3.5.8 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x (a+b x^3)} \, dx\) [408]

Optimal. Leaf size=258 \[ \frac {g x}{b}+\frac {h x^2}{2 b}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{5/3}}+\frac {c \log (x)}{a}+\frac {\left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{5/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 256, 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} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{2/3} b^{5/3}}-\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 e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{2/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac {c \log (x)}{a}+\frac {g x}{b}+\frac {h x^2}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.14, size = 258, normalized size = 1.00 \begin {gather*} \frac {6 a b^{2/3} g x+3 a b^{2/3} h x^2+2 \sqrt {3} \sqrt [3]{a} \left (-b^{4/3} d-\sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+6 b^{5/3} c \log (x)+2 \sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 b^{2/3} (b c-a f) \log \left (a+b x^3\right )}{6 a b^{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*(a + b*x^3)),x]

[Out]

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

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Maple [A]
time = 0.38, size = 259, normalized size = 1.00

method	result	size
default	\(\frac {\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\left (-a^{2} g +a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-a^{2} h +a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a b f -b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b a}+\frac {c \ln \left (x \right )}{a}\)	\(259\)
risch	\(\frac {h \,x^{2}}{2 b}+\frac {g x}{b}+\frac {\munderset {\textit {\_R} =\RootOf \left (a^{3} b^{2} \textit {\_Z}^{3}+\left (-3 a^{3} b^{2} f +3 a^{2} c \,b^{3}\right ) \textit {\_Z}^{2}+\left (3 a^{4} b g h -3 a^{3} b^{2} d h -3 a^{3} b^{2} e g +3 a^{3} b^{2} f^{2}-6 a^{2} b^{3} c f +3 a^{2} b^{3} d e +3 a \,b^{4} c^{2}\right ) \textit {\_Z} -a^{5} h^{3}+3 a^{4} b e \,h^{2}-3 a^{4} b f g h +a^{4} b \,g^{3}+3 a^{3} b^{2} c g h +3 a^{3} b^{2} d f h -3 a^{3} b^{2} d \,g^{2}-3 a^{3} b^{2} e^{2} h +3 a^{3} b^{2} e f g -a^{3} b^{2} f^{3}-3 a^{2} b^{3} c d h -3 a^{2} b^{3} c e g +3 a^{2} b^{3} c \,f^{2}+3 a^{2} b^{3} d^{2} g -3 a^{2} b^{3} d e f +a^{2} b^{3} e^{3}-3 a \,b^{4} c^{2} f +3 a \,b^{4} c d e -a \,b^{4} d^{3}+b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{2} b^{2}+\left (11 a^{2} b^{2} f -8 a \,b^{3} c \right ) \textit {\_R}^{2}+\left (-10 a^{3} b g h +10 a^{2} b^{2} d h +10 a^{2} b^{2} e g -10 a^{2} b^{2} f^{2}+14 a \,b^{3} c f -10 a \,b^{3} d e -4 b^{4} c^{2}\right ) \textit {\_R} +3 a^{4} h^{3}-9 a^{3} b e \,h^{2}+9 a^{3} b f g h -3 a^{3} b \,g^{3}-6 a^{2} b^{2} c g h -9 a^{2} b^{2} d f h +9 a^{2} b^{2} d \,g^{2}+9 a^{2} b^{2} e^{2} h -9 a^{2} b^{2} e f g +3 a^{2} b^{2} f^{3}+6 a \,b^{3} c d h +6 a \,b^{3} c e g -6 a \,b^{3} c \,f^{2}-9 a \,b^{3} d^{2} g +9 a \,b^{3} d e f -3 a \,b^{3} e^{3}+3 b^{4} c^{2} f -6 b^{4} c d e +3 b^{4} d^{3}\right ) x +\left (-a^{3} b h +a^{2} e \,b^{2}\right ) \textit {\_R}^{2}+\left (a^{3} b f h -a^{3} b \,g^{2}+2 a^{2} b^{2} c h +2 a^{2} b^{2} d g -a^{2} b^{2} e f -2 a \,b^{3} c e -a \,b^{3} d^{2}\right ) \textit {\_R} -3 a^{2} b^{2} c f h +3 a^{2} b^{2} c \,g^{2}+3 a \,b^{3} c^{2} h -6 a \,b^{3} c d g +3 a \,b^{3} c e f -3 b^{4} c^{2} e +3 b^{4} c \,d^{2}\right )}{3 b}+\frac {c \ln \left (x \right )}{a}\)	\(788\)

