3.5.9 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 (a+b x^3)} \, dx\) [409]

Optimal. Leaf size=253 \[ -\frac {c}{a x}+\frac {h x}{b}+\frac {\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{4/3}}+\frac {d \log (x)}{a}+\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} b^{4/3}}-\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} b^{4/3}}-\frac {(b d-a g) \log \left (a+b x^3\right )}{3 a b} \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 253, normalized size of antiderivative = 1.00, 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^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3} a^{4/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 a^{4/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 a^{4/3} b^{4/3}}-\frac {(b d-a g) \log \left (a+b x^3\right )}{3 a b}-\frac {c}{a x}+\frac {d \log (x)}{a}+\frac {h x}{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^2*(a + b*x^3)),x]

[Out]

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

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Mathematica [A]
time = 0.19, size = 257, normalized size = 1.02 \begin {gather*} \frac {1}{6} \left (-\frac {6 c}{a x}+\frac {6 h x}{b}+\frac {2 \sqrt {3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3} b^{4/3}}+\frac {6 d \log (x)}{a}+\frac {2 \left (b^{5/3} c+a^{2/3} b e-a b^{2/3} f-a^{5/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3} b^{4/3}}+\frac {\left (-b^{5/3} c-a^{2/3} b e+a b^{2/3} f+a^{5/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3} b^{4/3}}+\frac {2 (-b d+a g) \log \left (a+b x^3\right )}{a b}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.38, size = 260, normalized size = 1.03

method	result	size
default	\(\frac {h x}{b}+\frac {\left (-a^{2} h +a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a b f -b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a b g -b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b a}-\frac {c}{a x}+\frac {d \ln \left (x \right )}{a}\)	\(260\)
risch	\(\frac {h x}{b}-\frac {c}{a x}+\frac {\munderset {\textit {\_R} =\RootOf \left (a^{4} b \,\textit {\_Z}^{3}+\left (-3 a^{4} b g +3 d \,a^{3} b^{2}\right ) \textit {\_Z}^{2}+\left (-3 a^{4} b f h +3 a^{4} b \,g^{2}+3 a^{3} b^{2} c h -6 a^{3} b^{2} d g +3 a^{3} b^{2} e f -3 a^{2} b^{3} c e +3 a^{2} b^{3} d^{2}\right ) \textit {\_Z} +a^{5} h^{3}-3 a^{4} b e \,h^{2}+3 a^{4} b f g h -a^{4} b \,g^{3}-3 a^{3} b^{2} c g h -3 a^{3} b^{2} d f h +3 a^{3} b^{2} d \,g^{2}+3 a^{3} b^{2} e^{2} h -3 a^{3} b^{2} e f g +a^{3} b^{2} f^{3}+3 a^{2} b^{3} c d h +3 a^{2} b^{3} c e g -3 a^{2} b^{3} c \,f^{2}-3 a^{2} b^{3} d^{2} g +3 a^{2} b^{3} d e f -a^{2} b^{3} e^{3}+3 a \,b^{4} c^{2} f -3 a \,b^{4} c d e +a \,b^{4} d^{3}-b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{4} b +\left (11 a^{4} b g -8 d \,a^{3} b^{2}\right ) \textit {\_R}^{2}+\left (10 a^{4} b f h -10 a^{4} b \,g^{2}-10 a^{3} b^{2} c h +14 a^{3} b^{2} d g -10 a^{3} b^{2} e f +10 a^{2} b^{3} c e -4 a^{2} b^{3} d^{2}\right ) \textit {\_R} -3 a^{5} h^{3}+9 a^{4} b e \,h^{2}-9 a^{4} b f g h +3 a^{4} b \,g^{3}+9 a^{3} b^{2} c g h +6 a^{3} b^{2} d f h -6 a^{3} b^{2} d \,g^{2}-9 a^{3} b^{2} e^{2} h +9 a^{3} b^{2} e f g -3 a^{3} b^{2} f^{3}-6 a^{2} b^{3} c d h -9 a^{2} b^{3} c e g +9 a^{2} b^{3} c \,f^{2}+3 a^{2} b^{3} d^{2} g -6 a^{2} b^{3} d e f +3 a^{2} b^{3} e^{3}-9 a \,b^{4} c^{2} f +6 a \,b^{4} c d e +3 b^{5} c^{3}\right ) x +\left (a^{4} b f -a^{3} b^{2} c \right ) \textit {\_R}^{2}+\left (-a^{5} h^{2}+2 a^{4} b e h -a^{4} b f g +a^{3} b^{2} c g -2 a^{3} b^{2} d f -a^{3} b^{2} e^{2}+2 a^{2} b^{3} c d \right ) \textit {\_R} +3 a^{4} b d \,h^{2}-6 a^{3} b^{2} d e h +3 a^{3} b^{2} d f g -3 a^{2} b^{3} c d g -3 a^{2} b^{3} d^{2} f +3 a^{2} b^{3} d \,e^{2}+3 a \,b^{4} c \,d^{2}\right )}{3 b}+\frac {d \ln \left (-x \right )}{a}\)	\(812\)

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^2/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 293, normalized size = 1.16 \begin {gather*} \frac {h x}{b} + \frac {d \log \left (x\right )}{a} - \frac {\sqrt {3} {\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\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 {c}{a x} - \frac {{\left (2 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, 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}} - a^{2} h + a 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 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\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}} + a^{2} h - a b e\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^2/(b*x^3+a),x, algorithm="maxima")

