3.5.28 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 (a+b x^3)^3} \, dx\) [428]

Optimal. Leaf size=360 \[ -\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}+\frac {e \log (x)}{a^3}-\frac {\left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{2/3}}+\frac {\left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{2/3}}-\frac {e \log \left (a+b x^3\right )}{3 a^3} \]

[Out]

-1/2*c/a^3/x^2-d/a^3/x-1/6*x*(b*c-a*f+(-a*g+b*d)*x+(-a*h+b*e)*x^2)/a^2/(b*x^3+a)^2-1/18*x*(11*b*c-5*a*f+2*(-2*
a*g+5*b*d)*x+3*(-a*h+3*b*e)*x^2)/a^3/(b*x^3+a)+e*ln(x)/a^3-1/27*(5*b^(1/3)*(-a*f+4*b*c)-2*a^(1/3)*(-a*g+7*b*d)
)*ln(a^(1/3)+b^(1/3)*x)/a^(11/3)/b^(2/3)+1/54*(5*b^(1/3)*(-a*f+4*b*c)-2*a^(1/3)*(-a*g+7*b*d))*ln(a^(2/3)-a^(1/
3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/3)/b^(2/3)-1/3*e*ln(b*x^3+a)/a^3+1/27*(20*b^(4/3)*c+14*a^(1/3)*b*d-5*a*b^(1/3)
*f-2*a^(4/3)*g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(11/3)/b^(2/3)*3^(1/2)

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Rubi [A]
time = 0.56, antiderivative size = 357, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1843, 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 (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}-5 a f+20 b c\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}-\frac {x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac {e \log \left (a+b x^3\right )}{3 a^3}-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}+\frac {e \log (x)}{a^3}-\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*c/(a^3*x^2) - d/(a^3*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*a^2*(a + b*x^3)^2) - (x*(1
1*b*c - 5*a*f + 2*(5*b*d - 2*a*g)*x + 3*(3*b*e - a*h)*x^2))/(18*a^3*(a + b*x^3)) + ((20*b^(4/3)*c + 14*a^(1/3)
*b*d - 5*a*b^(1/3)*f - 2*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(
2/3)) + (e*Log[x])/a^3 - ((5*b^(1/3)*(4*b*c - a*f) - 2*a^(1/3)*(7*b*d - a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^
(11/3)*b^(2/3)) + ((20*b*c - 5*a*f - (2*a^(1/3)*(7*b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/(54*a^(11/3)*b^(1/3)) - (e*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 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*} \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 )^3} \, dx &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^2 c-6 b^2 d x-6 b^2 e x^2+5 b^2 \left (\frac {b c}{a}-f\right ) x^3+4 b^2 \left (\frac {b d}{a}-g\right ) x^4+3 b^2 \left (\frac {b e}{a}-h\right ) x^5}{x^3 \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^4 c+18 b^4 d x+18 b^4 e x^2-2 b^4 \left (\frac {11 b c}{a}-5 f\right ) x^3-2 b^4 \left (\frac {5 b d}{a}-2 g\right ) x^4}{x^3 \left (a+b x^3\right )} \, dx}{18 a^2 b^4}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^4 c}{a x^3}+\frac {18 b^4 d}{a x^2}+\frac {18 b^4 e}{a x}+\frac {2 b^4 \left (-5 (4 b c-a f)-2 (7 b d-a g) x-9 b e x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^4}\\ &=-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {e \log (x)}{a^3}+\frac {\int \frac {-5 (4 b c-a f)-2 (7 b d-a g) x-9 b e x^2}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {e \log (x)}{a^3}+\frac {\int \frac {-5 (4 b c-a f)-2 (7 b d-a g) x}{a+b x^3} \, dx}{9 a^3}-\frac {(b e) \int \frac {x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {e \log (x)}{a^3}-\frac {e \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (-10 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )+\sqrt [3]{b} \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} \sqrt [3]{b}}-\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}}\\ &=-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {e \log (x)}{a^3}-\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}-\frac {e \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 a^{4/3} g\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3} \sqrt [3]{b}}+\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 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}{54 a^{11/3} \sqrt [3]{b}}\\ &=-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {e \log (x)}{a^3}-\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 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 )}{54 a^{11/3} \sqrt [3]{b}}-\frac {e \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 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 )}{9 a^{11/3} b^{2/3}}\\ &=-\frac {c}{2 a^3 x^2}-\frac {d}{a^3 x}-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}+\frac {e \log (x)}{a^3}-\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (20 b c-5 a f-\frac {2 \sqrt [3]{a} (7 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 )}{54 a^{11/3} \sqrt [3]{b}}-\frac {e \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 337, normalized size = 0.94 \begin {gather*} -\frac {\frac {27 a c}{x^2}+\frac {54 a d}{x}-\frac {3 a (6 a e-b x (11 c+10 d x)+a x (5 f+4 g x))}{a+b x^3}+\frac {9 a^2 \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 )^2}+\frac {2 \sqrt {3} \sqrt [3]{a} \left (-20 b^{4/3} c-14 \sqrt [3]{a} b d+5 a \sqrt [3]{b} f+2 a^{4/3} g\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-54 a e \log (x)+\frac {2 \sqrt [3]{a} \left (20 b^{4/3} c-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+2 a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} \left (20 b^{4/3} c-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+2 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}}+18 a e \log \left (a+b x^3\right )}{54 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*((27*a*c)/x^2 + (54*a*d)/x - (3*a*(6*a*e - b*x*(11*c + 10*d*x) + a*x*(5*f + 4*g*x)))/(a + b*x^3) + (9*a^
2*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b*(a + b*x^3)^2) + (2*Sqrt[3]*a^(1/3)*(-20*b^(4/3)*c - 1
4*a^(1/3)*b*d + 5*a*b^(1/3)*f + 2*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - 54*a*e*Log
[x] + (2*a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/
3) - (a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/b^(2/3) + 18*a*e*Log[a + b*x^3])/a^4

