3.5.29 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 (a+b x^3)^3} \, dx\) [429]

Optimal. Leaf size=395 \[ -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}-\frac {(3 b c-a f) \log (x)}{a^4}-\frac {\left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{2/3}}+\frac {\left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{2/3}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4} \]

[Out]

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

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Rubi [A]
time = 0.67, antiderivative size = 392, 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} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\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 e-a h)}{\sqrt [3]{b}}-5 a g+20 b d\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 d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac {\log (x) (3 b c-a f)}{a^4}-\frac {x \left (-3 b x^2 \left (\frac {5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (-b x^2 \left (\frac {b c}{a}-f\right )+x (b e-a h)-a g+b d\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^4*(a + b*x^3)^3),x]

[Out]

-1/3*c/(a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d - a*g + (b*e - a*h)*x - b*((b*c)/a - f)*x^2))/(6*a^2*(a
+ b*x^3)^2) - (x*(11*b*d - 5*a*g + 2*(5*b*e - 2*a*h)*x - 3*b*((5*b*c)/a - 3*f)*x^2))/(18*a^3*(a + b*x^3)) + (
(20*b^(4/3)*d + 14*a^(1/3)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(9*Sqrt[3]*a^(11/3)*b^(2/3)) - ((3*b*c - a*f)*Log[x])/a^4 - ((5*b^(1/3)*(4*b*d - a*g) - 2*a^(1/3)*(7*b*e -
a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(2/3)) + ((20*b*d - 5*a*g - (2*a^(1/3)*(7*b*e - a*h))/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)) + ((3*b*c - a*f)*Log[a + b*x^3])/(3*a^4)

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

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Mathematica [A]
time = 0.37, size = 352, normalized size = 0.89 \begin {gather*} \frac {-\frac {18 a c}{x^3}-\frac {27 a d}{x^2}-\frac {54 a e}{x}+\frac {3 a (-12 b c+6 a f-b x (11 d+10 e x)+a x (5 g+4 h x))}{a+b x^3}+\frac {a^2 (-9 b (c+x (d+e x))+9 a (f+x (g+h x)))}{\left (a+b x^3\right )^2}+\frac {2 \sqrt {3} \sqrt [3]{a} \left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 a^{4/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+54 (-3 b c+a f) \log (x)-\frac {2 \sqrt [3]{a} \left (20 b^{4/3} d-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+2 a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {\sqrt [3]{a} \left (20 b^{4/3} d-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+2 a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+18 (3 b c-a f) \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^4*(a + b*x^3)^3),x]

[Out]

((-18*a*c)/x^3 - (27*a*d)/x^2 - (54*a*e)/x + (3*a*(-12*b*c + 6*a*f - b*x*(11*d + 10*e*x) + a*x*(5*g + 4*h*x)))
/(a + b*x^3) + (a^2*(-9*b*(c + x*(d + e*x)) + 9*a*(f + x*(g + h*x))))/(a + b*x^3)^2 + (2*Sqrt[3]*a^(1/3)*(20*b
^(4/3)*d + 14*a^(1/3)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3)
+ 54*(-3*b*c + a*f)*Log[x] - (2*a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(1
/3) + b^(1/3)*x])/b^(2/3) + (a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 18*(3*b*c - a*f)*Log[a + b*x^3])/(54*a^4)

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Maple [A]
time = 0.41, size = 394, normalized size = 1.00

