{4} b \,g^{3}+3 a^{3} b^{2} c g h +3 a^{3} b^{2} d f h -3 a^{3} b^{2} d \,g^{2}-3 a^{3} b^{2} e^{2} h +3 a^{3} b^{2} e f g -a^{3} b^{2} f^{3}-3 a^{2} b^{3} c d h -3 a^{2} b^{3} c e g +3 a^{2} b^{3} c \,f^{2}+3 a^{2} b^{3} d^{2} g -3 a^{2} b^{3} d e f +a^{2} b^{3} e^{3}-3 a \,b^{4} c^{2} f +3 a \,b^{4} c d e -a \,b^{4} d^{3}+b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{5} b^{3} \textit {\_R}^{3}+\left (11 a^{5} b^{2} h -8 a^{4} b^{3} e \right ) \textit {\_R}^{2}+\left (-10 a^{5} b \,h^{2}+14 a^{4} b^{2} e h -10 a^{4} b^{2} f g +10 a^{3} b^{3} c g +10 a^{3} b^{3} d f -4 a^{3} b^{3} e^{2}-10 a^{2} b^{4} c d \right ) \textit {\_R} +3 a^{5} h^{3}-6 a^{4} b e \,h^{2}+9 a^{4} b f g h -3 a^{4} b \,g^{3}-9 a^{3} b^{2} c g h -9 a^{3} b^{2} d f h +9 a^{3} b^{2} d \,g^{2}+3 a^{3} b^{2} e^{2} h -6 a^{3} b^{2} e f g +3 a^{3} b^{2} f^{3}+9 a^{2} b^{3} c d h +6 a^{2} b^{3} c e g -9 a^{2} b^{3} c \,f^{2}-9 a^{2} b^{3} d^{2} g +6 a^{2} b^{3} d e f +9 a \,b^{4} c^{2} f -6 a \,b^{4} c d e +3 a \,b^{4} d^{3}-3 b^{5} c^{3}\right ) x +\left (a^{5} b^{2} g -a^{4} b^{3} d \right ) \textit {\_R}^{2}+\left (-a^{5} b g h +a^{4} b^{2} d h -2 a^{4} b^{2} e g -a^{4} b^{2} f^{2}+2 a^{3} b^{3} c f +2 a^{3} b^{3} d e -a^{2} b^{4} c^{2}\right ) \textit {\_R} +3 a^{4} b e g h -3 a^{3} b^{2} d e h -3 a^{3} b^{2} e^{2} g +3 a^{3} b^{2} e \,f^{2}-6 a^{2} b^{3} c e f +3 a^{2} b^{3} d \,e^{2}+3 a \,b^{4} c^{2} e \right )\right )}{3}+\frac {e \ln \left (x \right )}{a}\)	\(827\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 274, normalized size = 1.05 \begin {gather*} \frac {e \log \left (x\right )}{a} - \frac {\sqrt {3} {\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b f \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} b} + \frac {{\left (2 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + b c - a f\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 (a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - b \left (\frac {a}{b}\right )^{\frac {2}{3}} e + b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - b c + a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, d x + c}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a),x, algorithm="maxima")

[Out]

e*log(x)/a - 1/3*sqrt(3)*(b^2*d*(a/b)^(2/3) - a*b*g*(a/b)^(2/3) + b^2*c*(a/b)^(1/3) - a*b*f*(a/b)^(1/3))*arcta
n(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b) + 1/6*(2*a*h*(a/b)^(2/3) - 2*b*(a/b)^(2/3)*e - b*d*(a/b
)^(1/3) + a*g*(a/b)^(1/3) + b*c - a*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b*(a/b)^(2/3)) + 1/3*(a*h*(a/
b)^(2/3) - b*(a/b)^(2/3)*e + b*d*(a/b)^(1/3) - a*g*(a/b)^(1/3) - b*c + a*f)*log(x + (a/b)^(1/3))/(a*b*(a/b)^(2
/3)) - 1/2*(2*d*x + c)/(a*x^2)