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Maple [A]
time = 0.40, size = 340, normalized size = 0.94

method	result	size
default	\(\frac {\frac {\left (\frac {2}{9} a b g -\frac {5}{9} b^{2} d \right ) x^{5}+\left (\frac {5}{18} a b f -\frac {11}{18} b^{2} c \right ) x^{4}+\frac {a b e \,x^{3}}{3}+\frac {a \left (7 a g -13 b d \right ) x^{2}}{18}+\frac {a \left (4 a f -7 b c \right ) x}{9}-\frac {a^{2} \left (a h -3 b e \right )}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 a f -20 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}+\frac {\left (2 a g -14 b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9}-\frac {e \ln \left (b \,x^{3}+a \right )}{3}}{a^{3}}-\frac {c}{2 a^{3} x^{2}}-\frac {d}{a^{3} x}+\frac {e \ln \left (x \right )}{a^{3}}\)	\(340\)
risch	\(\frac {\frac {2 b \left (a g -7 b d \right ) x^{7}}{9 a^{3}}+\frac {5 b \left (a f -4 b c \right ) x^{6}}{18 a^{3}}+\frac {b e \,x^{5}}{3 a^{2}}+\frac {7 \left (a g -7 b d \right ) x^{4}}{18 a^{2}}+\frac {4 \left (a f -4 b c \right ) x^{3}}{9 a^{2}}-\frac {\left (a h -3 b e \right ) x^{2}}{6 a b}-\frac {x d}{a}-\frac {c}{2 a}}{x^{2} \left (b \,x^{3}+a \right )^{2}}+\frac {e \ln \left (-x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{11} b^{2} \textit {\_Z}^{3}+27 a^{8} b^{2} e \,\textit {\_Z}^{2}+\left (30 a^{6} b f g -120 a^{5} b^{2} c g -210 a^{5} b^{2} d f +243 a^{5} b^{2} e^{2}+840 a^{4} b^{3} c d \right ) \textit {\_Z} +8 a^{4} g^{3}-168 a^{3} b d \,g^{2}+270 a^{3} b e f g -125 a^{3} b \,f^{3}-1080 a^{2} b^{2} c e g +1500 a^{2} b^{2} c \,f^{2}+1176 a^{2} b^{2} d^{2} g -1890 a^{2} b^{2} d e f +729 a^{2} b^{2} e^{3}-6000 a \,b^{3} c^{2} f +7560 a \,b^{3} c d e -2744 a \,b^{3} d^{3}+8000 b^{4} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b^{2}-72 \textit {\_R}^{2} a^{8} b^{2} e +\left (-100 a^{6} b f g +400 a^{5} b^{2} c g +700 a^{5} b^{2} d f -324 a^{5} b^{2} e^{2}-2800 a^{4} b^{3} c d \right ) \textit {\_R} -24 a^{4} g^{3}+504 a^{3} b d \,g^{2}-540 a^{3} b e f g +375 a^{3} b \,f^{3}+2160 a^{2} b^{2} c e g -4500 a^{2} b^{2} c \,f^{2}-3528 a^{2} b^{2} d^{2} g +3780 a^{2} b^{2} d e f +18000 a \,b^{3} c^{2} f -15120 a \,b^{3} c d e +8232 a \,b^{3} d^{3}-24000 b^{4} c^{3}\right ) x +\left (2 a^{9} b g -14 a^{8} b^{2} d \right ) \textit {\_R}^{2}+\left (-36 a^{6} b e g -25 a^{6} b \,f^{2}+200 a^{5} b^{2} c f +252 a^{5} b^{2} d e -400 a^{4} b^{3} c^{2}\right ) \textit {\_R} -486 a^{3} b \,e^{2} g +675 a^{3} b e \,f^{2}-5400 a^{2} b^{2} c e f +3402 a^{2} b^{2} d \,e^{2}+10800 a \,b^{3} c^{2} e \right )\right )}{27}\)	\(668\)

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

[Out]

1/a^3*(((2/9*a*b*g-5/9*b^2*d)*x^5+(5/18*a*b*f-11/18*b^2*c)*x^4+1/3*a*b*e*x^3+1/18*a*(7*a*g-13*b*d)*x^2+1/9*a*(
4*a*f-7*b*c)*x-1/6*a^2*(a*h-3*b*e)/b)/(b*x^3+a)^2+1/9*(5*a*f-20*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)))+1/9*(2*a*g-14*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))-1/2*c/a^3/x^2-d/a^3/
x+e*ln(x)/a^3