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/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 293, normalized size = 1.14 \begin {gather*} \frac {c \log \left (x\right )}{a} + \frac {h x^{2} + 2 \, g x}{2 \, b} - \frac {\sqrt {3} {\left (a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \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 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + a b d - a^{2} 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 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \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/(b*x^3+a),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 60.85, size = 15327, normalized size = 59.41 \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/(b*x^3+a),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [A]
time = 0.65, size = 281, normalized size = 1.09 \begin {gather*} \frac {c \log \left ({\left | x \right |}\right )}{a} - \frac {\sqrt {3} {\left (b^{2} d - a b g + \left (-a b^{2}\right )^{\frac {1}{3}} a h - \left (-a b^{2}\right )^{\frac {1}{3}} b 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}} b} - \frac {{\left (b^{2} d - a b g - \left (-a b^{2}\right )^{\frac {1}{3}} a h + \left (-a b^{2}\right )^{\frac {1}{3}} b 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}} b} - \frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} + \frac {b h x^{2} + 2 \, b g x}{2 \, b^{2}} + \frac {{\left (a^{3} b^{2} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - a^{2} b^{3} d + a^{3} b^{2} g\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^{3}} \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/(b*x^3+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.10, size = 1731, normalized size = 6.71 \begin {gather*} \left (\sum _{k=1}^3\ln \left (b^2\,c\,d^2-\mathrm {root}\left (27\,a^3\,b^5\,z^3-27\,a^3\,b^4\,f\,z^2+27\,a^2\,b^5\,c\,z^2+9\,a^4\,b^2\,g\,h\,z-9\,a^3\,b^3\,e\,g\,z-9\,a^3\,b^3\,d\,h\,z-18\,a^2\,b^4\,c\,f\,z+9\,a^2\,b^4\,d\,e\,z+9\,a\,b^5\,c^2\,z+9\,a^3\,b^3\,f^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\,\left (a^3\,g^2-\mathrm {root}\left (27\,a^3\,b^5\,z^3-27\,a^3\,b^4\,f\,z^2+27\,a^2\,b^5\,c\,z^2+9\,a^4\,b^2\,g\,h\,z-9\,a^3\,b^3\,e\,g\,z-9\,a^3\,b^3\,d\,h\,z-18\,a^2\,b^4\,c\,f\,z+9\,a^2\,b^4\,d\,e\,z+9\,a\,b^5\,c^2\,z+9\,a^3\,b^3\,f^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\,\left (\frac {x\,\left (33\,a^2\,b^4\,f-24\,a\,b^5\,c\right )}{b^2}+3\,a^2\,b^2\,e-3\,a^3\,b\,h-\mathrm {root}\left (27\,a^3\,b^5\,z^3-27\,a^3\,b^4\,f\,z^2+27\,a^2\,b^5\,c\,z^2+9\,a^4\,b^2\,g\,h\,z-9\,a^3\,b^3\,e\,g\,z-9\,a^3\,b^3\,d\,h\,z-18\,a^2\,b^4\,c\,f\,z+9\,a^2\,b^4\,d\,e\,z+9\,a\,b^5\,c^2\,z+9\,a^3\,b^3\,f^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\,a^2\,b^3\,x\,36\right )+\frac {x\,\left (4\,b^5\,c^2+10\,a^2\,b^3\,f^2-14\,a\,b^4\,c\,f+10\,a\,b^4\,d\,e-10\,a^2\,b^3\,d\,h-10\,a^2\,b^3\,e\,g+10\,a^3\,b^2\,g\,h\right )}{b^2}+a\,b^2\,d^2-a^3\,f\,h+2\,a\,b^2\,c\,e-2\,a^2\,b\,c\,h-2\,a^2\,b\,d\,g+a^2\,b\,e\,f\right )-b^2\,c^2\,e+a^2\,c\,g^2+\frac {x\,\left (a^4\,h^3-3\,a^3\,b\,e\,h^2+3\,a^3\,b\,f\,g\,h-a^3\,b\,g^3-2\,a^2\,b^2\,c\,g\,h-3\,a^2\,b^2\,d\,f\,h+3\,a^2\,b^2\,d\,g^2+3\,a^2\,b^2\,e^2\,h-3\,a^2\,b^2\,e\,f\,g+a^2\,b^2\,f^3+2\,a\,b^3\,c\,d\,h+2\,a\,b^3\,c\,e\,g-2\,a\,b^3\,c\,f^2-3\,a\,b^3\,d^2\,g+3\,a\,b^3\,d\,e\,f-a\,b^3\,e^3+b^4\,c^2\,f-2\,b^4\,c\,d\,e+b^4\,d^3\right )}{b^2}+a\,b\,c^2\,h-a^2\,c\,f\,h-2\,a\,b\,c\,d\,g+a\,b\,c\,e\,f\right )\,\mathrm {root}\left (27\,a^3\,b^5\,z^3-27\,a^3\,b^4\,f\,z^2+27\,a^2\,b^5\,c\,z^2+9\,a^4\,b^2\,g\,h\,z-9\,a^3\,b^3\,e\,g\,z-9\,a^3\,b^3\,d\,h\,z-18\,a^2\,b^4\,c\,f\,z+9\,a^2\,b^4\,d\,e\,z+9\,a\,b^5\,c^2\,z+9\,a^3\,b^3\,f^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\right )+\frac {h\,x^2}{2\,b}+\frac {c\,\ln \left (x\right )}{a}+\frac {g\,x}{b} \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*(a + b*x^3)),x)

