\(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 (a+b x^3)^2} \, dx\) [419]

Optimal result

Integrand size = 38, antiderivative size = 306 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f-a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} b^{2/3}}+\frac {e \log (x)}{a^2}-\frac {\left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} b^{2/3}}+\frac {\left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} b^{2/3}}-\frac {e \log \left (a+b x^3\right )}{3 a^2} \]

[Out]

-1/2*c/a^2/x^2-d/a^2/x-1/3*x*(b*c-a*f+(-a*g+b*d)*x+(-a*h+b*e)*x^2)/a^2/(b*x^3+a)+e*ln(x)/a^2-1/9*(b^(1/3)*(-2*
a*f+5*b*c)-a^(1/3)*(-a*g+4*b*d))*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^(2/3)+1/18*(b^(1/3)*(-2*a*f+5*b*c)-a^(1/3)*(-
a*g+4*b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)/b^(2/3)-1/3*e*ln(b*x^3+a)/a^2+1/9*(5*b^(4/3)*c+4
*a^(1/3)*b*d-2*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^(8/3)/b^(2/3)*3^(1/2
)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1843, 1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{3 \sqrt {3} a^{8/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}-2 a f+5 b c\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{9 a^{8/3} b^{2/3}}-\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a^2 \left (a+b x^3\right )}-\frac {e \log \left (a+b x^3\right )}{3 a^2}-\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}+\frac {e \log (x)}{a^2} \]

[In]

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

[Out]

-1/2*c/(a^2*x^2) - d/(a^2*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*a^2*(a + b*x^3)) + ((5*b^(
4/3)*c + 4*a^(1/3)*b*d - 2*a*b^(1/3)*f - a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt
[3]*a^(8/3)*b^(2/3)) + (e*Log[x])/a^2 - ((b^(1/3)*(5*b*c - 2*a*f) - a^(1/3)*(4*b*d - a*g))*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(8/3)*b^(2/3)) + ((5*b*c - 2*a*f - (a^(1/3)*(4*b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(8/3)*b^(1/3)) - (e*Log[a + b*x^3])/(3*a^2)

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 1843

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x] + S
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]]] /; FreeQ[{a, b}, x] && PolyQ[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

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*} \text {integral}& = -\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int \frac {-3 b^2 c-3 b^2 d x-3 b^2 e x^2+2 b^2 \left (\frac {b c}{a}-f\right ) x^3+b^2 \left (\frac {b d}{a}-g\right ) x^4}{x^3 \left (a+b x^3\right )} \, dx}{3 a b^2} \\ & = -\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^2 c}{a x^3}-\frac {3 b^2 d}{a x^2}-\frac {3 b^2 e}{a x}+\frac {b^2 \left (5 b c-2 a f+(4 b d-a g) x+3 b e x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{3 a b^2} \\ & = -\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {e \log (x)}{a^2}-\frac {\int \frac {5 b c-2 a f+(4 b d-a g) x+3 b e x^2}{a+b x^3} \, dx}{3 a^2} \\ & = -\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {e \log (x)}{a^2}-\frac {\int \frac {5 b c-2 a f+(4 b d-a g) x}{a+b x^3} \, dx}{3 a^2}-\frac {(b e) \int \frac {x^2}{a+b x^3} \, dx}{a^2} \\ & = -\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {e \log (x)}{a^2}-\frac {e \log \left (a+b x^3\right )}{3 a^2}-\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} (5 b c-2 a f)+\sqrt [3]{a} (4 b d-a g)\right )+\sqrt [3]{b} \left (-\sqrt [3]{b} (5 b c-2 a f)+\sqrt [3]{a} (4 b d-a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{8/3} \sqrt [3]{b}}-\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}} \\ & = -\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {e \log (x)}{a^2}-\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}-\frac {e \log \left (a+b x^3\right )}{3 a^2}-\frac {\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 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}{6 a^{7/3} \sqrt [3]{b}}+\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 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}{18 a^{8/3} \sqrt [3]{b}} \\ & = -\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {e \log (x)}{a^2}-\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 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 )}{18 a^{8/3} \sqrt [3]{b}}-\frac {e \log \left (a+b x^3\right )}{3 a^2}-\frac {\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 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 )}{3 a^{8/3} b^{2/3}} \\ & = -\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 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 )}{3 \sqrt {3} a^{8/3} b^{2/3}}+\frac {e \log (x)}{a^2}-\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac {\left (5 b c-2 a f-\frac {\sqrt [3]{a} (4 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 )}{18 a^{8/3} \sqrt [3]{b}}-\frac {e \log \left (a+b x^3\right )}{3 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {\frac {9 a c}{x^2}+\frac {18 a d}{x}+\frac {6 a \left (a^2 h+b^2 x (c+d x)-a b (e+x (f+g x))\right )}{b \left (a+b x^3\right )}+\frac {2 \sqrt {3} \sqrt [3]{a} \left (-5 b^{4/3} c-4 \sqrt [3]{a} b d+2 a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-18 a e \log (x)+\frac {2 \sqrt [3]{a} \left (5 b^{4/3} c-4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} \left (5 b^{4/3} c-4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+6 a e \log \left (a+b x^3\right )}{18 a^3} \]

