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

Optimal. Leaf size=264 \[ \frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{18 a^{5/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{9 a^{5/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.264159, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1858, 1411, 388, 200, 31, 634, 617, 204, 628} \[ \frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{18 a^{5/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{9 a^{5/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1411

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Simp[(c*x^(n + 1)*
(d + e*x^n)^(q + 1))/(e*(n*(q + 2) + 1)), x] + Dist[1/(e*(n*(q + 2) + 1)), Int[(d + e*x^n)^q*(a*e*(n*(q + 2) +
1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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}{\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}{3 a b^3 \left (a+b x^3\right )}-\frac{\int \frac{-2 b^3 c-a b^2 d+a^2 b e-a^3 f-3 a b (b e-a f) x^3-3 a b^2 f x^6}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac{f x^4}{4 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac{\int \frac{4 b \left (-2 b^3 c-a b^2 d+a^2 b e-a^3 f\right )-\left (-12 a^2 b^2 f+12 a b^2 (b e-a f)\right ) x^3}{a+b x^3} \, dx}{12 a b^4}\\ &=\frac{(b e-2 a f) x}{b^3}+\frac{f x^4}{4 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac{(b e-2 a f) x}{b^3}+\frac{f x^4}{4 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^3}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac{2 \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^{5/3} b^3}\\ &=\frac{(b e-2 a f) x}{b^3}+\frac{f x^4}{4 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 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^{5/3} b^{10/3}}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 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^{4/3} b^3}\\ &=\frac{(b e-2 a f) x}{b^3}+\frac{f x^4}{4 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 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^{5/3} b^{10/3}}\\ &=\frac{(b e-2 a f) x}{b^3}+\frac{f x^4}{4 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 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^{5/3} b^{10/3}}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}\\ \end{align*}

Mathematica [A]  time = 0.144892, size = 251, normalized size = 0.95 \[ \frac{\frac{12 \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{a^{5/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{a^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{a^{5/3}}+36 \sqrt [3]{b} x (b e-2 a f)+9 b^{4/3} f x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.009, size = 482, 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)/(b*x^3+a)^2,x)

[Out]

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

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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)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52419, size = 1917, normalized size = 7.26 \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)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.35866, size = 376, normalized size = 1.42 \begin{align*} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{10} - 343 a^{9} f^{3} + 588 a^{8} b e f^{2} - 147 a^{7} b^{2} d f^{2} - 336 a^{7} b^{2} e^{2} f - 294 a^{6} b^{3} c f^{2} + 168 a^{6} b^{3} d e f + 64 a^{6} b^{3} e^{3} + 336 a^{5} b^{4} c e f - 21 a^{5} b^{4} d^{2} f - 48 a^{5} b^{4} d e^{2} - 84 a^{4} b^{5} c d f - 96 a^{4} b^{5} c e^{2} + 12 a^{4} b^{5} d^{2} e - 84 a^{3} b^{6} c^{2} f + 48 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 48 a^{2} b^{7} c^{2} e - 6 a^{2} b^{7} c d^{2} - 12 a b^{8} c^{2} d - 8 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t a^{2} b^{3}}{7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c} + x \right )} \right )\right )} + \frac{f x^{4}}{4 b^{2}} - \frac{x \left (2 a f - b e\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07211, size = 433, normalized size = 1.64 \begin{align*} -\frac{{\left (2 \, b^{3} c + a b^{2} d + 7 \, a^{3} f - 4 \, a^{2} b 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^{2} b^{3}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 4 \, \left (-a b^{2}\right )^{\frac{1}{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^{2} b^{4}} + \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 4 \, \left (-a b^{2}\right )^{\frac{1}{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^{2} b^{4}} + \frac{b^{6} f x^{4} - 8 \, a b^{5} f x + 4 \, b^{6} x e}{4 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(2*b^3*c + a*b^2*d + 7*a^3*f - 4*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/9*sqrt(3)
*(2*(-a*b^2)^(1/3)*b^3*c + (-a*b^2)^(1/3)*a*b^2*d + 7*(-a*b^2)^(1/3)*a^3*f - 4*(-a*b^2)^(1/3)*a^2*b*e)*arctan(
1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^4) + 1/3*(b^3*c*x - a*b^2*d*x - a^3*f*x + a^2*b*x*e)/((b
*x^3 + a)*a*b^3) + 1/18*(2*(-a*b^2)^(1/3)*b^3*c + (-a*b^2)^(1/3)*a*b^2*d + 7*(-a*b^2)^(1/3)*a^3*f - 4*(-a*b^2)
^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^4) + 1/4*(b^6*f*x^4 - 8*a*b^5*f*x + 4*b^6*x*e)
/b^8