method	result	size
default	\(\frac {\frac {\left (\frac {2}{9} a^{2} b h -\frac {5}{9} a \,b^{2} e \right ) x^{5}+\left (\frac {5}{18} a^{2} b g -\frac {11}{18} a \,b^{2} d \right ) x^{4}+\left (\frac {1}{3} a^{2} b f -\frac {2}{3} a c \,b^{2}\right ) x^{3}+\frac {a^{2} \left (7 a h -13 b e \right ) x^{2}}{18}+\frac {a^{2} \left (4 a g -7 b d \right ) x}{9}+\frac {a^{3} f}{2}-\frac {5 c \,a^{2} b}{6}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 a^{2} g -20 a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}+\frac {\left (2 a^{2} h -14 a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9}+\frac {\left (-9 a b f +27 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{27 b}}{a^{4}}-\frac {d}{2 a^{3} x^{2}}-\frac {c}{3 a^{3} x^{3}}-\frac {e}{a^{3} x}+\frac {\left (a f -3 b c \right ) \ln \left (x \right )}{a^{4}}\)	\(394\)
risch	\(\frac {\frac {2 b \left (a h -7 b e \right ) x^{8}}{9 a^{3}}+\frac {5 b \left (a g -4 b d \right ) x^{7}}{18 a^{3}}+\frac {b \left (a f -3 b c \right ) x^{6}}{3 a^{3}}+\frac {7 \left (a h -7 b e \right ) x^{5}}{18 a^{2}}+\frac {4 \left (a g -4 b d \right ) x^{4}}{9 a^{2}}+\frac {\left (a f -3 b c \right ) x^{3}}{2 a^{2}}-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{12} b^{2} \textit {\_Z}^{3}+\left (27 a^{9} b^{2} f -81 a^{8} b^{3} c \right ) \textit {\_Z}^{2}+\left (30 a^{7} b g h -120 a^{6} b^{2} d h -210 a^{6} b^{2} e g +243 a^{6} b^{2} f^{2}-1458 a^{5} b^{3} c f +840 a^{5} b^{3} d e +2187 a^{4} b^{4} c^{2}\right ) \textit {\_Z} +8 a^{5} h^{3}-168 a^{4} b e \,h^{2}+270 a^{4} b f g h -125 a^{4} b \,g^{3}-810 a^{3} b^{2} c g h -1080 a^{3} b^{2} d f h +1500 a^{3} b^{2} d \,g^{2}+1176 a^{3} b^{2} e^{2} h -1890 a^{3} b^{2} e f g +729 a^{3} b^{2} f^{3}+3240 a^{2} b^{3} c d h +5670 a^{2} b^{3} c e g -6561 a^{2} b^{3} c \,f^{2}-6000 a^{2} b^{3} d^{2} g +7560 a^{2} b^{3} d e f -2744 a^{2} b^{3} e^{3}+19683 a \,b^{4} c^{2} f -22680 a \,b^{4} c d e +8000 a \,b^{4} d^{3}-19683 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b^{2}+\left (-72 a^{8} b^{2} f +216 a^{7} b^{3} c \right ) \textit {\_R}^{2}+\left (-100 a^{6} b g h +400 a^{5} b^{2} d h +700 a^{5} b^{2} e g -324 a^{5} b^{2} f^{2}+1944 a^{4} b^{3} c f -2800 a^{4} b^{3} d e -2916 a^{3} b^{4} c^{2}\right ) \textit {\_R} -24 a^{4} h^{3}+504 a^{3} b e \,h^{2}-540 a^{3} b f g h +375 a^{3} b \,g^{3}+1620 a^{2} b^{2} c g h +2160 a^{2} b^{2} d f h -4500 a^{2} b^{2} d \,g^{2}-3528 a^{2} b^{2} e^{2} h +3780 a^{2} b^{2} e f g -6480 a \,b^{3} c d h -11340 a \,b^{3} c e g +18000 a \,b^{3} d^{2} g -15120 a \,b^{3} d e f +8232 a \,b^{3} e^{3}+45360 b^{4} c d e -24000 b^{4} d^{3}\right ) x +\left (2 a^{9} b h -14 a^{8} b^{2} e \right ) \textit {\_R}^{2}+\left (-36 a^{6} b f h -25 a^{6} b \,g^{2}+108 a^{5} b^{2} c h +200 a^{5} b^{2} d g +252 a^{5} b^{2} e f -756 a^{4} b^{3} c e -400 a^{4} b^{3} d^{2}\right ) \textit {\_R} -486 a^{3} b \,f^{2} h +675 a^{3} b f \,g^{2}+2916 a^{2} b^{2} c f h -2025 a^{2} b^{2} c \,g^{2}-5400 a^{2} b^{2} d f g +3402 a^{2} b^{2} e \,f^{2}-4374 a \,b^{3} c^{2} h +16200 a \,b^{3} c d g -20412 a \,b^{3} c e f +10800 a \,b^{3} d^{2} f +30618 b^{4} c^{2} e -32400 b^{4} c \,d^{2}\right )\right )}{27}+\frac {\ln \left (-x \right ) f}{a^{3}}-\frac {3 \ln \left (-x \right ) b c}{a^{4}}\)	\(961\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^4*(((2/9*a^2*b*h-5/9*a*b^2*e)*x^5+(5/18*a^2*b*g-11/18*a*b^2*d)*x^4+(1/3*a^2*b*f-2/3*a*c*b^2)*x^3+1/18*a^2*
(7*a*h-13*b*e)*x^2+1/9*a^2*(4*a*g-7*b*d)*x+1/2*a^3*f-5/6*c*a^2*b)/(b*x^3+a)^2+1/9*(5*a^2*g-20*a*b*d)*(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)*arct
an(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))+1/9*(2*a^2*h-14*a*b*e)*(-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)))+1/2
7*(-9*a*b*f+27*b^2*c)*ln(b*x^3+a)/b)-1/2*d/a^3/x^2-1/3*c/a^3/x^3-e/a^3/x+(a*f-3*b*c)/a^4*ln(x)