[In]

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

[Out]

-1/18*((9*a*c)/x^2 + (18*a*d)/x + (6*a*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b*(a + b*x^3)) + (2
*Sqrt[3]*a^(1/3)*(-5*b^(4/3)*c - 4*a^(1/3)*b*d + 2*a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))
/Sqrt[3]])/b^(2/3) - 18*a*e*Log[x] + (2*a^(1/3)*(5*b^(4/3)*c - 4*a^(1/3)*b*d - 2*a*b^(1/3)*f + a^(4/3)*g)*Log[
a^(1/3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*(5*b^(4/3)*c - 4*a^(1/3)*b*d - 2*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) + 6*a*e*Log[a + b*x^3])/a^3

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {c}{2 a^{2} x^{2}}-\frac {d}{a^{2} x}+\frac {e \ln \left (x \right )}{a^{2}}+\frac {\frac {\left (\frac {a g}{3}-\frac {b d}{3}\right ) x^{2}+\left (\frac {a f}{3}-\frac {b c}{3}\right ) x -\frac {a \left (a h -b e \right )}{3 b}}{b \,x^{3}+a}+\frac {\left (2 a f -5 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {\left (a g -4 b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {e \ln \left (b \,x^{3}+a \right )}{3}}{a^{2}}\)	\(293\)
risch	\(\frac {\frac {\left (a g -4 b d \right ) x^{4}}{3 a^{2}}+\frac {\left (2 a f -5 b c \right ) x^{3}}{6 a^{2}}-\frac {\left (a h -b e \right ) x^{2}}{3 a b}-\frac {x d}{a}-\frac {c}{2 a}}{x^{2} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{8} b^{2} \textit {\_Z}^{3}+9 a^{6} b^{2} e \,\textit {\_Z}^{2}+\left (6 a^{5} b f g -15 a^{4} b^{2} c g -24 a^{4} b^{2} d f +27 a^{4} b^{2} e^{2}+60 a^{3} b^{3} c d \right ) \textit {\_Z} +a^{4} g^{3}-12 a^{3} b d \,g^{2}+18 a^{3} b e f g -8 a^{3} b \,f^{3}-45 a^{2} b^{2} c e g +60 a^{2} b^{2} c \,f^{2}+48 a^{2} b^{2} d^{2} g -72 a^{2} b^{2} d e f +27 a^{2} b^{2} e^{3}-150 a \,b^{3} c^{2} f +180 a \,b^{3} c d e -64 a \,b^{3} d^{3}+125 b^{4} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8} b^{2}-24 \textit {\_R}^{2} a^{6} b^{2} e +\left (-20 a^{5} b f g +50 a^{4} b^{2} c g +80 a^{4} b^{2} d f -36 a^{4} b^{2} e^{2}-200 a^{3} b^{3} c d \right ) \textit {\_R} -3 a^{4} g^{3}+36 a^{3} b d \,g^{2}-36 a^{3} b e f g +24 a^{3} b \,f^{3}+90 a^{2} b^{2} c e g -180 a^{2} b^{2} c \,f^{2}-144 a^{2} b^{2} d^{2} g +144 a^{2} b^{2} d e f +450 a \,b^{3} c^{2} f -360 a \,b^{3} c d e +192 a \,b^{3} d^{3}-375 b^{4} c^{3}\right ) x +\left (a^{7} b g -4 a^{6} b^{2} d \right ) \textit {\_R}^{2}+\left (-6 a^{5} b e g -4 a^{5} b \,f^{2}+20 a^{4} b^{2} c f +24 a^{4} b^{2} d e -25 a^{3} b^{3} c^{2}\right ) \textit {\_R} -27 a^{3} b \,e^{2} g +36 a^{3} b e \,f^{2}-180 a^{2} b^{2} c e f +108 a^{2} b^{2} d \,e^{2}+225 a \,b^{3} c^{2} e \right )\right )}{9}+\frac {e \ln \left (-x \right )}{a^{2}}\)	\(623\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 17.92 (sec) , antiderivative size = 12231, normalized size of antiderivative = 39.97 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {2 \, {\left (4 \, b^{2} d - a b g\right )} x^{4} + 6 \, a b d x + {\left (5 \, b^{2} c - 2 \, a b f\right )} x^{3} + 3 \, a b c - 2 \, {\left (a b e - a^{2} h\right )} x^{2}}{6 \, {\left (a^{2} b^{2} x^{5} + a^{3} b x^{2}\right )}} + \frac {e \log \left (x\right )}{a^{2}} - \frac {\sqrt {3} {\left (4 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a g \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {{\left (6 \, b e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, b c + 2 \, a f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, b c - 2 \, a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/6*(2*(4*b^2*d - a*b*g)*x^4 + 6*a*b*d*x + (5*b^2*c - 2*a*b*f)*x^3 + 3*a*b*c - 2*(a*b*e - a^2*h)*x^2)/(a^2*b^
2*x^5 + a^3*b*x^2) + e*log(x)/a^2 - 1/9*sqrt(3)*(4*b*d*(a/b)^(2/3) - a*g*(a/b)^(2/3) + 5*b*c*(a/b)^(1/3) - 2*a
*f*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^3 - 1/18*(6*b*e*(a/b)^(2/3) + 4*b*d*(a/b
)^(1/3) - a*g*(a/b)^(1/3) - 5*b*c + 2*a*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) - 1/9*(3
*b*e*(a/b)^(2/3) - 4*b*d*(a/b)^(1/3) + a*g*(a/b)^(1/3) + 5*b*c - 2*a*f)*log(x + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3
))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac {e \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {\sqrt {3} {\left (5 \, b^{2} c - 2 \, a b f - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} + \frac {{\left (5 \, b^{2} c - 2 \, a b f + 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} + \frac {{\left (4 \, a^{2} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b^{2} c - 2 \, a^{3} b f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{5} b} - \frac {2 \, {\left (4 \, b^{2} d - a b g\right )} x^{4} + 6 \, a b d x + {\left (5 \, b^{2} c - 2 \, a b f\right )} x^{3} + 3 \, a b c - 2 \, {\left (a b e - a^{2} h\right )} x^{2}}{6 \, {\left (b x^{3} + a\right )} a^{2} b x^{2}} \]

