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

Optimal. Leaf size=338 \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{18 a^{8/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt {3} a^{8/3} b^{2/3}}+\frac {(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac {\log (x) (2 b c-a f)}{a^3}-\frac {x \left (-b x^2 \left (\frac {b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.73, antiderivative size = 336, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}-2 a g+5 b d\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt {3} a^{8/3} b^{2/3}}-\frac {x \left (-b x^2 \left (\frac {b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}+\frac {(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac {\log (x) (2 b c-a f)}{a^3}-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]

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)^2),x]

[Out]

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

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 1829

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)*Coeff[R, x, i]*x^(i - m))/a, {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[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

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 )^2} \, dx &=-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) 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+3 b^2 \left (\frac {b c}{a}-f\right ) x^3+2 b^2 \left (\frac {b d}{a}-g\right ) x^4+b^2 \left (\frac {b e}{a}-h\right ) x^5}{x^4 \left (a+b x^3\right )} \, dx}{3 a b^2}\\ &=-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^2 c}{a x^4}-\frac {3 b^2 d}{a x^3}-\frac {3 b^2 e}{a x^2}-\frac {3 b^2 (-2 b c+a f)}{a^2 x}+\frac {b^2 \left (a (5 b d-2 a g)+a (4 b e-a h) x-3 b (2 b c-a f) x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^2}\\ &=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {(2 b c-a f) \log (x)}{a^3}-\frac {\int \frac {a (5 b d-2 a g)+a (4 b e-a h) x-3 b (2 b c-a f) x^2}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {(2 b c-a f) \log (x)}{a^3}-\frac {\int \frac {a (5 b d-2 a g)+a (4 b e-a h) x}{a+b x^3} \, dx}{3 a^3}+\frac {(b (2 b c-a f)) \int \frac {x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {(2 b c-a f) \log (x)}{a^3}+\frac {(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac {\int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} (5 b d-2 a g)+a^{4/3} (4 b e-a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (5 b d-2 a g)+a^{4/3} (4 b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{11/3} \sqrt [3]{b}}-\frac {\left (5 b d-2 a g-\frac {\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}}\\ &=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {(2 b c-a f) \log (x)}{a^3}-\frac {\left (5 b d-2 a g-\frac {\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac {(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (5 b^{4/3} d+4 \sqrt [3]{a} b e-2 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}{6 a^{7/3} \sqrt [3]{b}}+\frac {\left (5 b d-2 a g-\frac {\sqrt [3]{a} (4 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}{18 a^{8/3} \sqrt [3]{b}}\\ &=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {(2 b c-a f) \log (x)}{a^3}-\frac {\left (5 b d-2 a g-\frac {\sqrt [3]{a} (4 b e-a h)}{\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 d-2 a g-\frac {\sqrt [3]{a} (4 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 )}{18 a^{8/3} \sqrt [3]{b}}+\frac {(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (5 b^{4/3} d+4 \sqrt [3]{a} b e-2 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 )}{3 a^{8/3} b^{2/3}}\\ &=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\left (5 b^{4/3} d+4 \sqrt [3]{a} b e-2 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 )}{3 \sqrt {3} a^{8/3} b^{2/3}}-\frac {(2 b c-a f) \log (x)}{a^3}-\frac {\left (5 b d-2 a g-\frac {\sqrt [3]{a} (4 b e-a h)}{\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 d-2 a g-\frac {\sqrt [3]{a} (4 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 )}{18 a^{8/3} \sqrt [3]{b}}+\frac {(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.62, size = 303, normalized size = 0.90 \[ \frac {\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac {2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{2/3}}+\frac {a (6 a (f+x (g+h x))-6 b (c+x (d+e x)))}{a+b x^3}+6 (2 b c-a f) \log \left (a+b x^3\right )+18 \log (x) (a f-2 b c)-\frac {6 a c}{x^3}-\frac {9 a d}{x^2}-\frac {18 a e}{x}}{18 a^3} \]

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)^2),x]

[Out]

