3.269 \(\int \frac{c+d x^3+e x^6+f x^9}{x^5 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=269 \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}}+\frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4} \]

[Out]

-c/(4*a^2*x^4) + (2*b*c - a*d)/(a^3*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^3*b*(a + b*x^3)) - ((7
*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(10/3)
*b^(5/3)) - ((7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(10/3)*b^(5/3)) + ((7*b^
3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(10/3)*b^(5/3))

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Rubi [A]  time = 0.288055, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}}+\frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]

[Out]

-c/(4*a^2*x^4) + (2*b*c - a*d)/(a^3*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^3*b*(a + b*x^3)) - ((7
*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(10/3)
*b^(5/3)) - ((7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(10/3)*b^(5/3)) + ((7*b^
3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(10/3)*b^(5/3))

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 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

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]

Rubi steps

\begin{align*} \int \frac{c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^2} \, dx &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+3 b^3 \left (\frac{b c}{a}-d\right ) x^3-b^2 \left (\frac{b^3 c}{a^2}-\frac{b^2 d}{a}+b e+2 a f\right ) x^6}{x^5 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^5}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^2}-\frac{b^2 \left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) x}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^3 b}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{10/3} b^{4/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{10/3} b^{4/3}}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\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^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^3 b^{4/3}}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\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^{10/3} b^{5/3}}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3} b^{5/3}}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3} b^{5/3}}\\ \end{align*}

Mathematica [A]  time = 0.168302, size = 255, normalized size = 0.95 \[ \frac{-\frac{12 \sqrt [3]{a} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{9 a^{4/3} c}{x^4}-\frac{36 \sqrt [3]{a} (a d-2 b c)}{x}}{36 a^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]

[Out]

((-9*a^(4/3)*c)/x^4 - (36*a^(1/3)*(-2*b*c + a*d))/x - (12*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/
(b*(a + b*x^3)) - (4*Sqrt[3]*(7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt
[3]])/b^(5/3) - (4*(7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*(7*b^3*c -
4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3))/(36*a^(10/3))

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Maple [B]  time = 0.013, size = 486, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^5/(b*x^3+a)^2,x)

[Out]

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

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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((f*x^9+e*x^6+d*x^3+c)/x^5/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52471, size = 1968, normalized size = 7.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^5/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[-1/36*(9*a^3*b^3*c - 12*(7*a*b^5*c - 4*a^2*b^4*d + a^3*b^3*e - a^4*b^2*f)*x^6 - 9*(7*a^2*b^4*c - 4*a^3*b^3*d)
*x^3 - 6*sqrt(1/3)*((7*a*b^5*c - 4*a^2*b^4*d + a^3*b^3*e + 2*a^4*b^2*f)*x^7 + (7*a^2*b^4*c - 4*a^3*b^3*d + a^4
*b^2*e + 2*a^5*b*f)*x^4)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x
^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) - 2*((7*b^4*c - 4*a*b^3*d + a
^2*b^2*e + 2*a^3*b*f)*x^7 + (7*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*f)*x^4)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-
a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 4*((7*b^4*c - 4*a*b^3*d + a^2*b^2*e + 2*a^3*b*f)*x^7 + (7*a*b^3*c - 4*a^2
*b^2*d + a^3*b*e + 2*a^4*f)*x^4)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^4*b^4*x^7 + a^5*b^3*x^4), -1/36*
(9*a^3*b^3*c - 12*(7*a*b^5*c - 4*a^2*b^4*d + a^3*b^3*e - a^4*b^2*f)*x^6 - 9*(7*a^2*b^4*c - 4*a^3*b^3*d)*x^3 -
12*sqrt(1/3)*((7*a*b^5*c - 4*a^2*b^4*d + a^3*b^3*e + 2*a^4*b^2*f)*x^7 + (7*a^2*b^4*c - 4*a^3*b^3*d + a^4*b^2*e
+ 2*a^5*b*f)*x^4)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b
) - 2*((7*b^4*c - 4*a*b^3*d + a^2*b^2*e + 2*a^3*b*f)*x^7 + (7*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*f)*x^4)*
(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 4*((7*b^4*c - 4*a*b^3*d + a^2*b^2*e + 2*a^
3*b*f)*x^7 + (7*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*f)*x^4)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^4
*b^4*x^7 + a^5*b^3*x^4)]