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Maxima [A]
time = 0.50, size = 448, normalized size = 1.13 \begin {gather*} \frac {4 \, {\left (a b h - 7 \, b^{2} e\right )} x^{8} - 5 \, {\left (4 \, b^{2} d - a b g\right )} x^{7} - 6 \, {\left (3 \, b^{2} c - a b f\right )} x^{6} + 7 \, {\left (a^{2} h - 7 \, a b e\right )} x^{5} - 8 \, {\left (4 \, a b d - a^{2} g\right )} x^{4} - 18 \, a^{2} x^{2} e - 9 \, a^{2} d x - 9 \, {\left (3 \, a b c - a^{2} f\right )} x^{3} - 6 \, a^{2} c}{18 \, {\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} - \frac {{\left (3 \, b c - a f\right )} \log \left (x\right )}{a^{4}} + \frac {\sqrt {3} {\left (2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - 14 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - 20 \, a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, 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 )}{27 \, a^{5}} + \frac {{\left (54 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 18 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + 20 \, a b d - 5 \, 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 )}{54 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (27 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 9 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 14 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - 20 \, a b d + 5 \, a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} 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^4/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 72.05, size = 16697, normalized size = 42.27 \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^4/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/108*(24*(7*a*b^2*e - a^2*b*h)*x^8 + 30*(4*a*b^2*d - a^2*b*g)*x^7 + 36*(3*a*b^2*c - a^2*b*f)*x^6 + 108*a^3*e
*x^2 + 42*(7*a^2*b*e - a^3*h)*x^5 + 54*a^3*d*x + 48*(4*a^2*b*d - a^3*g)*x^4 + 36*a^3*c + 54*(3*a^2*b*c - a^3*f
)*x^3 + 2*(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*(3*b*c - a*f)^2/a^8 - (729
*b^3*c^2 + 10*a^3*g*h + (81*f^2 - 70*e*g - 40*d*h)*a^2*b + 2*(140*d*e - 243*c*f)*a*b^2)/(a^8*b))/(1458*(3*b*c
- a*f)^3/a^12 - 27*(729*b^3*c^2 + 10*a^3*g*h + (81*f^2 - 70*e*g - 40*d*h)*a^2*b + 2*(140*d*e - 243*c*f)*a*b^2)
*(3*b*c - a*f)/(a^12*b) - (8000*b^4*d^3 + 2744*a*b^3*e^3 - 6000*a*b^3*d^2*g + 1500*a^2*b^2*d*g^2 - 125*a^3*b*g
^3 - 1176*a^2*b^2*e^2*h + 168*a^3*b*e*h^2 - 8*a^4*h^3)/(a^11*b^2) + (19683*b^5*c^3 - 8*a^5*h^3 + (125*g^3 - 27
0*f*g*h + 168*e*h^2)*a^4*b - 3*(243*f^3 - 630*e*f*g + 392*e^2*h - 270*c*g*h + 20*(25*g^2 - 18*f*h)*d)*a^3*b^2
+ (2744*e^3 - 7560*d*e*f + 6000*d^2*g + 81*(81*f^2 - 70*e*g - 40*d*h)*c)*a^2*b^3 - (8000*d^3 - 22680*c*d*e + 1
9683*c^2*f ...

