3.5.20 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 (a+b x^3)^2} \, dx\) [420]

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

[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 = 0.49, 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 = {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 (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 {\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 {(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} \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)^2),x]

[Out]

-1/3*c/(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 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 )^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 ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{8/3} b^{2/3}}\\ &=-\frac {c}{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 = 0.30, size = 303, normalized size = 0.90 \begin {gather*} \frac {-\frac {6 a c}{x^3}-\frac {9 a d}{x^2}-\frac {18 a e}{x}+\frac {a (-6 b (c+x (d+e x))+6 a (f+x (g+h x)))}{a+b x^3}-\frac {2 \sqrt {3} \sqrt [3]{a} \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 {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+18 (-2 b c+a f) \log (x)-\frac {2 \sqrt [3]{a} \left (5 b^{4/3} d-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {\sqrt [3]{a} \left (5 b^{4/3} d-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+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}}+6 (2 b c-a f) \log \left (a+b x^3\right )}{18 a^3} \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)^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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Maple [A]
time = 0.58, size = 332, normalized size = 0.98

method	result	size
default	\(\frac {\frac {\left (\frac {1}{3} a^{2} h -\frac {1}{3} a b e \right ) x^{2}+\left (\frac {1}{3} a^{2} g -\frac {1}{3} a b d \right ) x +\frac {a \left (a f -b c \right )}{3}}{b \,x^{3}+a}+\frac {\left (2 a^{2} g -5 a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {\left (a^{2} h -4 a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {\left (-3 a b f +6 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{9 b}}{a^{3}}-\frac {e}{a^{2} x}-\frac {c}{3 a^{2} x^{3}}-\frac {d}{2 a^{2} x^{2}}+\frac {\left (a f -2 b c \right ) \ln \left (x \right )}{a^{3}}\)	\(332\)
risch	\(\frac {\frac {\left (a h -4 b e \right ) x^{5}}{3 a^{2}}+\frac {\left (2 a g -5 b d \right ) x^{4}}{6 a^{2}}+\frac {\left (a f -2 b c \right ) x^{3}}{3 a^{2}}-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{9} b^{2} \textit {\_Z}^{3}+\left (9 a^{7} b^{2} f -18 a^{6} b^{3} c \right ) \textit {\_Z}^{2}+\left (6 a^{6} b g h -15 a^{5} b^{2} d h -24 a^{5} b^{2} e g +27 a^{5} b^{2} f^{2}-108 a^{4} b^{3} c f +60 a^{4} b^{3} d e +108 a^{3} b^{4} c^{2}\right ) \textit {\_Z} +a^{5} h^{3}-12 a^{4} b e \,h^{2}+18 a^{4} b f g h -8 a^{4} b \,g^{3}-36 a^{3} b^{2} c g h -45 a^{3} b^{2} d f h +60 a^{3} b^{2} d \,g^{2}+48 a^{3} b^{2} e^{2} h -72 a^{3} b^{2} e f g +27 a^{3} b^{2} f^{3}+90 a^{2} b^{3} c d h +144 a^{2} b^{3} c e g -162 a^{2} b^{3} c \,f^{2}-150 a^{2} b^{3} d^{2} g +180 a^{2} b^{3} d e f -64 a^{2} b^{3} e^{3}+324 a \,b^{4} c^{2} f -360 a \,b^{4} c d e +125 a \,b^{4} d^{3}-216 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8} b^{2}+\left (-24 a^{6} b^{2} f +48 a^{5} b^{3} c \right ) \textit {\_R}^{2}+\left (-20 a^{5} b g h +50 a^{4} b^{2} d h +80 a^{4} b^{2} e g -36 a^{4} b^{2} f^{2}+144 a^{3} b^{3} c f -200 a^{3} b^{3} d e -144 a^{2} b^{4} c^{2}\right ) \textit {\_R} -3 a^{4} h^{3}+36 a^{3} b e \,h^{2}-36 a^{3} b f g h +24 a^{3} b \,g^{3}+72 a^{2} b^{2} c g h +90 a^{2} b^{2} d f h -180 a^{2} b^{2} d \,g^{2}-144 a^{2} b^{2} e^{2} h +144 a^{2} b^{2} e f g -180 a \,b^{3} c d h -288 a \,b^{3} c e g +450 a \,b^{3} d^{2} g -360 a \,b^{3} d e f +192 a \,b^{3} e^{3}+720 b^{4} c d e -375 b^{4} d^{3}\right ) x +\left (a^{7} b h -4 a^{6} b^{2} e \right ) \textit {\_R}^{2}+\left (-6 a^{5} b f h -4 a^{5} b \,g^{2}+12 a^{4} b^{2} c h +20 a^{4} b^{2} d g +24 a^{4} b^{2} e f -48 a^{3} b^{3} c e -25 a^{3} b^{3} d^{2}\right ) \textit {\_R} -27 a^{3} b \,f^{2} h +36 a^{3} b f \,g^{2}+108 a^{2} b^{2} c f h -72 a^{2} b^{2} c \,g^{2}-180 a^{2} b^{2} d f g +108 a^{2} b^{2} e \,f^{2}-108 a \,b^{3} c^{2} h +360 a \,b^{3} c d g -432 a \,b^{3} c e f +225 a \,b^{3} d^{2} f +432 b^{4} c^{2} e -450 b^{4} c \,d^{2}\right )\right )}{9}+\frac {\ln \left (-x \right ) f}{a^{2}}-\frac {2 \ln \left (-x \right ) b c}{a^{3}}\)	\(909\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 369, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (a h - 4 \, b e\right )} x^{5} - {\left (5 \, b d - 2 \, a g\right )} x^{4} - 2 \, {\left (2 \, b c - a f\right )} x^{3} - 6 \, a x^{2} e - 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 \left (x\right )}{a^{3}} + \frac {\sqrt {3} {\left (a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - 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}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + 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}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 4 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - 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}}} \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)^2,x, algorithm="maxima")

