3.5.18 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 (a+b x^3)^2} \, dx\) [418]

Optimal. Leaf size=301 \[ -\frac {c}{a^2 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a^2 b \left (a+b x^3\right )}+\frac {\left (4 b^{5/3} c-2 a^{2/3} b e-a b^{2/3} f-a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{4/3}}+\frac {d \log (x)}{a^2}+\frac {\left (b^{2/3} (4 b c-a f)+a^{2/3} (2 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{4/3}}-\frac {\left (b^{2/3} (4 b c-a f)+a^{2/3} (2 b e+a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{4/3}}-\frac {d \log \left (a+b x^3\right )}{3 a^2} \]

[Out]

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

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Rubi [A]
time = 0.55, antiderivative size = 301, normalized size of antiderivative = 1.00, 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 (-2 a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+4 b^{5/3} c\right )}{3 \sqrt {3} a^{7/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{18 a^{7/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{9 a^{7/3} b^{4/3}}+\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^2}-\frac {c}{a^2 x}+\frac {d \log (x)}{a^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^2*(a + b*x^3)^2),x]

[Out]

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

Rule 31

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

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

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Mathematica [A]
time = 0.20, size = 285, normalized size = 0.95 \begin {gather*} -\frac {\frac {18 a c}{x}+\frac {6 a \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{b \left (a+b x^3\right )}+\frac {2 \sqrt {3} a^{2/3} \left (-4 b^{5/3} c+2 a^{2/3} b e+a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{4/3}}-18 a d \log (x)-\frac {2 a^{2/3} \left (4 b^{5/3} c+2 a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}+\frac {a^{2/3} \left (4 b^{5/3} c+2 a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{4/3}}+6 a d \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^2*(a + b*x^3)^2),x]

[Out]

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

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Maple [A]
time = 0.43, size = 298, normalized size = 0.99

method	result	size
default	\(\frac {\frac {\left (\frac {a f}{3}-\frac {b c}{3}\right ) x^{2}-\frac {a \left (a h -b e \right ) x}{3 b}-\frac {a \left (a g -b d \right )}{3 b}}{b \,x^{3}+a}+\frac {\left (a^{2} h +2 a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a b f -4 b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-b d \ln \left (b \,x^{3}+a \right )}{3 b}}{a^{2}}-\frac {c}{a^{2} x}+\frac {d \ln \left (x \right )}{a^{2}}\)	\(298\)
risch	\(\frac {\frac {\left (a f -4 b c \right ) x^{3}}{3 a^{2}}-\frac {\left (a h -b e \right ) x^{2}}{3 a b}-\frac {\left (a g -b d \right ) x}{3 a b}-\frac {c}{a}}{x \left (b \,x^{3}+a \right )}+\frac {d \ln \left (x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{7} b^{4} \textit {\_Z}^{3}+9 a^{5} b^{4} d \,\textit {\_Z}^{2}+\left (3 a^{5} b^{2} f h -12 a^{4} b^{3} c h +6 a^{4} b^{3} e f -24 a^{3} b^{4} c e +27 a^{3} b^{4} d^{2}\right ) \textit {\_Z} -a^{5} h^{3}-6 a^{4} b e \,h^{2}+9 a^{3} b^{2} d f h -12 a^{3} b^{2} e^{2} h +a^{3} b^{2} f^{3}-36 a^{2} b^{3} c d h -12 a^{2} b^{3} c \,f^{2}+18 a^{2} b^{3} d e f -8 a^{2} b^{3} e^{3}+48 a \,b^{4} c^{2} f -72 a \,b^{4} c d e +27 a \,b^{4} d^{3}-64 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{7} b^{4}-24 \textit {\_R}^{2} a^{5} b^{4} d +\left (-10 a^{5} b^{2} f h +40 a^{4} b^{3} c h -20 a^{4} b^{3} e f +80 a^{3} b^{4} c e -36 a^{3} b^{4} d^{2}\right ) \textit {\_R} +3 a^{5} h^{3}+18 a^{4} b e \,h^{2}-18 a^{3} b^{2} d f h +36 a^{3} b^{2} e^{2} h -3 a^{3} b^{2} f^{3}+72 a^{2} b^{3} c d h +36 a^{2} b^{3} c \,f^{2}-36 a^{2} b^{3} d e f +24 a^{2} b^{3} e^{3}-144 a \,b^{4} c^{2} f +144 a \,b^{4} c d e +192 b^{5} c^{3}\right ) x +\left (a^{6} b^{3} f -4 b^{4} c \,a^{5}\right ) \textit {\_R}^{2}+\left (-a^{6} h^{2} b -4 b^{2} e h \,a^{5}-6 a^{4} b^{3} d f -4 a^{4} b^{3} e^{2}+24 b^{4} c d \,a^{3}\right ) \textit {\_R} +9 a^{4} b d \,h^{2}+36 a^{3} b^{2} d e h -27 a^{2} b^{3} d^{2} f +36 a^{2} b^{3} d \,e^{2}+108 a \,b^{4} c \,d^{2}\right )\right )}{9}\)	\(634\)