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

[Out]

Timed out

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Giac [A]
time = 0.50, size = 431, normalized size = 1.09 \begin {gather*} \frac {\sqrt {3} {\left (20 \, b^{2} d - 5 \, a b g + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h - 14 \, \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 )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} + \frac {{\left (20 \, b^{2} d - 5 \, a b g - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h + 14 \, \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 )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} + \frac {{\left (3 \, b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} - \frac {{\left (3 \, b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (2 \, a^{6} b h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{5} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - 20 \, a^{5} b^{2} d + 5 \, a^{6} 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 )}{27 \, a^{9} b} + \frac {4 \, {\left (a^{2} b h - 7 \, a b^{2} e\right )} x^{8} - 5 \, {\left (4 \, a b^{2} d - a^{2} b g\right )} x^{7} - 6 \, {\left (3 \, a b^{2} c - a^{2} b f\right )} x^{6} + 7 \, {\left (a^{3} h - 7 \, a^{2} b e\right )} x^{5} - 18 \, a^{3} x^{2} e - 9 \, a^{3} d x - 8 \, {\left (4 \, a^{2} b d - a^{3} g\right )} x^{4} - 6 \, a^{3} c - 9 \, {\left (3 \, a^{2} b c - a^{3} f\right )} x^{3}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4} x^{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^4/(b*x^3+a)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 6.32, size = 1994, normalized size = 5.05 \begin {gather*} \left (\sum _{k=1}^3\ln \left (-\frac {18\,h\,a^3\,b^2\,f^2-25\,a^3\,b^2\,f\,g^2-108\,h\,a^2\,b^3\,c\,f+75\,a^2\,b^3\,c\,g^2+200\,a^2\,b^3\,d\,f\,g-126\,e\,a^2\,b^3\,f^2+162\,h\,a\,b^4\,c^2-600\,a\,b^4\,c\,d\,g+756\,e\,a\,b^4\,c\,f-400\,a\,b^4\,d^2\,f-1134\,e\,b^5\,c^2+1200\,b^5\,c\,d^2}{81\,a^9}-\mathrm {root}\left (19683\,a^{12}\,b^2\,z^3+19683\,a^9\,b^2\,f\,z^2-59049\,a^8\,b^3\,c\,z^2+810\,a^7\,b\,g\,h\,z-5670\,a^6\,b^2\,e\,g\,z-3240\,a^6\,b^2\,d\,h\,z-39366\,a^5\,b^3\,c\,f\,z+22680\,a^5\,b^3\,d\,e\,z+6561\,a^6\,b^2\,f^2\,z+59049\,a^4\,b^4\,c^2\,z+270\,a^4\,b\,f\,g\,h-22680\,a\,b^4\,c\,d\,e-1890\,a^3\,b^2\,e\,f\,g-1080\,a^3\,b^2\,d\,f\,h-810\,a^3\,b^2\,c\,g\,h+7560\,a^2\,b^3\,d\,e\,f+5670\,a^2\,b^3\,c\,e\,g+3240\,a^2\,b^3\,c\,d\,h-168\,a^4\,b\,e\,h^2+19683\,a\,b^4\,c^2\,f+1176\,a^3\,b^2\,e^2\,h-6000\,a^2\,b^3\,d^2\,g+1500\,a^3\,b^2\,d\,g^2-6561\,a^2\,b^3\,c\,f^2+729\,a^3\,b^2\,f^3-2744\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8000\,a\,b^4\,d^3+8\,a^5\,h^3-19683\,b^5\,c^3,z,k\right )\,\left (\frac {400\,a^4\,b^4\,d^2+25\,a^6\,b^2\,g^2+756\,a^4\,b^4\,c\,e-108\,a^5\,b^3\,c\,h-200\,a^5\,b^3\,d\,g-252\,a^5\,b^3\,e\,f+36\,a^6\,b^2\,f\,h}{81\,a^9}+\mathrm {root}\left (19683\,a^{12}\,b^2\,z^3+19683\,a^9\,b^2\,f\,z^2-59049\,a^8\,b^3\,c\,z^2+810\,a^7\,b\,g\,h\,z-5670\,a^6\,b^2\,e\,g\,z-3240\,a^6\,b^2\,d\,h\,z-39366\,a^5\,b^3\,c\,f\,z+22680\,a^5\,b^3\,d\,e\,z+6561\,a^6\,b^2\,f^2\,z+59049\,a^4\,b^4\,c^2\,z+270\,a^4\,b\,f\,g\,h-22680\,a\,b^4\,c\,d\,e-1890\,a^3\,b^2\,e\,f\,g-1080\,a^3\,b^2\,d\,f\,h-810\,a^3\,b^2\,c\,g\,h+7560\,a^2\,b^3\,d\,e\,f+5670\,a^2\,b^3\,c\,e\,g+3240\,a^2\,b^3\,c\,d\,h-168\,a^4\,b\,e\,h^2+19683\,a\,b^4\,c^2\,f+1176\,a^3\,b^2\,e^2\,h-6000\,a^2\,b^3\,d^2\,g+1500\,a^3\,b^2\,d\,g^2-6561\,a^2\,b^3\,c\,f^2+729\,a^3\,b^2\,f^3-2744\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8000\,a\,b^4\,d^3+8\,a^5\,h^3-19683\,b^5\,c^3,z,k\right )\,\left (\frac {378\,a^8\,b^3\,e-54\,a^9\,b^2\,h}{81\,a^9}-\frac {x\,\left (52488\,a^7\,b^4\,c-17496\,a^8\,b^3\,f\right )}{729\,a^9}+\mathrm {root}\left (19683\,a^{12}\,b^2\,z^3+19683\,a^9\,b^2\,f\,z^2-59049\,a^8\,b^3\,c\,z^2+810\,a^7\,b\,g\,h\,z-5670\,a^6\,b^2\,e\,g\,z-3240\,a^6\,b^2\,d\,h\,z-39366\,a^5\,b^3\,c\,f\,z+22680\,a^5\,b^3\,d\,e\,z+6561\,a^6\,b^2\,f^2\,z+59049\,a^4\,b^4\,c^2\,z+270\,a^4\,b\,f\,g\,h-22680\,a\,b^4\,c\,d\,e-1890\,a^3\,b^2\,e\,f\,g-1080\,a^3\,b^2\,d\,f\,h-810\,a^3\,b^2\,c\,g\,h+7560\,a^2\,b^3\,d\,e\,f+5670\,a^2\,b^3\,c\,e\,g+3240\,a^2\,b^3\,c\,d\,h-168\,a^4\,b\,e\,h^2+19683\,a\,b^4\,c^2\,f+1176\,a^3\,b^2\,e^2\,h-6000\,a^2\,b^3\,d^2\,g+1500\,a^3\,b^2\,d\,g^2-6561\,a^2\,b^3\,c\,f^2+729\,a^3\,b^2\,f^3-2744\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8000\,a\,b^4\,d^3+8\,a^5\,h^3-19683\,b^5\,c^3,z,k\right )\,a^2\,b^3\,x\,36\right )+\frac {x\,\left (26244\,a^3\,b^5\,c^2+2916\,a^5\,b^3\,f^2-17496\,a^4\,b^4\,c\,f+25200\,a^4\,b^4\,d\,e-3600\,a^5\,b^3\,d\,h-6300\,a^5\,b^3\,e\,g+900\,a^6\,b^2\,g\,h\right )}{729\,a^9}\right )-\frac {x\,\left (8\,a^4\,b\,h^3-168\,a^3\,b^2\,e\,h^2-125\,a^3\,b^2\,g^3+180\,f\,a^3\,b^2\,g\,h+1500\,a^2\,b^3\,d\,g^2-720\,f\,a^2\,b^3\,d\,h+1176\,a^2\,b^3\,e^2\,h-1260\,f\,a^2\,b^3\,e\,g-540\,c\,a^2\,b^3\,g\,h-6000\,a\,b^4\,d^2\,g+5040\,f\,a\,b^4\,d\,e+2160\,c\,a\,b^4\,d\,h-2744\,a\,b^4\,e^3+3780\,c\,a\,b^4\,e\,g+8000\,b^5\,d^3-15120\,c\,b^5\,d\,e\right )}{729\,a^9}\right )\,\mathrm {root}\left (19683\,a^{12}\,b^2\,z^3+19683\,a^9\,b^2\,f\,z^2-59049\,a^8\,b^3\,c\,z^2+810\,a^7\,b\,g\,h\,z-5670\,a^6\,b^2\,e\,g\,z-3240\,a^6\,b^2\,d\,h\,z-39366\,a^5\,b^3\,c\,f\,z+22680\,a^5\,b^3\,d\,e\,z+6561\,a^6\,b^2\,f^2\,z+59049\,a^4\,b^4\,c^2\,z+270\,a^4\,b\,f\,g\,h-22680\,a\,b^4\,c\,d\,e-1890\,a^3\,b^2\,e\,f\,g-1080\,a^3\,b^2\,d\,f\,h-810\,a^3\,b^2\,c\,g\,h+7560\,a^2\,b^3\,d\,e\,f+5670\,a^2\,b^3\,c\,e\,g+3240\,a^2\,b^3\,c\,d\,h-168\,a^4\,b\,e\,h^2+19683\,a\,b^4\,c^2\,f+1176\,a^3\,b^2\,e^2\,h-6000\,a^2\,b^3\,d^2\,g+1500\,a^3\,b^2\,d\,g^2-6561\,a^2\,b^3\,c\,f^2+729\,a^3\,b^2\,f^3-2744\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8000\,a\,b^4\,d^3+8\,a^5\,h^3-19683\,b^5\,c^3,z,k\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {x^3\,\left (3\,b\,c-a\,f\right )}{2\,a^2}+\frac {4\,x^4\,\left (4\,b\,d-a\,g\right )}{9\,a^2}+\frac {7\,x^5\,\left (7\,b\,e-a\,h\right )}{18\,a^2}+\frac {d\,x}{2\,a}+\frac {b\,x^6\,\left (3\,b\,c-a\,f\right )}{3\,a^3}+\frac {5\,b\,x^7\,\left (4\,b\,d-a\,g\right )}{18\,a^3}+\frac {2\,b\,x^8\,\left (7\,b\,e-a\,h\right )}{9\,a^3}}{a^2\,x^3+2\,a\,b\,x^6+b^2\,x^9}-\frac {\ln \left (x\right )\,\left (3\,b\,c-a\,f\right )}{a^4} \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^4*(a + b*x^3)^3),x)