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Fricas [C] Result contains complex when optimal does not.
time = 35.63, size = 15424, normalized size = 59.32 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a),x, algorithm="fricas")

[Out]

-1/36*(2*((-I*sqrt(3) + 1)*((b*e - a*h)^2/(a^2*b^2) - (b^3*c*d + a^3*h^2 + (f*g - 2*e*h)*a^2*b + (e^2 - d*f -
c*g)*a*b^2)/(a^3*b^2))/(-1/27*(b*e - a*h)^3/(a^3*b^3) + 1/18*(b^3*c*d + a^3*h^2 + (f*g - 2*e*h)*a^2*b + (e^2 -
d*f - c*g)*a*b^2)*(b*e - a*h)/(a^4*b^3) - 1/54*(b^4*c^3 + a*b^3*d^3 - 3*a*b^3*c^2*f + 3*a^2*b^2*c*f^2 - a^3*b
*f^3 - 3*a^2*b^2*d^2*g + 3*a^3*b*d*g^2 - a^4*g^3)/(a^5*b^2) - 1/54*(b^5*c^3 - a^5*h^3 + (g^3 - 3*f*g*h + 3*e*h
^2)*a^4*b - (f^3 - 3*e*f*g + 3*e^2*h - 3*c*g*h + 3*(g^2 - f*h)*d)*a^3*b^2 + (e^3 - 3*d*e*f + 3*d^2*g + 3*(f^2
- e*g - d*h)*c)*a^2*b^3 - (d^3 - 3*c*d*e + 3*c^2*f)*a*b^4)/(a^5*b^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*(b*e -
a*h)^3/(a^3*b^3) + 1/18*(b^3*c*d + a^3*h^2 + (f*g - 2*e*h)*a^2*b + (e^2 - d*f - c*g)*a*b^2)*(b*e - a*h)/(a^4*b
^3) - 1/54*(b^4*c^3 + a*b^3*d^3 - 3*a*b^3*c^2*f + 3*a^2*b^2*c*f^2 - a^3*b*f^3 - 3*a^2*b^2*d^2*g + 3*a^3*b*d*g^
2 - a^4*g^3)/(a^5*b^2) - 1/54*(b^5*c^3 - a^5*h^3 + (g^3 - 3*f*g*h + 3*e*h^2)*a^4*b - (f^3 - 3*e*f*g + 3*e^2*h
- 3*c*g*h  ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3/(b*x**3+a),x)

[Out]

Timed out

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Giac [A]
time = 0.48, size = 269, normalized size = 1.03 \begin {gather*} \frac {e \log \left ({\left | x \right |}\right )}{a} + \frac {\sqrt {3} {\left (b^{2} c - a b f - \left (-a b^{2}\right )^{\frac {1}{3}} b d + \left (-a b^{2}\right )^{\frac {1}{3}} a g\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}} a} + \frac {{\left (b^{2} c - a b f + \left (-a b^{2}\right )^{\frac {1}{3}} b d - \left (-a b^{2}\right )^{\frac {1}{3}} a g\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}} a} + \frac {{\left (a h - b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} + \frac {{\left (a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} c - a^{2} b f\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} - \frac {2 \, d x + c}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.20, size = 2500, normalized size = 9.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)),x)

[Out]