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Maxima [A]
time = 0.50, size = 394, normalized size = 1.09 \begin {gather*} \frac {6 \, a b^{2} x^{5} e - 4 \, {\left (7 \, b^{3} d - a b^{2} g\right )} x^{7} - 5 \, {\left (4 \, b^{3} c - a b^{2} f\right )} x^{6} - 18 \, a^{2} b d x - 7 \, {\left (7 \, a b^{2} d - a^{2} b g\right )} x^{4} - 9 \, a^{2} b c - 8 \, {\left (4 \, a b^{2} c - a^{2} b f\right )} x^{3} - 3 \, {\left (a^{3} h - 3 \, a^{2} b e\right )} x^{2}}{18 \, {\left (a^{3} b^{3} x^{8} + 2 \, a^{4} b^{2} x^{5} + a^{5} b x^{2}\right )}} + \frac {e \log \left (x\right )}{a^{3}} - \frac {\sqrt {3} {\left (14 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a g \left (\frac {a}{b}\right )^{\frac {2}{3}} + 20 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, 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 )}{27 \, a^{4}} - \frac {{\left (18 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}} e + 14 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, b c + 5 \, 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 )}{54 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - 14 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, b c - 5 \, a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \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^3/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/18*(6*a*b^2*x^5*e - 4*(7*b^3*d - a*b^2*g)*x^7 - 5*(4*b^3*c - a*b^2*f)*x^6 - 18*a^2*b*d*x - 7*(7*a*b^2*d - a^
2*b*g)*x^4 - 9*a^2*b*c - 8*(4*a*b^2*c - a^2*b*f)*x^3 - 3*(a^3*h - 3*a^2*b*e)*x^2)/(a^3*b^3*x^8 + 2*a^4*b^2*x^5
+ a^5*b*x^2) + e*log(x)/a^3 - 1/27*sqrt(3)*(14*b*d*(a/b)^(2/3) - 2*a*g*(a/b)^(2/3) + 20*b*c*(a/b)^(1/3) - 5*a
*f*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^4 - 1/54*(18*b*(a/b)^(2/3)*e + 14*b*d*(a
/b)^(1/3) - 2*a*g*(a/b)^(1/3) - 20*b*c + 5*a*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(2/3)) - 1
/27*(9*b*(a/b)^(2/3)*e - 14*b*d*(a/b)^(1/3) + 2*a*g*(a/b)^(1/3) + 20*b*c - 5*a*f)*log(x + (a/b)^(1/3))/(a^3*b*
(a/b)^(2/3))

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Fricas [C] Result contains complex when optimal does not.
time = 17.06, size = 12435, normalized size = 34.54 \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^3/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/2916*(972*a*b^2*e*x^5 - 648*(7*b^3*d - a*b^2*g)*x^7 - 810*(4*b^3*c - a*b^2*f)*x^6 - 2916*a^2*b*d*x - 1134*(7
*a*b^2*d - a^2*b*g)*x^4 - 1458*a^2*b*c - 1296*(4*a*b^2*c - a^2*b*f)*x^3 + 486*(3*a^2*b*e - a^3*h)*x^2 - 2*(a^3
*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70
*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*
b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3
- 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3
- 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(34
3*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2
*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000
*a*b^3*c^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**3/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.47, size = 399, normalized size = 1.11 \begin {gather*} -\frac {e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {e \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {\sqrt {3} {\left (20 \, b^{2} c - 5 \, a b f - 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 2 \, \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 )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} + \frac {{\left (20 \, b^{2} c - 5 \, a b f + 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 2 \, \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 )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {28 \, b^{3} d x^{7} - 4 \, a b^{2} g x^{7} + 20 \, b^{3} c x^{6} - 5 \, a b^{2} f x^{6} - 6 \, a b^{2} x^{5} e + 49 \, a b^{2} d x^{4} - 7 \, a^{2} b g x^{4} + 32 \, a b^{2} c x^{3} - 8 \, a^{2} b f x^{3} + 3 \, a^{3} h x^{2} - 9 \, a^{2} b x^{2} e + 18 \, a^{2} b d x + 9 \, a^{2} b c}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3} b} + \frac {{\left (14 \, a^{3} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{4} b g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, a^{3} b^{2} c - 5 \, a^{4} 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 )}{27 \, a^{7} b} \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^3/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/3*e*log(abs(b*x^3 + a))/a^3 + e*log(abs(x))/a^3 + 1/27*sqrt(3)*(20*b^2*c - 5*a*b*f - 14*(-a*b^2)^(1/3)*b*d
+ 2*(-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^3) + 1/54*(20*
b^2*c - 5*a*b*f + 14*(-a*b^2)^(1/3)*b*d - 2*(-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^3) - 1/18*(28*b^3*d*x^7 - 4*a*b^2*g*x^7 + 20*b^3*c*x^6 - 5*a*b^2*f*x^6 - 6*a*b^2*x^5*e + 49*a*b^2
*d*x^4 - 7*a^2*b*g*x^4 + 32*a*b^2*c*x^3 - 8*a^2*b*f*x^3 + 3*a^3*h*x^2 - 9*a^2*b*x^2*e + 18*a^2*b*d*x + 9*a^2*b
*c)/((b*x^4 + a*x)^2*a^3*b) + 1/27*(14*a^3*b^2*d*(-a/b)^(1/3) - 2*a^4*b*g*(-a/b)^(1/3) + 20*a^3*b^2*c - 5*a^4*
b*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b)