[Out]

1/6*(2*(a*h - 4*b*e)*x^5 - (5*b*d - 2*a*g)*x^4 - 2*(2*b*c - a*f)*x^3 - 6*a*x^2*e - 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)*(a^2*h*(a/b)^(2/3) - 4*a*b*(a/b)^(2/3)*e - 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) + a^2*h*(a/b)^(1/3) - 4*a*b*(a/b)^(1/3)*e + 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) - a^2*h*(a/b)^(1/3) + 4
*a*b*(a/b)^(1/3)*e - 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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Fricas [C] Result contains complex when optimal does not.
time = 67.33, size = 16568, normalized size = 49.02 \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)^2,x, algorithm="fricas")

[Out]

-1/36*(12*(4*a*b*e - a^2*h)*x^5 + 36*a^2*e*x^2 + 6*(5*a*b*d - 2*a^2*g)*x^4 + 18*a^2*d*x + 12*(2*a*b*c - a^2*f)
*x^3 + 12*a^2*c + 2*(a^3*b*x^6 + a^4*x^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*(2*b*c - a*f)^2/a^6 - (36*b^3*c^2
+ 2*a^3*g*h + (9*f^2 - 8*e*g - 5*d*h)*a^2*b + 4*(5*d*e - 9*c*f)*a*b^2)/(a^6*b))/(54*(2*b*c - a*f)^3/a^9 - 9*(
36*b^3*c^2 + 2*a^3*g*h + (9*f^2 - 8*e*g - 5*d*h)*a^2*b + 4*(5*d*e - 9*c*f)*a*b^2)*(2*b*c - a*f)/(a^9*b) - (125
*b^4*d^3 + 64*a*b^3*e^3 - 150*a*b^3*d^2*g + 60*a^2*b^2*d*g^2 - 8*a^3*b*g^3 - 48*a^2*b^2*e^2*h + 12*a^3*b*e*h^2
- a^4*h^3)/(a^8*b^2) + (216*b^5*c^3 - a^5*h^3 + 2*(4*g^3 - 9*f*g*h + 6*e*h^2)*a^4*b - 3*(9*f^3 - 24*e*f*g + 1
6*e^2*h - 12*c*g*h + 5*(4*g^2 - 3*f*h)*d)*a^3*b^2 + 2*(32*e^3 - 90*d*e*f + 75*d^2*g + 9*(9*f^2 - 8*e*g - 5*d*h
)*c)*a^2*b^3 - (125*d^3 - 360*c*d*e + 324*c^2*f)*a*b^4)/(a^9*b^2))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*(2*
b*c - a*f)^3/a^9 - 9*(36*b^3*c^2 + 2*a^3*g*h + (9*f^2 - 8*e*g - 5*d*h)*a^2*b + 4*(5*d*e - 9*c*f)*a*b^2)*(2*b*c
- a*f)/(a ...

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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)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.49, size = 363, normalized size = 1.07 \begin {gather*} \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}} \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)^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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Mupad [B]
time = 5.96, size = 1924, normalized size = 5.69 \begin {gather*} \left (\sum _{k=1}^3\ln \left (-\frac {3\,h\,a^3\,b^2\,f^2-4\,a^3\,b^2\,f\,g^2-12\,h\,a^2\,b^3\,c\,f+8\,a^2\,b^3\,c\,g^2+20\,a^2\,b^3\,d\,f\,g-12\,e\,a^2\,b^3\,f^2+12\,h\,a\,b^4\,c^2-40\,a\,b^4\,c\,d\,g+48\,e\,a\,b^4\,c\,f-25\,a\,b^4\,d^2\,f-48\,e\,b^5\,c^2+50\,b^5\,c\,d^2}{9\,a^6}-\mathrm {root}\left (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\right )\,\left (\frac {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}+\mathrm {root}\left (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\right )\,\left (\frac {36\,a^6\,b^3\,e-9\,a^7\,b^2\,h}{9\,a^6}-\frac {x\,\left (1296\,a^5\,b^4\,c-648\,a^6\,b^3\,f\right )}{27\,a^6}+\mathrm {root}\left (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\right )\,a^2\,b^3\,x\,36\right )+\frac {x\,\left (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\right )}{27\,a^6}\right )-\frac {x\,\left (a^4\,b\,h^3-12\,a^3\,b^2\,e\,h^2-8\,a^3\,b^2\,g^3+12\,f\,a^3\,b^2\,g\,h+60\,a^2\,b^3\,d\,g^2-30\,f\,a^2\,b^3\,d\,h+48\,a^2\,b^3\,e^2\,h-48\,f\,a^2\,b^3\,e\,g-24\,c\,a^2\,b^3\,g\,h-150\,a\,b^4\,d^2\,g+120\,f\,a\,b^4\,d\,e+60\,c\,a\,b^4\,d\,h-64\,a\,b^4\,e^3+96\,c\,a\,b^4\,e\,g+125\,b^5\,d^3-240\,c\,b^5\,d\,e\right )}{27\,a^6}\right )\,\mathrm {root}\left (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\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {x^3\,\left (2\,b\,c-a\,f\right )}{3\,a^2}+\frac {x^4\,\left (5\,b\,d-2\,a\,g\right )}{6\,a^2}+\frac {x^5\,\left (4\,b\,e-a\,h\right )}{3\,a^2}+\frac {d\,x}{2\,a}}{b\,x^6+a\,x^3}-\frac {\ln \left (x\right )\,\left (2\,b\,c-a\,f\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^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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