[In]

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

[Out]

-1/3*e*log(abs(b*x^3 + a))/a^2 + e*log(abs(x))/a^2 + 1/9*sqrt(3)*(5*b^2*c - 2*a*b*f - 4*(-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^2) + 1/18*(5*b^2*c
- 2*a*b*f + 4*(-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^2) + 1/9*(4*a^2*b^2*d*(-a/b)^(1/3) - a^3*b*g*(-a/b)^(1/3) + 5*a^2*b^2*c - 2*a^3*b*f)*(-a/b)^(1/3)*log(abs(
x - (-a/b)^(1/3)))/(a^5*b) - 1/6*(2*(4*b^2*d - a*b*g)*x^4 + 6*a*b*d*x + (5*b^2*c - 2*a*b*f)*x^3 + 3*a*b*c - 2*
(a*b*e - a^2*h)*x^2)/((b*x^3 + a)*a^2*b*x^2)

Mupad [B] (verification not implemented)

Time = 9.83 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.33 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

symsum(log((b^2*e*(25*b^2*c^2 + 4*a^2*f^2 - 3*a^2*e*g - 20*a*b*c*f + 12*a*b*d*e))/(9*a^5) - (root(729*a^8*b^2*
z^3 + 729*a^6*b^2*e*z^2 + 54*a^5*b*f*g*z - 216*a^4*b^2*d*f*z - 135*a^4*b^2*c*g*z + 540*a^3*b^3*c*d*z + 243*a^4
*b^2*e^2*z + 18*a^3*b*e*f*g + 180*a*b^3*c*d*e - 72*a^2*b^2*d*e*f - 45*a^2*b^2*c*e*g - 12*a^3*b*d*g^2 - 150*a*b
^3*c^2*f + 48*a^2*b^2*d^2*g + 60*a^2*b^2*c*f^2 + 27*a^2*b^2*e^3 - 8*a^3*b*f^3 - 64*a*b^3*d^3 + 125*b^4*c^3 + a
^4*g^3, z, k)*b^2*(25*b^2*c^2 + 4*a^2*f^2 - 9*root(729*a^8*b^2*z^3 + 729*a^6*b^2*e*z^2 + 54*a^5*b*f*g*z - 216*
a^4*b^2*d*f*z - 135*a^4*b^2*c*g*z + 540*a^3*b^3*c*d*z + 243*a^4*b^2*e^2*z + 18*a^3*b*e*f*g + 180*a*b^3*c*d*e -
72*a^2*b^2*d*e*f - 45*a^2*b^2*c*e*g - 12*a^3*b*d*g^2 - 150*a*b^3*c^2*f + 48*a^2*b^2*d^2*g + 60*a^2*b^2*c*f^2
+ 27*a^2*b^2*e^3 - 8*a^3*b*f^3 - 64*a*b^3*d^3 + 125*b^4*c^3 + a^4*g^3, z, k)*a^4*g + 6*a^2*e*g + 36*root(729*a
^8*b^2*z^3 + 729*a^6*b^2*e*z^2 + 54*a^5*b*f*g*z - 216*a^4*b^2*d*f*z - 135*a^4*b^2*c*g*z + 540*a^3*b^3*c*d*z +
243*a^4*b^2*e^2*z + 18*a^3*b*e*f*g + 180*a*b^3*c*d*e - 72*a^2*b^2*d*e*f - 45*a^2*b^2*c*e*g - 12*a^3*b*d*g^2 -