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Mupad [B]
time = 5.66, size = 1697, normalized size = 4.71 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {b^2\,e\,\left (25\,a^2\,f^2-18\,e\,g\,a^2-200\,a\,b\,c\,f+126\,d\,e\,a\,b+400\,b^2\,c^2\right )}{81\,a^8}-\frac {\mathrm {root}\left (19683\,a^{11}\,b^2\,z^3+19683\,a^8\,b^2\,e\,z^2+810\,a^6\,b\,f\,g\,z-5670\,a^5\,b^2\,d\,f\,z-3240\,a^5\,b^2\,c\,g\,z+22680\,a^4\,b^3\,c\,d\,z+6561\,a^5\,b^2\,e^2\,z+270\,a^3\,b\,e\,f\,g+7560\,a\,b^3\,c\,d\,e-1890\,a^2\,b^2\,d\,e\,f-1080\,a^2\,b^2\,c\,e\,g-168\,a^3\,b\,d\,g^2-6000\,a\,b^3\,c^2\,f+1176\,a^2\,b^2\,d^2\,g+1500\,a^2\,b^2\,c\,f^2+729\,a^2\,b^2\,e^3-125\,a^3\,b\,f^3-2744\,a\,b^3\,d^3+8\,a^4\,g^3+8000\,b^4\,c^3,z,k\right )\,b^2\,\left (400\,b^2\,c^2+25\,a^2\,f^2-\mathrm {root}\left (19683\,a^{11}\,b^2\,z^3+19683\,a^8\,b^2\,e\,z^2+810\,a^6\,b\,f\,g\,z-5670\,a^5\,b^2\,d\,f\,z-3240\,a^5\,b^2\,c\,g\,z+22680\,a^4\,b^3\,c\,d\,z+6561\,a^5\,b^2\,e^2\,z+270\,a^3\,b\,e\,f\,g+7560\,a\,b^3\,c\,d\,e-1890\,a^2\,b^2\,d\,e\,f-1080\,a^2\,b^2\,c\,e\,g-168\,a^3\,b\,d\,g^2-6000\,a\,b^3\,c^2\,f+1176\,a^2\,b^2\,d^2\,g+1500\,a^2\,b^2\,c\,f^2+729\,a^2\,b^2\,e^3-125\,a^3\,b\,f^3-2744\,a\,b^3\,d^3+8\,a^4\,g^3+8000\,b^4\,c^3,z,k\right )\,a^5\,g\,54+36\,a^2\,e\,g+\mathrm {root}\left (19683\,a^{11}\,b^2\,z^3+19683\,a^8\,b^2\,e\,z^2+810\,a^6\,b\,f\,g\,z-5670\,a^5\,b^2\,d\,f\,z-3240\,a^5\,b^2\,c\,g\,z+22680\,a^4\,b^3\,c\,d\,z+6561\,a^5\,b^2\,e^2\,z+270\,a^3\,b\,e\,f\,g+7560\,a\,b^3\,c\,d\,e-1890\,a^2\,b^2\,d\,e\,f-1080\,a^2\,b^2\,c\,e\,g-168\,a^3\,b\,d\,g^2-6000\,a\,b^3\,c^2\,f+1176\,a^2\,b^2\,d^2\,g+1500\,a^2\,b^2\,c\,f^2+729\,a^2\,b^2\,e^3-125\,a^3\,b\,f^3-2744\,a\,b^3\,d^3+8\,a^4\,g^3+8000\,b^4\,c^3,z,k\right )\,a^4\,b\,d\,378+324\,a\,b\,e^2\,x+2800\,b^2\,c\,d\,x+100\,a^2\,f\,g\,x+{\mathrm {root}\left (19683\,a^{11}\,b^2\,z^3+19683\,a^8\,b^2\,e\,z^2+810\,a^6\,b\,f\,g\,z-5670\,a^5\,b^2\,d\,f\,z-3240\,a^5\,b^2\,c\,g\,z+22680\,a^4\,b^3\,c\,d\,z+6561\,a^5\,b^2\,e^2\,z+270\,a^3\,b\,e\,f\,g+7560\,a\,b^3\,c\,d\,e-1890\,a^2\,b^2\,d\,e\,f-1080\,a^2\,b^2\,c\,e\,g-168\,a^3\,b\,d\,g^2-6000\,a\,b^3\,c^2\,f+1176\,a^2\,b^2\,d^2\,g+1500\,a^2\,b^2\,c\,f^2+729\,a^2\,b^2\,e^3-125\,a^3\,b\,f^3-2744\,a\,b^3\,d^3+8\,a^4\,g^3+8000\,b^4\,c^3,z,k\right )}^2\,a^7\,b\,x\,2916-200\,a\,b\,c\,f-252\,a\,b\,d\,e-400\,a\,b\,c\,g\,x-700\,a\,b\,d\,f\,x+\mathrm {root}\left (19683\,a^{11}\,b^2\,z^3+19683\,a^8\,b^2\,e\,z^2+810\,a^6\,b\,f\,g\,z-5670\,a^5\,b^2\,d\,f\,z-3240\,a^5\,b^2\,c\,g\,z+22680\,a^4\,b^3\,c\,d\,z+6561\,a^5\,b^2\,e^2\,z+270\,a^3\,b\,e\,f\,g+7560\,a\,b^3\,c\,d\,e-1890\,a^2\,b^2\,d\,e\,f-1080\,a^2\,b^2\,c\,e\,g-168\,a^3\,b\,d\,g^2-6000\,a\,b^3\,c^2\,f+1176\,a^2\,b^2\,d^2\,g+1500\,a^2\,b^2\,c\,f^2+729\,a^2\,b^2\,e^3-125\,a^3\,b\,f^3-2744\,a\,b^3\,d^3+8\,a^4\,g^3+8000\,b^4\,c^3,z,k\right )\,a^4\,b\,e\,x\,1944\right )}{a^5\,81}-\frac {b\,x\,\left (8\,a^4\,g^3-168\,a^3\,b\,d\,g^2-125\,a^3\,b\,f^3+180\,e\,a^3\,b\,f\,g+1500\,a^2\,b^2\,c\,f^2-720\,e\,a^2\,b^2\,c\,g+1176\,a^2\,b^2\,d^2\,g-1260\,e\,a^2\,b^2\,d\,f-6000\,a\,b^3\,c^2\,f+5040\,e\,a\,b^3\,c\,d-2744\,a\,b^3\,d^3+8000\,b^4\,c^3\right )}{729\,a^9}\right )\,\mathrm {root}\left (19683\,a^{11}\,b^2\,z^3+19683\,a^8\,b^2\,e\,z^2+810\,a^6\,b\,f\,g\,z-5670\,a^5\,b^2\,d\,f\,z-3240\,a^5\,b^2\,c\,g\,z+22680\,a^4\,b^3\,c\,d\,z+6561\,a^5\,b^2\,e^2\,z+270\,a^3\,b\,e\,f\,g+7560\,a\,b^3\,c\,d\,e-1890\,a^2\,b^2\,d\,e\,f-1080\,a^2\,b^2\,c\,e\,g-168\,a^3\,b\,d\,g^2-6000\,a\,b^3\,c^2\,f+1176\,a^2\,b^2\,d^2\,g+1500\,a^2\,b^2\,c\,f^2+729\,a^2\,b^2\,e^3-125\,a^3\,b\,f^3-2744\,a\,b^3\,d^3+8\,a^4\,g^3+8000\,b^4\,c^3,z,k\right )\right )-\frac {\frac {c}{2\,a}+\frac {4\,x^3\,\left (4\,b\,c-a\,f\right )}{9\,a^2}+\frac {7\,x^4\,\left (7\,b\,d-a\,g\right )}{18\,a^2}+\frac {d\,x}{a}+\frac {5\,b\,x^6\,\left (4\,b\,c-a\,f\right )}{18\,a^3}+\frac {2\,b\,x^7\,\left (7\,b\,d-a\,g\right )}{9\,a^3}-\frac {x^2\,\left (3\,b\,e-a\,h\right )}{6\,a\,b}-\frac {b\,e\,x^5}{3\,a^2}}{a^2\,x^2+2\,a\,b\,x^5+b^2\,x^8}+\frac {e\,\ln \left (x\right )}{a^3} \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^3*(a + b*x^3)^3),x)