[Out]

symsum(log(- (1200*b^5*c*d^2 - 1134*b^5*c^2*e + 75*a^2*b^3*c*g^2 - 126*a^2*b^3*e*f^2 - 25*a^3*b^2*f*g^2 + 18*a
^3*b^2*f^2*h - 400*a*b^4*d^2*f + 162*a*b^4*c^2*h - 108*a^2*b^3*c*f*h + 200*a^2*b^3*d*f*g - 600*a*b^4*c*d*g + 7
56*a*b^4*c*e*f)/(81*a^9) - root(19683*a^12*b^2*z^3 + 19683*a^9*b^2*f*z^2 - 59049*a^8*b^3*c*z^2 + 810*a^7*b*g*h
*z - 5670*a^6*b^2*e*g*z - 3240*a^6*b^2*d*h*z - 39366*a^5*b^3*c*f*z + 22680*a^5*b^3*d*e*z + 6561*a^6*b^2*f^2*z
+ 59049*a^4*b^4*c^2*z + 270*a^4*b*f*g*h - 22680*a*b^4*c*d*e - 1890*a^3*b^2*e*f*g - 1080*a^3*b^2*d*f*h - 810*a^
3*b^2*c*g*h + 7560*a^2*b^3*d*e*f + 5670*a^2*b^3*c*e*g + 3240*a^2*b^3*c*d*h - 168*a^4*b*e*h^2 + 19683*a*b^4*c^2
*f + 1176*a^3*b^2*e^2*h - 6000*a^2*b^3*d^2*g + 1500*a^3*b^2*d*g^2 - 6561*a^2*b^3*c*f^2 + 729*a^3*b^2*f^3 - 274
4*a^2*b^3*e^3 - 125*a^4*b*g^3 + 8000*a*b^4*d^3 + 8*a^5*h^3 - 19683*b^5*c^3, z, k)*((400*a^4*b^4*d^2 + 25*a^6*b
^2*g^2 + 756*a^4*b^4*c*e - 108*a^5*b^3*c*h - 200*a^5*b^3*d*g - 252*a^5*b^3*e*f + 36*a^6*b^2*f*h)/(81*a^9) + ro
ot(19683*a^12*b^2*z^3 + 19683*a^9*b^2*f*z^2 - 59049*a^8*b^3*c*z^2 + 810*a^7*b*g*h*z - 5670*a^6*b^2*e*g*z - 324
0*a^6*b^2*d*h*z - 39366*a^5*b^3*c*f*z + 22680*a^5*b^3*d*e*z + 6561*a^6*b^2*f^2*z + 59049*a^4*b^4*c^2*z + 270*a
^4*b*f*g*h - 22680*a*b^4*c*d*e - 1890*a^3*b^2*e*f*g - 1080*a^3*b^2*d*f*h - 810*a^3*b^2*c*g*h + 7560*a^2*b^3*d*
e*f + 5670*a^2*b^3*c*e*g + 3240*a^2*b^3*c*d*h - 168*a^4*b*e*h^2 + 19683*a*b^4*c^2*f + 1176*a^3*b^2*e^2*h - 600
0*a^2*b^3*d^2*g + 1500*a^3*b^2*d*g^2 - 6561*a^2*b^3*c*f^2 + 729*a^3*b^2*f^3 - 2744*a^2*b^3*e^3 - 125*a^4*b*g^3
+ 8000*a*b^4*d^3 + 8*a^5*h^3 - 19683*b^5*c^3, z, k)*((378*a^8*b^3*e - 54*a^9*b^2*h)/(81*a^9) - (x*(52488*a^7*
b^4*c - 17496*a^8*b^3*f))/(729*a^9) + 36*root(19683*a^12*b^2*z^3 + 19683*a^9*b^2*f*z^2 - 59049*a^8*b^3*c*z^2 +
810*a^7*b*g*h*z - 5670*a^6*b^2*e*g*z - 3240*a^6*b^2*d*h*z - 39366*a^5*b^3*c*f*z + 22680*a^5*b^3*d*e*z + 6561*