150*a*b^3*c^2*f + 48*a^2*b^2*d^2*g + 60*a^2*b^2*c*f^2 + 27*a^2*b^2*e^3 - 8*a^3*b*f^3 - 64*a*b^3*d^3 + 125*b^4*
c^3 + a^4*g^3, z, k)*a^3*b*d + 36*a*b*e^2*x + 200*b^2*c*d*x + 20*a^2*f*g*x + 324*root(729*a^8*b^2*z^3 + 729*a^
6*b^2*e*z^2 + 54*a^5*b*f*g*z - 216*a^4*b^2*d*f*z - 135*a^4*b^2*c*g*z + 540*a^3*b^3*c*d*z + 243*a^4*b^2*e^2*z +
18*a^3*b*e*f*g + 180*a*b^3*c*d*e - 72*a^2*b^2*d*e*f - 45*a^2*b^2*c*e*g - 12*a^3*b*d*g^2 - 150*a*b^3*c^2*f + 4
8*a^2*b^2*d^2*g + 60*a^2*b^2*c*f^2 + 27*a^2*b^2*e^3 - 8*a^3*b*f^3 - 64*a*b^3*d^3 + 125*b^4*c^3 + a^4*g^3, z, k
)^2*a^5*b*x - 20*a*b*c*f - 24*a*b*d*e - 50*a*b*c*g*x - 80*a*b*d*f*x + 216*root(729*a^8*b^2*z^3 + 729*a^6*b^2*e
*z^2 + 54*a^5*b*f*g*z - 216*a^4*b^2*d*f*z - 135*a^4*b^2*c*g*z + 540*a^3*b^3*c*d*z + 243*a^4*b^2*e^2*z + 18*a^3
*b*e*f*g + 180*a*b^3*c*d*e - 72*a^2*b^2*d*e*f - 45*a^2*b^2*c*e*g - 12*a^3*b*d*g^2 - 150*a*b^3*c^2*f + 48*a^2*b
^2*d^2*g + 60*a^2*b^2*c*f^2 + 27*a^2*b^2*e^3 - 8*a^3*b*f^3 - 64*a*b^3*d^3 + 125*b^4*c^3 + a^4*g^3, z, k)*a^3*b
*e*x))/(9*a^3) - (b*x*(125*b^4*c^3 + a^4*g^3 - 64*a*b^3*d^3 - 8*a^3*b*f^3 + 60*a^2*b^2*c*f^2 + 48*a^2*b^2*d^2*
g - 150*a*b^3*c^2*f - 12*a^3*b*d*g^2 - 30*a^2*b^2*c*e*g - 48*a^2*b^2*d*e*f + 120*a*b^3*c*d*e + 12*a^3*b*e*f*g)
)/(27*a^6))*root(729*a^8*b^2*z^3 + 729*a^6*b^2*e*z^2 + 54*a^5*b*f*g*z - 216*a^4*b^2*d*f*z - 135*a^4*b^2*c*g*z
+ 540*a^3*b^3*c*d*z + 243*a^4*b^2*e^2*z + 18*a^3*b*e*f*g + 180*a*b^3*c*d*e - 72*a^2*b^2*d*e*f - 45*a^2*b^2*c*e
*g - 12*a^3*b*d*g^2 - 150*a*b^3*c^2*f + 48*a^2*b^2*d^2*g + 60*a^2*b^2*c*f^2 + 27*a^2*b^2*e^3 - 8*a^3*b*f^3 - 6
4*a*b^3*d^3 + 125*b^4*c^3 + a^4*g^3, z, k), k, 1, 3) - (c/(2*a) + (x^3*(5*b*c - 2*a*f))/(6*a^2) + (x^4*(4*b*d
- a*g))/(3*a^2) + (d*x)/a - (x^2*(b*e - a*h))/(3*a*b))/(a*x^2 + b*x^5) + (e*log(x))/a^2