[Out]

symsum(log((b^2*e*(400*b^2*c^2 + 25*a^2*f^2 - 18*a^2*e*g - 200*a*b*c*f + 126*a*b*d*e))/(81*a^8) - (root(19683*
a^11*b^2*z^3 + 19683*a^8*b^2*e*z^2 + 810*a^6*b*f*g*z - 5670*a^5*b^2*d*f*z - 3240*a^5*b^2*c*g*z + 22680*a^4*b^3
*c*d*z + 6561*a^5*b^2*e^2*z + 270*a^3*b*e*f*g + 7560*a*b^3*c*d*e - 1890*a^2*b^2*d*e*f - 1080*a^2*b^2*c*e*g - 1
68*a^3*b*d*g^2 - 6000*a*b^3*c^2*f + 1176*a^2*b^2*d^2*g + 1500*a^2*b^2*c*f^2 + 729*a^2*b^2*e^3 - 125*a^3*b*f^3
- 2744*a*b^3*d^3 + 8*a^4*g^3 + 8000*b^4*c^3, z, k)*b^2*(400*b^2*c^2 + 25*a^2*f^2 - 54*root(19683*a^11*b^2*z^3
+ 19683*a^8*b^2*e*z^2 + 810*a^6*b*f*g*z - 5670*a^5*b^2*d*f*z - 3240*a^5*b^2*c*g*z + 22680*a^4*b^3*c*d*z + 6561
*a^5*b^2*e^2*z + 270*a^3*b*e*f*g + 7560*a*b^3*c*d*e - 1890*a^2*b^2*d*e*f - 1080*a^2*b^2*c*e*g - 168*a^3*b*d*g^
2 - 6000*a*b^3*c^2*f + 1176*a^2*b^2*d^2*g + 1500*a^2*b^2*c*f^2 + 729*a^2*b^2*e^3 - 125*a^3*b*f^3 - 2744*a*b^3*
d^3 + 8*a^4*g^3 + 8000*b^4*c^3, z, k)*a^5*g + 36*a^2*e*g + 378*root(19683*a^11*b^2*z^3 + 19683*a^8*b^2*e*z^2 +
810*a^6*b*f*g*z - 5670*a^5*b^2*d*f*z - 3240*a^5*b^2*c*g*z + 22680*a^4*b^3*c*d*z + 6561*a^5*b^2*e^2*z + 270*a^
3*b*e*f*g + 7560*a*b^3*c*d*e - 1890*a^2*b^2*d*e*f - 1080*a^2*b^2*c*e*g - 168*a^3*b*d*g^2 - 6000*a*b^3*c^2*f +
1176*a^2*b^2*d^2*g + 1500*a^2*b^2*c*f^2 + 729*a^2*b^2*e^3 - 125*a^3*b*f^3 - 2744*a*b^3*d^3 + 8*a^4*g^3 + 8000*
b^4*c^3, z, k)*a^4*b*d + 324*a*b*e^2*x + 2800*b^2*c*d*x + 100*a^2*f*g*x + 2916*root(19683*a^11*b^2*z^3 + 19683
*a^8*b^2*e*z^2 + 810*a^6*b*f*g*z - 5670*a^5*b^2*d*f*z - 3240*a^5*b^2*c*g*z + 22680*a^4*b^3*c*d*z + 6561*a^5*b^
2*e^2*z + 270*a^3*b*e*f*g + 7560*a*b^3*c*d*e - 1890*a^2*b^2*d*e*f - 1080*a^2*b^2*c*e*g - 168*a^3*b*d*g^2 - 600