symsum(log(-(b^5*c^3*x - a^5*h^3*x - a^2*b^3*d*e^2 + 36*root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*
z^2 - 18*a^4*b^2*e*h*z + 9*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z
+ 9*a^3*b^3*e^2*z - 3*a^4*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a
^2*b^3*d*e*f - 3*a^2*b^3*c*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2
*d*g^2 + 3*a^2*b^3*d^2*g + 3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5
*c^3, z, k)^3*a^5*b^3*x - a^3*b^2*e*f^2 + a^3*b^2*e^2*g - a^3*b^2*f^3*x - a*b^4*c^2*e - a*b^4*d^3*x + a^4*b*g^
3*x + root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^4*b^2*e*h*z + 9*a^4*b^2*f*g*z - 9*a^3*b
^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^3*e^2*z - 3*a^4*b*f*g*h + 3*a*b^4*c*d*e
+ 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*f - 3*a^2*b^3*c*e*g - 3*a^2*b^3*c*d*h +
3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*a^2*b^3*d^2*g + 3*a^2*b^3*c*f^2 - a^3*b
^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)*a^2*b^4*c^2 + 3*root(27*a^5*b^3*z^3 -
27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^4*b^2*e*h*z + 9*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z +
9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^3*e^2*z - 3*a^4*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b
^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*f - 3*a^2*b^3*c*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2
*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*a^2*b^3*d^2*g + 3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3
+ a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)^2*a^4*b^3*d + root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*
e*z^2 - 18*a^4*b^2*e*h*z + 9*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2
*z + 9*a^3*b^3*e^2*z - 3*a^4*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3
*a^2*b^3*d*e*f - 3*a^2*b^3*c*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b
^2*d*g^2 + 3*a^2*b^3*d^2*g + 3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b
^5*c^3, z, k)*a^4*b^2*f^2 - 3*root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^4*b^2*e*h*z + 9
*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^3*e^2*z - 3*a^4
*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*f - 3*a^2*b^3*c
*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*a^2*b^3*d^2*g +
3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)^2*a^5*b^2*g +
2*a^2*b^3*c*e*f + a^3*b^2*d*e*h + 10*root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^4*b^2*e*
h*z + 9*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^3*e^2*z
- 3*a^4*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*f - 3*a^
2*b^3*c*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*a^2*b^3*
d^2*g + 3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)*a^5*b*h
^2*x - 3*a*b^4*c^2*f*x + 2*a^4*b*e*h^2*x + 4*root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^
4*b^2*e*h*z + 9*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^
3*e^2*z - 3*a^4*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*
f - 3*a^2*b^3*c*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*
a^2*b^3*d^2*g + 3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)
*a^3*b^3*e^2*x + 24*root(27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^4*b^2*e*h*z + 9*a^4*b^2*f
*g*z - 9*a^3*b^3*d*f*z - 9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^3*e^2*z - 3*a^4*b*f*g*h +
3*a*b^4*c*d*e + 3*a^3*b^2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*f - 3*a^2*b^3*c*e*g - 3*a
^2*b^3*c*d*h + 3*a^4*b*e*h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*a^2*b^3*d^2*g + 3*a^2*b^3
*c*f^2 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)^2*a^4*b^3*e*x - 33*root(
27*a^5*b^3*z^3 - 27*a^5*b^2*h*z^2 + 27*a^4*b^3*e*z^2 - 18*a^4*b^2*e*h*z + 9*a^4*b^2*f*g*z - 9*a^3*b^3*d*f*z -
9*a^3*b^3*c*g*z + 9*a^2*b^4*c*d*z + 9*a^5*b*h^2*z + 9*a^3*b^3*e^2*z - 3*a^4*b*f*g*h + 3*a*b^4*c*d*e + 3*a^3*b^
2*e*f*g + 3*a^3*b^2*d*f*h + 3*a^3*b^2*c*g*h - 3*a^2*b^3*d*e*f - 3*a^2*b^3*c*e*g - 3*a^2*b^3*c*d*h + 3*a^4*b*e*
h^2 - 3*a*b^4*c^2*f - 3*a^3*b^2*e^2*h - 3*a^3*b^2*d*g^2 + 3*a^2*b^3*d^2*g + 3*a^2*b^3*c*f^2 - a^3*b^2*f^3 - a*
b^4*d^3 - a^5*h^3 + a^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3, z, k)^2*a^5*b^2*h*x + 3*a^2*b^3*c*f^2*x + 3*a^2*b^3*d^2
*g*x - 3*a^3*b^2*d*g^2*x - a^3*b^2*e^2*h*x + ro...

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