a^6*b^2*f^2*z + 59049*a^4*b^4*c^2*z + 270*a^4*b*f*g*h - 22680*a*b^4*c*d*e - 1890*a^3*b^2*e*f*g - 1080*a^3*b^2*
d*f*h - 810*a^3*b^2*c*g*h + 7560*a^2*b^3*d*e*f + 5670*a^2*b^3*c*e*g + 3240*a^2*b^3*c*d*h - 168*a^4*b*e*h^2 + 1
9683*a*b^4*c^2*f + 1176*a^3*b^2*e^2*h - 6000*a^2*b^3*d^2*g + 1500*a^3*b^2*d*g^2 - 6561*a^2*b^3*c*f^2 + 729*a^3
*b^2*f^3 - 2744*a^2*b^3*e^3 - 125*a^4*b*g^3 + 8000*a*b^4*d^3 + 8*a^5*h^3 - 19683*b^5*c^3, z, k)*a^2*b^3*x) + (
x*(26244*a^3*b^5*c^2 + 2916*a^5*b^3*f^2 - 17496*a^4*b^4*c*f + 25200*a^4*b^4*d*e - 3600*a^5*b^3*d*h - 6300*a^5*
b^3*e*g + 900*a^6*b^2*g*h))/(729*a^9)) - (x*(8000*b^5*d^3 - 2744*a*b^4*e^3 + 8*a^4*b*h^3 - 125*a^3*b^2*g^3 + 1
500*a^2*b^3*d*g^2 + 1176*a^2*b^3*e^2*h - 168*a^3*b^2*e*h^2 - 15120*b^5*c*d*e - 6000*a*b^4*d^2*g - 540*a^2*b^3*
c*g*h - 720*a^2*b^3*d*f*h - 1260*a^2*b^3*e*f*g + 180*a^3*b^2*f*g*h + 2160*a*b^4*c*d*h + 3780*a*b^4*c*e*g + 504
0*a*b^4*d*e*f))/(729*a^9))*root(19683*a^12*b^2*z^3 + 19683*a^9*b^2*f*z^2 - 59049*a^8*b^3*c*z^2 + 810*a^7*b*g*h
*z - 5670*a^6*b^2*e*g*z - 3240*a^6*b^2*d*h*z - 39366*a^5*b^3*c*f*z + 22680*a^5*b^3*d*e*z + 6561*a^6*b^2*f^2*z
+ 59049*a^4*b^4*c^2*z + 270*a^4*b*f*g*h - 22680*a*b^4*c*d*e - 1890*a^3*b^2*e*f*g - 1080*a^3*b^2*d*f*h - 810*a^
3*b^2*c*g*h + 7560*a^2*b^3*d*e*f + 5670*a^2*b^3*c*e*g + 3240*a^2*b^3*c*d*h - 168*a^4*b*e*h^2 + 19683*a*b^4*c^2
*f + 1176*a^3*b^2*e^2*h - 6000*a^2*b^3*d^2*g + 1500*a^3*b^2*d*g^2 - 6561*a^2*b^3*c*f^2 + 729*a^3*b^2*f^3 - 274
4*a^2*b^3*e^3 - 125*a^4*b*g^3 + 8000*a*b^4*d^3 + 8*a^5*h^3 - 19683*b^5*c^3, z, k), k, 1, 3) - (c/(3*a) + (e*x^
2)/a + (x^3*(3*b*c - a*f))/(2*a^2) + (4*x^4*(4*b*d - a*g))/(9*a^2) + (7*x^5*(7*b*e - a*h))/(18*a^2) + (d*x)/(2
*a) + (b*x^6*(3*b*c - a*f))/(3*a^3) + (5*b*x^7*(4*b*d - a*g))/(18*a^3) + (2*b*x^8*(7*b*e - a*h))/(9*a^3))/(a^2
*x^3 + b^2*x^9 + 2*a*b*x^6) - (log(x)*(3*b*c - a*f))/a^4

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