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

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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)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.23, size = 363, normalized size = 1.07 \[ \frac {\sqrt {3} {\left (5 \, b^{2} d - 2 \, a b g + \left (-a b^{2}\right )^{\frac {1}{3}} a h - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} + \frac {{\left (5 \, b^{2} d - 2 \, a b g - \left (-a b^{2}\right )^{\frac {1}{3}} a h + 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} + \frac {{\left (2 \, b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac {{\left (2 \, b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (a^{5} b h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a^{4} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - 5 \, a^{4} b^{2} d + 2 \, a^{5} b g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{7} b} + \frac {2 \, {\left (a^{2} h - 4 \, a b e\right )} x^{5} - {\left (5 \, a b d - 2 \, a^{2} g\right )} x^{4} - 6 \, a^{2} x^{2} e - 3 \, a^{2} d x - 2 \, {\left (2 \, a b c - a^{2} f\right )} x^{3} - 2 \, a^{2} c}{6 \, {\left (b x^{3} + a\right )} a^{3} x^{3}} \]

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)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(5*b^2*d - 2*a*b*g + (-a*b^2)^(1/3)*a*h - 4*(-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^2) + 1/18*(5*b^2*d - 2*a*b*g - (-a*b^2)^(1/3)*a*h + 4*(-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^2) + 1/3*(2*b*c - a*f)*log(abs(b*x^3 + a))/a^3 -
(2*b*c - a*f)*log(abs(x))/a^3 - 1/9*(a^5*b*h*(-a/b)^(1/3) - 4*a^4*b^2*(-a/b)^(1/3)*e - 5*a^4*b^2*d + 2*a^5*b*g
)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b) + 1/6*(2*(a^2*h - 4*a*b*e)*x^5 - (5*a*b*d - 2*a^2*g)*x^4 - 6
*a^2*x^2*e - 3*a^2*d*x - 2*(2*a*b*c - a^2*f)*x^3 - 2*a^2*c)/((b*x^3 + a)*a^3*x^3)

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maple [B]  time = 0.06, size = 561, normalized size = 1.66 \[ \frac {h \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}-\frac {b e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {g x}{3 \left (b \,x^{3}+a \right ) a}-\frac {b d x}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {2 \sqrt {3}\, g \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {2 g \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a 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 )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a 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 )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {h \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a 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 )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {f}{3 \left (b \,x^{3}+a \right ) a}-\frac {b c}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {5 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {5 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {4 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {4 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {f \ln \relax (x )}{a^{2}}-\frac {f \ln \left (b \,x^{3}+a \right )}{3 a^{2}}-\frac {2 b c \ln \relax (x )}{a^{3}}+\frac {2 b c \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {e}{a^{2} x}-\frac {d}{2 a^{2} x^{2}}-\frac {c}{3 a^{2} 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)^2,x)

[Out]

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

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maxima [A]  time = 3.08, size = 365, normalized size = 1.08 \[ -\frac {2 \, {\left (4 \, b e - a h\right )} x^{5} + {\left (5 \, b d - 2 \, a g\right )} x^{4} + 6 \, a e x^{2} + 2 \, {\left (2 \, b c - a f\right )} x^{3} + 3 \, a d x + 2 \, a c}{6 \, {\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} - \frac {{\left (2 \, b c - a f\right )} \log \relax (x)}{a^{3}} - \frac {\sqrt {3} {\left (4 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} + \frac {{\left (12 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 6 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b d - 2 \, a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (6 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b d + 2 \, a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

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)^2,x, algorithm="maxima")

[Out]

-1/6*(2*(4*b*e - a*h)*x^5 + (5*b*d - 2*a*g)*x^4 + 6*a*e*x^2 + 2*(2*b*c - a*f)*x^3 + 3*a*d*x + 2*a*c)/(a^2*b*x^
6 + a^3*x^3) - (2*b*c - a*f)*log(x)/a^3 - 1/9*sqrt(3)*(4*a*b*e*(a/b)^(2/3) - a^2*h*(a/b)^(2/3) + 5*a*b*d*(a/b)
^(1/3) - 2*a^2*g*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^4 + 1/18*(12*b^2*c*(a/b)^(
2/3) - 6*a*b*f*(a/b)^(2/3) - 4*a*b*e*(a/b)^(1/3) + a^2*h*(a/b)^(1/3) + 5*a*b*d - 2*a^2*g)*log(x^2 - x*(a/b)^(1
/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(2/3)) + 1/9*(6*b^2*c*(a/b)^(2/3) - 3*a*b*f*(a/b)^(2/3) + 4*a*b*e*(a/b)^(1/3)
- a^2*h*(a/b)^(1/3) - 5*a*b*d + 2*a^2*g)*log(x + (a/b)^(1/3))/(a^3*b*(a/b)^(2/3))

