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

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

[Out]

-c/(3*a^2*x^3) - d/(2*a^2*x^2) - e/(a^2*x) - (x*(b*d - a*g + (b*e - a*h)*x - b*((b*c)/a - f)*x^2))/(3*a^2*(a +
b*x^3)) + ((5*b^(4/3)*d + 4*a^(1/3)*b*e - 2*a*b^(1/3)*g - a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*
a^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^(2/3)) - ((2*b*c - a*f)*Log[x])/a^3 - ((b^(1/3)*(5*b*d - 2*a*g) - a^(1/3)*(4*b
*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)*b^(2/3)) + ((b^(1/3)*(5*b*d - 2*a*g) - a^(1/3)*(4*b*e - a*h))*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)*b^(2/3)) + ((2*b*c - a*f)*Log[a + b*x^3])/(3*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.727317, antiderivative size = 336, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}-2 a g+5 b d\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(3*a^2*x^3) - d/(2*a^2*x^2) - e/(a^2*x) - (x*(b*d - a*g + (b*e - a*h)*x - b*((b*c)/a - f)*x^2))/(3*a^2*(a +
b*x^3)) + ((5*b^(4/3)*d + 4*a^(1/3)*b*e - 2*a*b^(1/3)*g - a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*
a^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^(2/3)) - ((2*b*c - a*f)*Log[x])/a^3 - ((b^(1/3)*(5*b*d - 2*a*g) - a^(1/3)*(4*b
*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)*b^(2/3)) + ((5*b*d - 2*a*g - (a^(1/3)*(4*b*e - a*h))/b^(1/3))*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)*b^(1/3)) + ((2*b*c - a*f)*Log[a + b*x^3])/(3*a^3)

Rule 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.454282, size = 303, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{2/3}}+\frac{a (6 a (f+x (g+h x))-6 b (c+x (d+e x)))}{a+b x^3}+6 (2 b c-a f) \log \left (a+b x^3\right )+18 \log (x) (a f-2 b c)-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}}{18 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 561, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^4/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.09594, size = 520, normalized size = 1.54 \begin{align*} \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{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - 4 \, \left (-a b^{2}\right )^{\frac{2}{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 \, a^{3} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + 4 \, \left (-a b^{2}\right )^{\frac{2}{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 \, a^{3} b^{2}} - \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{align*}

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]

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