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^2/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^2*(((1/3*a*f-1/3*b*c)*x^2-1/3*a*(a*h-b*e)/b*x-1/3*a*(a*g-b*d)/b)/(b*x^3+a)+1/3/b*((a^2*h+2*a*b*e)*(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)*arc
tan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))+(a*b*f-4*b^2*c)*(-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)))-b*d*ln(b*
x^3+a)))-c/a^2/x+d*ln(x)/a^2

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Maxima [A]
time = 0.50, size = 332, normalized size = 1.10 \begin {gather*} -\frac {{\left (4 \, b^{2} c - a b f\right )} x^{3} + 3 \, a b c + {\left (a^{2} h - a b e\right )} x^{2} - {\left (a b d - a^{2} g\right )} x}{3 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x\right )}} + \frac {d \log \left (x\right )}{a^{2}} - \frac {\sqrt {3} {\left (4 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} 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 \, a^{3} b} - \frac {{\left (6 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h + 2 \, a 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 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h - 2 \, a b e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b^{2} \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^2/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/3*((4*b^2*c - a*b*f)*x^3 + 3*a*b*c + (a^2*h - a*b*e)*x^2 - (a*b*d - a^2*g)*x)/(a^2*b^2*x^4 + a^3*b*x) + d*l
og(x)/a^2 - 1/9*sqrt(3)*(4*b^2*c*(a/b)^(2/3) - a*b*f*(a/b)^(2/3) - a^2*h*(a/b)^(1/3) - 2*a*b*(a/b)^(1/3)*e)*ar
ctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b) - 1/18*(6*b^2*d*(a/b)^(2/3) + 4*b^2*c*(a/b)^(1/3) -
a*b*f*(a/b)^(1/3) + a^2*h + 2*a*b*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b^2*(a/b)^(2/3)) - 1/9*(3*b^2
*d*(a/b)^(2/3) - 4*b^2*c*(a/b)^(1/3) + a*b*f*(a/b)^(1/3) - a^2*h - 2*a*b*e)*log(x + (a/b)^(1/3))/(a^2*b^2*(a/b
)^(2/3))

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Fricas [C] Result contains complex when optimal does not.
time = 21.85, size = 12556, normalized size = 41.71 \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^2/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