[Out]

symsum(log(b^2*c*d^2 - root(27*a^3*b^5*z^3 - 27*a^3*b^4*f*z^2 + 27*a^2*b^5*c*z^2 + 9*a^4*b^2*g*h*z - 9*a^3*b^3
*e*g*z - 9*a^3*b^3*d*h*z - 18*a^2*b^4*c*f*z + 9*a^2*b^4*d*e*z + 9*a*b^5*c^2*z + 9*a^3*b^3*f^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^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3, z, k)*(a^3*g^2 - root(27*a^
3*b^5*z^3 - 27*a^3*b^4*f*z^2 + 27*a^2*b^5*c*z^2 + 9*a^4*b^2*g*h*z - 9*a^3*b^3*e*g*z - 9*a^3*b^3*d*h*z - 18*a^2
*b^4*c*f*z + 9*a^2*b^4*d*e*z + 9*a*b^5*c^2*z + 9*a^3*b^3*f^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^2*b^3*e^3 + a^4*b*g
^3 + b^5*c^3 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3, z, k)*((x*(33*a^2*b^4*f - 24*a*b^5*c))/b^2 + 3*a^2*b^2*e - 3
*a^3*b*h - 36*root(27*a^3*b^5*z^3 - 27*a^3*b^4*f*z^2 + 27*a^2*b^5*c*z^2 + 9*a^4*b^2*g*h*z - 9*a^3*b^3*e*g*z -
9*a^3*b^3*d*h*z - 18*a^2*b^4*c*f*z + 9*a^2*b^4*d*e*z + 9*a*b^5*c^2*z + 9*a^3*b^3*f^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^2*b^3*e^3 + a^4*b*g^3 + b^5*c^3 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3, z, k)*a^2*b^3*x) + (x*(4*b^5*c^2 + 1
0*a^2*b^3*f^2 - 14*a*b^4*c*f + 10*a*b^4*d*e - 10*a^2*b^3*d*h - 10*a^2*b^3*e*g + 10*a^3*b^2*g*h))/b^2 + a*b^2*d
^2 - a^3*f*h + 2*a*b^2*c*e - 2*a^2*b*c*h - 2*a^2*b*d*g + a^2*b*e*f) - b^2*c^2*e + a^2*c*g^2 + (x*(b^4*d^3 + a^
4*h^3 - a*b^3*e^3 - a^3*b*g^3 + b^4*c^2*f + a^2*b^2*f^3 + 3*a^2*b^2*d*g^2 + 3*a^2*b^2*e^2*h - 2*b^4*c*d*e - 2*
a*b^3*c*f^2 - 3*a*b^3*d^2*g - 3*a^3*b*e*h^2 - 2*a^2*b^2*c*g*h - 3*a^2*b^2*d*f*h - 3*a^2*b^2*e*f*g + 2*a*b^3*c*
d*h + 2*a*b^3*c*e*g + 3*a*b^3*d*e*f + 3*a^3*b*f*g*h))/b^2 + a*b*c^2*h - a^2*c*f*h - 2*a*b*c*d*g + a*b*c*e*f)*r
oot(27*a^3*b^5*z^3 - 27*a^3*b^4*f*z^2 + 27*a^2*b^5*c*z^2 + 9*a^4*b^2*g*h*z - 9*a^3*b^3*e*g*z - 9*a^3*b^3*d*h*z
- 18*a^2*b^4*c*f*z + 9*a^2*b^4*d*e*z + 9*a*b^5*c^2*z + 9*a^3*b^3*f^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^2*b^3*e^3
+ a^4*b*g^3 + b^5*c^3 - a^3*b^2*f^3 - a*b^4*d^3 - a^5*h^3, z, k), k, 1, 3) + (h*x^2)/(2*b) + (c*log(x))/a + (g
*x)/b

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