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

Optimal. Leaf size=276 \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 a^{5/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{5/3} b^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{5/3} b^{2/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\log (x) (b c-a f)}{a^2}-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x} \]

[Out]

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

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Rubi [A]  time = 0.44, antiderivative size = 274, 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 = {1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{5/3} b^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{5/3} b^{2/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\log (x) (b c-a f)}{a^2}-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(3*a*x^3) - d/(2*a*x^2) - e/(a*x) + ((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^(5/3)*b^(2/3)) - ((b*c - a*f)*Log[x])/a^2 - ((b^(1/3)*(b*d - a*g)
- a^(1/3)*(b*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)*b^(2/3)) + ((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^(5/3)*b^(1/3)) + ((b*c - a*f)*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 204

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rubi steps

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

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Mathematica [A]  time = 0.55, size = 264, normalized size = 0.96 \[ -\frac {-\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{b^{2/3}}+\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{b^{2/3}}+\frac {2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )}{b^{2/3}}-2 (b c-a f) \log \left (a+b x^3\right )+6 \log (x) (b c-a f)+\frac {2 a c}{x^3}+\frac {3 a d}{x^2}+\frac {6 a e}{x}}{6 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.23, size = 291, normalized size = 1.05 \[ \frac {\sqrt {3} {\left (b^{2} d - a b g + \left (-a b^{2}\right )^{\frac {1}{3}} a h - \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} + \frac {{\left (b^{2} d - a b g - \left (-a b^{2}\right )^{\frac {1}{3}} a h + \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} + \frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac {{\left (b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {{\left (a^{4} b h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - a^{3} b^{2} d + a^{4} b g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{5} b} - \frac {6 \, a x^{2} e + 3 \, a d x + 2 \, a c}{6 \, a^{2} x^{3}} \]

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^4/(b*x^3+a),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.06, size = 442, normalized size = 1.60 \[ -\frac {\sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {\sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}+\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}-\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}+\frac {f \ln \relax (x )}{a}-\frac {f \ln \left (b \,x^{3}+a \right )}{3 a}-\frac {b c \ln \relax (x )}{a^{2}}+\frac {b c \ln \left (b \,x^{3}+a \right )}{3 a^{2}}+\frac {\sqrt {3}\, g \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {g \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {g \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {\sqrt {3}\, h \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}-\frac {h \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}+\frac {h \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}-\frac {e}{a x}-\frac {d}{2 a \,x^{2}}-\frac {c}{3 a \,x^{3}} \]

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^4/(b*x^3+a),x)

[Out]

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

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maxima [A]  time = 3.08, size = 302, normalized size = 1.09 \[ -\frac {{\left (b c - a f\right )} \log \relax (x)}{a^{2}} - \frac {\sqrt {3} {\left (a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b d - a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, e x^{2} + 3 \, d x + 2 \, c}{6 \, a x^{3}} \]

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^4/(b*x^3+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.87, size = 1842, normalized size = 6.67 \[ \text {result too large to display} \]

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^4*(a + b*x^3)),x)

[Out]

symsum(log(- (b^5*c*d^2 - b^5*c^2*e + a^2*b^3*c*g^2 - a^2*b^3*e*f^2 - a^3*b^2*f*g^2 + a^3*b^2*f^2*h - a*b^4*d^
2*f + a*b^4*c^2*h - 2*a^2*b^3*c*f*h + 2*a^2*b^3*d*f*g - 2*a*b^4*c*d*g + 2*a*b^4*c*e*f)/a^3 - root(27*a^6*b^2*z
^3 + 27*a^5*b^2*f*z^2 - 27*a^4*b^3*c*z^2 + 9*a^5*b*g*h*z - 9*a^4*b^2*e*g*z - 9*a^4*b^2*d*h*z - 18*a^3*b^3*c*f*
z + 9*a^3*b^3*d*e*z + 9*a^4*b^2*f^2*z + 9*a^2*b^4*c^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^4*d^2 + a^4*b^2*g^2 + 2*a^2*b^4*c*e - 2*a^3*b^3*c*h -
2*a^3*b^3*d*g - 2*a^3*b^3*e*f + 2*a^4*b^2*f*h)/a^3 + root(27*a^6*b^2*z^3 + 27*a^5*b^2*f*z^2 - 27*a^4*b^3*c*z^
2 + 9*a^5*b*g*h*z - 9*a^4*b^2*e*g*z - 9*a^4*b^2*d*h*z - 18*a^3*b^3*c*f*z + 9*a^3*b^3*d*e*z + 9*a^4*b^2*f^2*z +
9*a^2*b^4*c^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^4*b^3*e - 3*a^5*b^2*h)/a^3 - (x*(24*a^3*b^4*c - 24*a^4*b^3*f))/a^3 + 36*root(27*a^6*b^2*z^3 +
27*a^5*b^2*f*z^2 - 27*a^4*b^3*c*z^2 + 9*a^5*b*g*h*z - 9*a^4*b^2*e*g*z - 9*a^4*b^2*d*h*z - 18*a^3*b^3*c*f*z + 9
*a^3*b^3*d*e*z + 9*a^4*b^2*f^2*z + 9*a^2*b^4*c^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*a*b^5*c^2 + 4*a^3*b^3*f^2 - 8*a^2*b^4*c*f + 10*
a^2*b^4*d*e - 10*a^3*b^3*d*h - 10*a^3*b^3*e*g + 10*a^4*b^2*g*h))/a^3) - (x*(b^5*d^3 - a*b^4*e^3 + a^4*b*h^3 -
a^3*b^2*g^3 + 3*a^2*b^3*d*g^2 + 3*a^2*b^3*e^2*h - 3*a^3*b^2*e*h^2 - 2*b^5*c*d*e - 3*a*b^4*d^2*g - 2*a^2*b^3*c*
g*h - 2*a^2*b^3*d*f*h - 2*a^2*b^3*e*f*g + 2*a^3*b^2*f*g*h + 2*a*b^4*c*d*h + 2*a*b^4*c*e*g + 2*a*b^4*d*e*f))/a^
3)*root(27*a^6*b^2*z^3 + 27*a^5*b^2*f*z^2 - 27*a^4*b^3*c*z^2 + 9*a^5*b*g*h*z - 9*a^4*b^2*e*g*z - 9*a^4*b^2*d*h
*z - 18*a^3*b^3*c*f*z + 9*a^3*b^3*d*e*z + 9*a^4*b^2*f^2*z + 9*a^2*b^4*c^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) - (c/(3*a) + (e*x^2)/a + (d*x)/
(2*a))/x^3 - (log(x)*(b*c - a*f))/a^2

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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