-1/324*(108*(4*b^2*c - a*b*f)*x^3 + 324*a*b*c - 108*(a*b*e - a^2*h)*x^2 + 2*(a^2*b^2*x^4 + a^3*b*x)*((-I*sqrt(
3) + 1)*(9*d^2/a^4 - (a^2*f*h + 2*(e*f - 2*c*h)*a*b + (9*d^2 - 8*c*e)*b^2)/(a^4*b^2))/(-1/27*d^3/a^6 + 1/162*(
a^2*f*h + 2*(e*f - 2*c*h)*a*b + (9*d^2 - 8*c*e)*b^2)*d/(a^6*b^2) - 1/1458*(64*b^5*c^3 - 8*a^2*b^3*e^3 - 48*a*b
^4*c^2*f + 12*a^2*b^3*c*f^2 - a^3*b^2*f^3 - 12*a^3*b^2*e^2*h - 6*a^4*b*e*h^2 - a^5*h^3)/(a^7*b^4) + 1/1458*(64
*b^5*c^3 + 6*a^4*b*e*h^2 + a^5*h^3 - (f^3 - 12*e^2*h + 9*d*f*h)*a^3*b^2 + 2*(4*e^3 - 9*d*e*f + 6*(f^2 + 3*d*h)
*c)*a^2*b^3 - 3*(9*d^3 - 24*c*d*e + 16*c^2*f)*a*b^4)/(a^7*b^4))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*d^3/a^6 + 1/
162*(a^2*f*h + 2*(e*f - 2*c*h)*a*b + (9*d^2 - 8*c*e)*b^2)*d/(a^6*b^2) - 1/1458*(64*b^5*c^3 - 8*a^2*b^3*e^3 - 4
8*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 - a^3*b^2*f^3 - 12*a^3*b^2*e^2*h - 6*a^4*b*e*h^2 - a^5*h^3)/(a^7*b^4) + 1/145
8*(64*b^5*c^3 + 6*a^4*b*e*h^2 + a^5*h^3 - (f^3 - 12*e^2*h + 9*d*f*h)*a^3*b^2 + 2*(4*e^3 - 9*d*e*f + 6*(f^2 + 3
*d*h)*c)*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**2/(b*x**3+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.47, size = 328, normalized size = 1.09 \begin {gather*} -\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {\sqrt {3} {\left (a^{2} h + 2 \, a b e + 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {1}{3}} a f\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 (a^{2} h + 2 \, a b e - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {1}{3}} 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 )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} - \frac {4 \, b^{2} c x^{3} - a b f x^{3} + a^{2} h x^{2} - a b x^{2} e - a b d x + a^{2} g x + 3 \, a b c}{3 \, {\left (b x^{4} + a x\right )} a^{2} b} + \frac {{\left (4 \, a^{2} b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} b^{2} h - 2 \, a^{3} b^{3} e\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^{5} b^{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^2/(b*x^3+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.77, size = 1684, normalized size = 5.59 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {d\,\left (a^3\,h^2+4\,a^2\,b\,e\,h+4\,a\,b^2\,e^2-3\,d\,f\,a\,b^2+12\,c\,d\,b^3\right )}{9\,a^4}-\frac {\mathrm {root}\left (729\,a^7\,b^4\,z^3+729\,a^5\,b^4\,d\,z^2+27\,a^5\,b^2\,f\,h\,z-108\,a^4\,b^3\,c\,h\,z+54\,a^4\,b^3\,e\,f\,z-216\,a^3\,b^4\,c\,e\,z+243\,a^3\,b^4\,d^2\,z-72\,a\,b^4\,c\,d\,e+9\,a^3\,b^2\,d\,f\,h-36\,a^2\,b^3\,c\,d\,h+18\,a^2\,b^3\,d\,e\,f-6\,a^4\,b\,e\,h^2+48\,a\,b^4\,c^2\,f-12\,a^3\,b^2\,e^2\,h-12\,a^2\,b^3\,c\,f^2-8\,a^2\,b^3\,e^3+27\,a\,b^4\,d^3-a^5\,h^3-64\,b^5\,c^3+a^3\,b^2\,f^3,z,k\right )\,\left (a^3\,h^2+4\,a\,b^2\,e^2+36\,b^3\,d^2\,x-24\,b^3\,c\,d+{\mathrm {root}\left (729\,a^7\,b^4\,z^3+729\,a^5\,b^4\,d\,z^2+27\,a^5\,b^2\,f\,h\,z-108\,a^4\,b^3\,c\,h\,z+54\,a^4\,b^3\,e\,f\,z-216\,a^3\,b^4\,c\,e\,z+243\,a^3\,b^4\,d^2\,z-72\,a\,b^4\,c\,d\,e+9\,a^3\,b^2\,d\,f\,h-36\,a^2\,b^3\,c\,d\,h+18\,a^2\,b^3\,d\,e\,f-6\,a^4\,b\,e\,h^2+48\,a\,b^4\,c^2\,f-12\,a^3\,b^2\,e^2\,h-12\,a^2\,b^3\,c\,f^2-8\,a^2\,b^3\,e^3+27\,a\,b^4\,d^3-a^5\,h^3-64\,b^5\,c^3+a^3\,b^2\,f^3,z,k\right )}^2\,a^4\,b^3\,x\,324+6\,a\,b^2\,d\,f+4\,a^2\,b\,e\,h-80\,b^3\,c\,e\,x+\mathrm {root}\left (729\,a^7\,b^4\,z^3+729\,a^5\,b^4\,d\,z^2+27\,a^5\,b^2\,f\,h\,z-108\,a^4\,b^3\,c\,h\,z+54\,a^4\,b^3\,e\,f\,z-216\,a^3\,b^4\,c\,e\,z+243\,a^3\,b^4\,d^2\,z-72\,a\,b^4\,c\,d\,e+9\,a^3\,b^2\,d\,f\,h-36\,a^2\,b^3\,c\,d\,h+18\,a^2\,b^3\,d\,e\,f-6\,a^4\,b\,e\,h^2+48\,a\,b^4\,c^2\,f-12\,a^3\,b^2\,e^2\,h-12\,a^2\,b^3\,c\,f^2-8\,a^2\,b^3\,e^3+27\,a\,b^4\,d^3-a^5\,h^3-64\,b^5\,c^3+a^3\,b^2\,f^3,z,k\right )\,a^2\,b^3\,c\,36-\mathrm {root}\left (729\,a^7\,b^4\,z^3+729\,a^5\,b^4\,d\,z^2+27\,a^5\,b^2\,f\,h\,z-108\,a^4\,b^3\,c\,h\,z+54\,a^4\,b^3\,e\,f\,z-216\,a^3\,b^4\,c\,e\,z+243\,a^3\,b^4\,d^2\,z-72\,a\,b^4\,c\,d\,e+9\,a^3\,b^2\,d\,f\,h-36\,a^2\,b^3\,c\,d\,h+18\,a^2\,b^3\,d\,e\,f-6\,a^4\,b\,e\,h^2+48\,a\,b^4\,c^2\,f-12\,a^3\,b^2\,e^2\,h-12\,a^2\,b^3\,c\,f^2-8\,a^2\,b^3\,e^3+27\,a\,b^4\,d^3-a^5\,h^3-64\,b^5\,c^3+a^3\,b^2\,f^3,z,k\right )\,a^3\,b^2\,f\,9+\mathrm {root}\left (729\,a^7\,b^4\,z^3+729\,a^5\,b^4\,d\,z^2+27\,a^5\,b^2\,f\,h\,z-108\,a^4\,b^3\,c\,h\,z+54\,a^4\,b^3\,e\,f\,z-216\,a^3\,b^4\,c\,e\,z+243\,a^3\,b^4\,d^2\,z-72\,a\,b^4\,c\,d\,e+9\,a^3\,b^2\,d\,f\,h-36\,a^2\,b^3\,c\,d\,h+18\,a^2\,b^3\,d\,e\,f-6\,a^4\,b\,e\,h^2+48\,a\,b^4\,c^2\,f-12\,a^3\,b^2\,e^2\,h-12\,a^2\,b^3\,c\,f^2-8\,a^2\,b^3\,e^3+27\,a\,b^4\,d^3-a^5\,h^3-64\,b^5\,c^3+a^3\,b^2\,f^3,z,k\right )\,a^2\,b^3\,d\,x\,216-40\,a\,b^2\,c\,h\,x+20\,a\,b^2\,e\,f\,x+10\,a^2\,b\,f\,h\,x\right )}{a^2\,9}+\frac {x\,\left (a^5\,h^3+6\,a^4\,b\,e\,h^2+12\,a^3\,b^2\,e^2\,h-a^3\,b^2\,f^3-6\,d\,a^3\,b^2\,f\,h+12\,a^2\,b^3\,c\,f^2+24\,d\,a^2\,b^3\,c\,h+8\,a^2\,b^3\,e^3-12\,d\,a^2\,b^3\,e\,f-48\,a\,b^4\,c^2\,f+48\,d\,a\,b^4\,c\,e+64\,b^5\,c^3\right )}{27\,a^5\,b}\right )\,\mathrm {root}\left (729\,a^7\,b^4\,z^3+729\,a^5\,b^4\,d\,z^2+27\,a^5\,b^2\,f\,h\,z-108\,a^4\,b^3\,c\,h\,z+54\,a^4\,b^3\,e\,f\,z-216\,a^3\,b^4\,c\,e\,z+243\,a^3\,b^4\,d^2\,z-72\,a\,b^4\,c\,d\,e+9\,a^3\,b^2\,d\,f\,h-36\,a^2\,b^3\,c\,d\,h+18\,a^2\,b^3\,d\,e\,f-6\,a^4\,b\,e\,h^2+48\,a\,b^4\,c^2\,f-12\,a^3\,b^2\,e^2\,h-12\,a^2\,b^3\,c\,f^2-8\,a^2\,b^3\,e^3+27\,a\,b^4\,d^3-a^5\,h^3-64\,b^5\,c^3+a^3\,b^2\,f^3,z,k\right )\right )-\frac {\frac {c}{a}+\frac {x^3\,\left (4\,b\,c-a\,f\right )}{3\,a^2}-\frac {x\,\left (b\,d-a\,g\right )}{3\,a\,b}-\frac {x^2\,\left (b\,e-a\,h\right )}{3\,a\,b}}{b\,x^4+a\,x}+\frac {d\,\ln \left (x\right )}{a^2} \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^2*(a + b*x^3)^2),x)