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Sympy [A]  time = 138.391, size = 473, normalized size = 1.76 \begin{align*} \operatorname{RootSum}{\left (729 t^{3} a^{10} b^{5} + 8 a^{9} f^{3} + 12 a^{8} b e f^{2} - 48 a^{7} b^{2} d f^{2} + 6 a^{7} b^{2} e^{2} f + 84 a^{6} b^{3} c f^{2} - 48 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} + 84 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f - 12 a^{5} b^{4} d e^{2} - 336 a^{4} b^{5} c d f + 21 a^{4} b^{5} c e^{2} + 48 a^{4} b^{5} d^{2} e + 294 a^{3} b^{6} c^{2} f - 168 a^{3} b^{6} c d e - 64 a^{3} b^{6} d^{3} + 147 a^{2} b^{7} c^{2} e + 336 a^{2} b^{7} c d^{2} - 588 a b^{8} c^{2} d + 343 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{7} b^{3}}{4 a^{6} f^{2} + 4 a^{5} b e f - 16 a^{4} b^{2} d f + a^{4} b^{2} e^{2} + 28 a^{3} b^{3} c f - 8 a^{3} b^{3} d e + 14 a^{2} b^{4} c e + 16 a^{2} b^{4} d^{2} - 56 a b^{5} c d + 49 b^{6} c^{2}} + x \right )} \right )\right )} - \frac{3 a^{2} b c + x^{6} \left (4 a^{3} f - 4 a^{2} b e + 16 a b^{2} d - 28 b^{3} c\right ) + x^{3} \left (12 a^{2} b d - 21 a b^{2} c\right )}{12 a^{4} b x^{4} + 12 a^{3} b^{2} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**5/(b*x**3+a)**2,x)

[Out]

RootSum(729*_t**3*a**10*b**5 + 8*a**9*f**3 + 12*a**8*b*e*f**2 - 48*a**7*b**2*d*f**2 + 6*a**7*b**2*e**2*f + 84*
a**6*b**3*c*f**2 - 48*a**6*b**3*d*e*f + a**6*b**3*e**3 + 84*a**5*b**4*c*e*f + 96*a**5*b**4*d**2*f - 12*a**5*b*
*4*d*e**2 - 336*a**4*b**5*c*d*f + 21*a**4*b**5*c*e**2 + 48*a**4*b**5*d**2*e + 294*a**3*b**6*c**2*f - 168*a**3*
b**6*c*d*e - 64*a**3*b**6*d**3 + 147*a**2*b**7*c**2*e + 336*a**2*b**7*c*d**2 - 588*a*b**8*c**2*d + 343*b**9*c*
*3, Lambda(_t, _t*log(81*_t**2*a**7*b**3/(4*a**6*f**2 + 4*a**5*b*e*f - 16*a**4*b**2*d*f + a**4*b**2*e**2 + 28*
a**3*b**3*c*f - 8*a**3*b**3*d*e + 14*a**2*b**4*c*e + 16*a**2*b**4*d**2 - 56*a*b**5*c*d + 49*b**6*c**2) + x)))
- (3*a**2*b*c + x**6*(4*a**3*f - 4*a**2*b*e + 16*a*b**2*d - 28*b**3*c) + x**3*(12*a**2*b*d - 21*a*b**2*c))/(12
*a**4*b*x**4 + 12*a**3*b**2*x**7)

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Giac [A]  time = 1.08189, size = 483, normalized size = 1.8 \begin{align*} -\frac{{\left (7 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{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^{4} b} + \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{3} b} - \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} 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^{4} b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} 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^{4} b^{3}} + \frac{8 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^5/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/9*(7*b^3*c*(-a/b)^(1/3) - 4*a*b^2*d*(-a/b)^(1/3) + 2*a^3*f*(-a/b)^(1/3) + a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3
)*log(abs(x - (-a/b)^(1/3)))/(a^4*b) + 1/3*(b^3*c*x^2 - a*b^2*d*x^2 - a^3*f*x^2 + a^2*b*x^2*e)/((b*x^3 + a)*a^
3*b) - 1/9*sqrt(3)*(7*(-a*b^2)^(2/3)*b^3*c - 4*(-a*b^2)^(2/3)*a*b^2*d + 2*(-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3
)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^3) + 1/18*(7*(-a*b^2)^(2/3)*b^3*c - 4*
(-a*b^2)^(2/3)*a*b^2*d + 2*(-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/(a^4*b^3) + 1/4*(8*b*c*x^3 - 4*a*d*x^3 - a*c)/(a^3*x^4)