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mupad [B]  time = 5.96, size = 1924, normalized size = 5.69 \[ \text {result too large to display} \]

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)^2),x)

[Out]

symsum(log(- (50*b^5*c*d^2 - 48*b^5*c^2*e + 8*a^2*b^3*c*g^2 - 12*a^2*b^3*e*f^2 - 4*a^3*b^2*f*g^2 + 3*a^3*b^2*f
^2*h - 25*a*b^4*d^2*f + 12*a*b^4*c^2*h - 12*a^2*b^3*c*f*h + 20*a^2*b^3*d*f*g - 40*a*b^4*c*d*g + 48*a*b^4*c*e*f
)/(9*a^6) - root(729*a^9*b^2*z^3 + 729*a^7*b^2*f*z^2 - 1458*a^6*b^3*c*z^2 + 54*a^6*b*g*h*z - 216*a^5*b^2*e*g*z
- 135*a^5*b^2*d*h*z - 972*a^4*b^3*c*f*z + 540*a^4*b^3*d*e*z + 243*a^5*b^2*f^2*z + 972*a^3*b^4*c^2*z + 18*a^4*
b*f*g*h - 360*a*b^4*c*d*e - 72*a^3*b^2*e*f*g - 45*a^3*b^2*d*f*h - 36*a^3*b^2*c*g*h + 180*a^2*b^3*d*e*f + 144*a
^2*b^3*c*e*g + 90*a^2*b^3*c*d*h - 12*a^4*b*e*h^2 + 324*a*b^4*c^2*f + 48*a^3*b^2*e^2*h - 150*a^2*b^3*d^2*g + 60
*a^3*b^2*d*g^2 - 162*a^2*b^3*c*f^2 + 27*a^3*b^2*f^3 - 64*a^2*b^3*e^3 - 8*a^4*b*g^3 + 125*a*b^4*d^3 - 216*b^5*c
^3 + a^5*h^3, z, k)*((25*a^3*b^4*d^2 + 4*a^5*b^2*g^2 + 48*a^3*b^4*c*e - 12*a^4*b^3*c*h - 20*a^4*b^3*d*g - 24*a
^4*b^3*e*f + 6*a^5*b^2*f*h)/(9*a^6) + root(729*a^9*b^2*z^3 + 729*a^7*b^2*f*z^2 - 1458*a^6*b^3*c*z^2 + 54*a^6*b
*g*h*z - 216*a^5*b^2*e*g*z - 135*a^5*b^2*d*h*z - 972*a^4*b^3*c*f*z + 540*a^4*b^3*d*e*z + 243*a^5*b^2*f^2*z + 9
72*a^3*b^4*c^2*z + 18*a^4*b*f*g*h - 360*a*b^4*c*d*e - 72*a^3*b^2*e*f*g - 45*a^3*b^2*d*f*h - 36*a^3*b^2*c*g*h +
180*a^2*b^3*d*e*f + 144*a^2*b^3*c*e*g + 90*a^2*b^3*c*d*h - 12*a^4*b*e*h^2 + 324*a*b^4*c^2*f + 48*a^3*b^2*e^2*
h - 150*a^2*b^3*d^2*g + 60*a^3*b^2*d*g^2 - 162*a^2*b^3*c*f^2 + 27*a^3*b^2*f^3 - 64*a^2*b^3*e^3 - 8*a^4*b*g^3 +
125*a*b^4*d^3 - 216*b^5*c^3 + a^5*h^3, z, k)*((36*a^6*b^3*e - 9*a^7*b^2*h)/(9*a^6) - (x*(1296*a^5*b^4*c - 648
*a^6*b^3*f))/(27*a^6) + 36*root(729*a^9*b^2*z^3 + 729*a^7*b^2*f*z^2 - 1458*a^6*b^3*c*z^2 + 54*a^6*b*g*h*z - 21