[Out]

symsum(log((d*(a^3*h^2 + 4*a*b^2*e^2 + 12*b^3*c*d - 3*a*b^2*d*f + 4*a^2*b*e*h))/(9*a^4) - (root(729*a^7*b^4*z^
3 + 729*a^5*b^4*d*z^2 + 27*a^5*b^2*f*h*z - 108*a^4*b^3*c*h*z + 54*a^4*b^3*e*f*z - 216*a^3*b^4*c*e*z + 243*a^3*
b^4*d^2*z - 72*a*b^4*c*d*e + 9*a^3*b^2*d*f*h - 36*a^2*b^3*c*d*h + 18*a^2*b^3*d*e*f - 6*a^4*b*e*h^2 + 48*a*b^4*
c^2*f - 12*a^3*b^2*e^2*h - 12*a^2*b^3*c*f^2 - 8*a^2*b^3*e^3 + 27*a*b^4*d^3 - a^5*h^3 - 64*b^5*c^3 + a^3*b^2*f^
3, z, k)*(a^3*h^2 + 4*a*b^2*e^2 + 36*b^3*d^2*x - 24*b^3*c*d + 324*root(729*a^7*b^4*z^3 + 729*a^5*b^4*d*z^2 + 2
7*a^5*b^2*f*h*z - 108*a^4*b^3*c*h*z + 54*a^4*b^3*e*f*z - 216*a^3*b^4*c*e*z + 243*a^3*b^4*d^2*z - 72*a*b^4*c*d*
e + 9*a^3*b^2*d*f*h - 36*a^2*b^3*c*d*h + 18*a^2*b^3*d*e*f - 6*a^4*b*e*h^2 + 48*a*b^4*c^2*f - 12*a^3*b^2*e^2*h
- 12*a^2*b^3*c*f^2 - 8*a^2*b^3*e^3 + 27*a*b^4*d^3 - a^5*h^3 - 64*b^5*c^3 + a^3*b^2*f^3, z, k)^2*a^4*b^3*x + 6*
a*b^2*d*f + 4*a^2*b*e*h - 80*b^3*c*e*x + 36*root(729*a^7*b^4*z^3 + 729*a^5*b^4*d*z^2 + 27*a^5*b^2*f*h*z - 108*
a^4*b^3*c*h*z + 54*a^4*b^3*e*f*z - 216*a^3*b^4*c*e*z + 243*a^3*b^4*d^2*z - 72*a*b^4*c*d*e + 9*a^3*b^2*d*f*h -
36*a^2*b^3*c*d*h + 18*a^2*b^3*d*e*f - 6*a^4*b*e*h^2 + 48*a*b^4*c^2*f - 12*a^3*b^2*e^2*h - 12*a^2*b^3*c*f^2 - 8
*a^2*b^3*e^3 + 27*a*b^4*d^3 - a^5*h^3 - 64*b^5*c^3 + a^3*b^2*f^3, z, k)*a^2*b^3*c - 9*root(729*a^7*b^4*z^3 + 7
29*a^5*b^4*d*z^2 + 27*a^5*b^2*f*h*z - 108*a^4*b^3*c*h*z + 54*a^4*b^3*e*f*z - 216*a^3*b^4*c*e*z + 243*a^3*b^4*d
^2*z - 72*a*b^4*c*d*e + 9*a^3*b^2*d*f*h - 36*a^2*b^3*c*d*h + 18*a^2*b^3*d*e*f - 6*a^4*b*e*h^2 + 48*a*b^4*c^2*f
- 12*a^3*b^2*e^2*h - 12*a^2*b^3*c*f^2 - 8*a^2*b^3*e^3 + 27*a*b^4*d^3 - a^5*h^3 - 64*b^5*c^3 + a^3*b^2*f^3, z,
k)*a^3*b^2*f + 216*root(729*a^7*b^4*z^3 + 729*a^5*b^4*d*z^2 + 27*a^5*b^2*f*h*z - 108*a^4*b^3*c*h*z + 54*a^4*b
^3*e*f*z - 216*a^3*b^4*c*e*z + 243*a^3*b^4*d^2*z - 72*a*b^4*c*d*e + 9*a^3*b^2*d*f*h - 36*a^2*b^3*c*d*h + 18*a^
2*b^3*d*e*f - 6*a^4*b*e*h^2 + 48*a*b^4*c^2*f - 12*a^3*b^2*e^2*h - 12*a^2*b^3*c*f^2 - 8*a^2*b^3*e^3 + 27*a*b^4*
d^3 - a^5*h^3 - 64*b^5*c^3 + a^3*b^2*f^3, z, k)*a^2*b^3*d*x - 40*a*b^2*c*h*x + 20*a*b^2*e*f*x + 10*a^2*b*f*h*x
))/(9*a^2) + (x*(64*b^5*c^3 + a^5*h^3 + 8*a^2*b^3*e^3 - a^3*b^2*f^3 + 12*a^2*b^3*c*f^2 + 12*a^3*b^2*e^2*h - 48
*a*b^4*c^2*f + 6*a^4*b*e*h^2 + 24*a^2*b^3*c*d*h - 12*a^2*b^3*d*e*f - 6*a^3*b^2*d*f*h + 48*a*b^4*c*d*e))/(27*a^
5*b))*root(729*a^7*b^4*z^3 + 729*a^5*b^4*d*z^2 + 27*a^5*b^2*f*h*z - 108*a^4*b^3*c*h*z + 54*a^4*b^3*e*f*z - 216
*a^3*b^4*c*e*z + 243*a^3*b^4*d^2*z - 72*a*b^4*c*d*e + 9*a^3*b^2*d*f*h - 36*a^2*b^3*c*d*h + 18*a^2*b^3*d*e*f -
6*a^4*b*e*h^2 + 48*a*b^4*c^2*f - 12*a^3*b^2*e^2*h - 12*a^2*b^3*c*f^2 - 8*a^2*b^3*e^3 + 27*a*b^4*d^3 - a^5*h^3
- 64*b^5*c^3 + a^3*b^2*f^3, z, k), k, 1, 3) - (c/a + (x^3*(4*b*c - a*f))/(3*a^2) - (x*(b*d - a*g))/(3*a*b) - (
x^2*(b*e - a*h))/(3*a*b))/(a*x + b*x^4) + (d*log(x))/a^2

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