[Out]

h*x/b + d*log(x)/a - 1/3*sqrt(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)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b) - c/(a*x) - 1/6*(2*b^2*d*(a/b)^(2/3) - 2*a*b*g*
(a/b)^(2/3) + b^2*c*(a/b)^(1/3) - a*b*f*(a/b)^(1/3) - a^2*h + a*b*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a
*b^2*(a/b)^(2/3)) - 1/3*(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) + a^2*h
- a*b*e)*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 = 65.20, size = 15238, normalized size = 60.23 \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^2/(b*x^3+a),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [A]
time = 0.54, size = 277, normalized size = 1.09 \begin {gather*} \frac {h x}{b} + \frac {d \log \left ({\left | x \right |}\right )}{a} + \frac {\sqrt {3} {\left (a^{2} h - a b e - \left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {1}{3}} a f\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 (a^{2} h - a b e + \left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {1}{3}} 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 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (b d - a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} - \frac {c}{a x} + \frac {{\left (a b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} b^{2} h - a^{2} b^{3} e\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^2/(b*x^3+a),x, algorithm="giac")

[Out]

h*x/b + d*log(abs(x))/a + 1/3*sqrt(3)*(a^2*h - a*b*e - (-a*b^2)^(1/3)*b*c + (-a*b^2)^(1/3)*a*f)*arctan(1/3*sqr
t(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a) + 1/6*(a^2*h - a*b*e + (-a*b^2)^(1/3)*b*c - (-a*b^2
)^(1/3)*a*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a) - 1/3*(b*d - a*g)*log(abs(b*x^3 + a))
/(a*b) - c/(a*x) + 1/3*(a*b^4*c*(-a/b)^(1/3) - a^2*b^3*f*(-a/b)^(1/3) + a^3*b^2*h - a^2*b^3*e)*(-a/b)^(1/3)*lo
g(abs(x - (-a/b)^(1/3)))/(a^3*b^3)

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Mupad [B]
time = 5.09, size = 1802, normalized size = 7.12 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {a^3\,d\,h^2-2\,a^2\,b\,d\,e\,h+f\,g\,a^2\,b\,d-f\,a\,b^2\,d^2+a\,b^2\,d\,e^2-c\,g\,a\,b^2\,d+c\,b^3\,d^2}{a}-\mathrm {root}\left (27\,a^4\,b^4\,z^3-27\,a^4\,b^3\,g\,z^2+27\,a^3\,b^4\,d\,z^2-9\,a^4\,b^2\,f\,h\,z-18\,a^3\,b^3\,d\,g\,z+9\,a^3\,b^3\,e\,f\,z+9\,a^3\,b^3\,c\,h\,z-9\,a^2\,b^4\,c\,e\,z+9\,a^4\,b^2\,g^2\,z+9\,a^2\,b^4\,d^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 (\mathrm {root}\left (27\,a^4\,b^4\,z^3-27\,a^4\,b^3\,g\,z^2+27\,a^3\,b^4\,d\,z^2-9\,a^4\,b^2\,f\,h\,z-18\,a^3\,b^3\,d\,g\,z+9\,a^3\,b^3\,e\,f\,z+9\,a^3\,b^3\,c\,h\,z-9\,a^2\,b^4\,c\,e\,z+9\,a^4\,b^2\,g^2\,z+9\,a^2\,b^4\,d^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 {3\,a^2\,b^3\,c-3\,a^3\,b^2\,f}{a}+\frac {x\,\left (24\,a^3\,b^4\,d-33\,a^4\,b^3\,g\right )}{a^2\,b}+\mathrm {root}\left (27\,a^4\,b^4\,z^3-27\,a^4\,b^3\,g\,z^2+27\,a^3\,b^4\,d\,z^2-9\,a^4\,b^2\,f\,h\,z-18\,a^3\,b^3\,d\,g\,z+9\,a^3\,b^3\,e\,f\,z+9\,a^3\,b^3\,c\,h\,z-9\,a^2\,b^4\,c\,e\,z+9\,a^4\,b^2\,g^2\,z+9\,a^2\,b^4\,d^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 {a^4\,h^2+a^2\,b^2\,e^2-2\,a\,b^3\,c\,d-2\,a^3\,b\,e\,h+a^3\,b\,f\,g-a^2\,b^2\,c\,g+2\,a^2\,b^2\,d\,f}{a}+\frac {x\,\left (4\,a^2\,b^4\,d^2+10\,a^4\,b^2\,g^2-10\,a^2\,b^4\,c\,e+10\,a^3\,b^3\,c\,h-14\,a^3\,b^3\,d\,g+10\,a^3\,b^3\,e\,f-10\,a^4\,b^2\,f\,h\right )}{a^2\,b}\right )+\frac {x\,\left (-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+2\,a^3\,b^2\,d\,f\,h-2\,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^3-2\,a^2\,b^3\,c\,d\,h-3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,f^2+a^2\,b^3\,d^2\,g-2\,a^2\,b^3\,d\,e\,f+a^2\,b^3\,e^3-3\,a\,b^4\,c^2\,f+2\,a\,b^4\,c\,d\,e+b^5\,c^3\right )}{a^2\,b}\right )\,\mathrm {root}\left (27\,a^4\,b^4\,z^3-27\,a^4\,b^3\,g\,z^2+27\,a^3\,b^4\,d\,z^2-9\,a^4\,b^2\,f\,h\,z-18\,a^3\,b^3\,d\,g\,z+9\,a^3\,b^3\,e\,f\,z+9\,a^3\,b^3\,c\,h\,z-9\,a^2\,b^4\,c\,e\,z+9\,a^4\,b^2\,g^2\,z+9\,a^2\,b^4\,d^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}{b}-\frac {c}{a\,x}+\frac {d\,\ln \left (x\right )}{a} \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^2*(a + b*x^3)),x)

[Out]

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

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