6*a^5*b^2*e*g*z - 135*a^5*b^2*d*h*z - 972*a^4*b^3*c*f*z + 540*a^4*b^3*d*e*z + 243*a^5*b^2*f^2*z + 972*a^3*b^4*
c^2*z + 18*a^4*b*f*g*h - 360*a*b^4*c*d*e - 72*a^3*b^2*e*f*g - 45*a^3*b^2*d*f*h - 36*a^3*b^2*c*g*h + 180*a^2*b^
3*d*e*f + 144*a^2*b^3*c*e*g + 90*a^2*b^3*c*d*h - 12*a^4*b*e*h^2 + 324*a*b^4*c^2*f + 48*a^3*b^2*e^2*h - 150*a^2
*b^3*d^2*g + 60*a^3*b^2*d*g^2 - 162*a^2*b^3*c*f^2 + 27*a^3*b^2*f^3 - 64*a^2*b^3*e^3 - 8*a^4*b*g^3 + 125*a*b^4*
d^3 - 216*b^5*c^3 + a^5*h^3, z, k)*a^2*b^3*x) + (x*(432*a^2*b^5*c^2 + 108*a^4*b^3*f^2 - 432*a^3*b^4*c*f + 600*
a^3*b^4*d*e - 150*a^4*b^3*d*h - 240*a^4*b^3*e*g + 60*a^5*b^2*g*h))/(27*a^6)) - (x*(125*b^5*d^3 - 64*a*b^4*e^3
+ a^4*b*h^3 - 8*a^3*b^2*g^3 + 60*a^2*b^3*d*g^2 + 48*a^2*b^3*e^2*h - 12*a^3*b^2*e*h^2 - 240*b^5*c*d*e - 150*a*b
^4*d^2*g - 24*a^2*b^3*c*g*h - 30*a^2*b^3*d*f*h - 48*a^2*b^3*e*f*g + 12*a^3*b^2*f*g*h + 60*a*b^4*c*d*h + 96*a*b
^4*c*e*g + 120*a*b^4*d*e*f))/(27*a^6))*root(729*a^9*b^2*z^3 + 729*a^7*b^2*f*z^2 - 1458*a^6*b^3*c*z^2 + 54*a^6*
b*g*h*z - 216*a^5*b^2*e*g*z - 135*a^5*b^2*d*h*z - 972*a^4*b^3*c*f*z + 540*a^4*b^3*d*e*z + 243*a^5*b^2*f^2*z +
972*a^3*b^4*c^2*z + 18*a^4*b*f*g*h - 360*a*b^4*c*d*e - 72*a^3*b^2*e*f*g - 45*a^3*b^2*d*f*h - 36*a^3*b^2*c*g*h
+ 180*a^2*b^3*d*e*f + 144*a^2*b^3*c*e*g + 90*a^2*b^3*c*d*h - 12*a^4*b*e*h^2 + 324*a*b^4*c^2*f + 48*a^3*b^2*e^2
*h - 150*a^2*b^3*d^2*g + 60*a^3*b^2*d*g^2 - 162*a^2*b^3*c*f^2 + 27*a^3*b^2*f^3 - 64*a^2*b^3*e^3 - 8*a^4*b*g^3
+ 125*a*b^4*d^3 - 216*b^5*c^3 + a^5*h^3, z, k), k, 1, 3) - (c/(3*a) + (e*x^2)/a + (x^3*(2*b*c - a*f))/(3*a^2)
+ (x^4*(5*b*d - 2*a*g))/(6*a^2) + (x^5*(4*b*e - a*h))/(3*a^2) + (d*x)/(2*a))/(a*x^3 + b*x^6) - (log(x)*(2*b*c
- a*f))/a^3

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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)**2,x)

[Out]

Timed out

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