0*a*b^3*c^2*f + 1176*a^2*b^2*d^2*g + 1500*a^2*b^2*c*f^2 + 729*a^2*b^2*e^3 - 125*a^3*b*f^3 - 2744*a*b^3*d^3 + 8
*a^4*g^3 + 8000*b^4*c^3, z, k)^2*a^7*b*x - 200*a*b*c*f - 252*a*b*d*e - 400*a*b*c*g*x - 700*a*b*d*f*x + 1944*ro
ot(19683*a^11*b^2*z^3 + 19683*a^8*b^2*e*z^2 + 810*a^6*b*f*g*z - 5670*a^5*b^2*d*f*z - 3240*a^5*b^2*c*g*z + 2268
0*a^4*b^3*c*d*z + 6561*a^5*b^2*e^2*z + 270*a^3*b*e*f*g + 7560*a*b^3*c*d*e - 1890*a^2*b^2*d*e*f - 1080*a^2*b^2*
c*e*g - 168*a^3*b*d*g^2 - 6000*a*b^3*c^2*f + 1176*a^2*b^2*d^2*g + 1500*a^2*b^2*c*f^2 + 729*a^2*b^2*e^3 - 125*a
^3*b*f^3 - 2744*a*b^3*d^3 + 8*a^4*g^3 + 8000*b^4*c^3, z, k)*a^4*b*e*x))/(81*a^5) - (b*x*(8000*b^4*c^3 + 8*a^4*
g^3 - 2744*a*b^3*d^3 - 125*a^3*b*f^3 + 1500*a^2*b^2*c*f^2 + 1176*a^2*b^2*d^2*g - 6000*a*b^3*c^2*f - 168*a^3*b*
d*g^2 - 720*a^2*b^2*c*e*g - 1260*a^2*b^2*d*e*f + 5040*a*b^3*c*d*e + 180*a^3*b*e*f*g))/(729*a^9))*root(19683*a^
11*b^2*z^3 + 19683*a^8*b^2*e*z^2 + 810*a^6*b*f*g*z - 5670*a^5*b^2*d*f*z - 3240*a^5*b^2*c*g*z + 22680*a^4*b^3*c
*d*z + 6561*a^5*b^2*e^2*z + 270*a^3*b*e*f*g + 7560*a*b^3*c*d*e - 1890*a^2*b^2*d*e*f - 1080*a^2*b^2*c*e*g - 168
*a^3*b*d*g^2 - 6000*a*b^3*c^2*f + 1176*a^2*b^2*d^2*g + 1500*a^2*b^2*c*f^2 + 729*a^2*b^2*e^3 - 125*a^3*b*f^3 -
2744*a*b^3*d^3 + 8*a^4*g^3 + 8000*b^4*c^3, z, k), k, 1, 3) - (c/(2*a) + (4*x^3*(4*b*c - a*f))/(9*a^2) + (7*x^4
*(7*b*d - a*g))/(18*a^2) + (d*x)/a + (5*b*x^6*(4*b*c - a*f))/(18*a^3) + (2*b*x^7*(7*b*d - a*g))/(9*a^3) - (x^2
*(3*b*e - a*h))/(6*a*b) - (b*e*x^5)/(3*a^2))/(a^2*x^2 + b^2*x^8 + 2*a*b*x^5) + (e*log(x))/a^3

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