3.22 \(\int \frac{(c+d x^3)^3}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=234 \[ -\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{3 a b^3 \left (a+b x^3\right )}+\frac{d^3 x^4}{4 b^2} \]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^4)/(4*b^2) + ((b*c - a*d)^3*x)/(3*a*b^3*(a + b*x^3)) - ((b*c - a*d)^2*(2*
b*c + 7*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(10/3)) + ((b*c - a*d)^2*
(2*b*c + 7*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(10/3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*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 = 0.219666, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {390, 385, 200, 31, 634, 617, 204, 628} \[ -\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{3 a b^3 \left (a+b x^3\right )}+\frac{d^3 x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^3/(a + b*x^3)^2,x]

[Out]

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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{\left (c+d x^3\right )^3}{\left (a+b x^3\right )^2} \, dx &=\int \left (\frac{d^2 (3 b c-2 a d)}{b^3}+\frac{d^3 x^3}{b^2}+\frac{(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^3}{b^3 \left (a+b x^3\right )^2}\right ) \, dx\\ &=\frac{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^4}{4 b^2}+\frac{\int \frac{(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^3}{\left (a+b x^3\right )^2} \, dx}{b^3}\\ &=\frac{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^4}{4 b^2}+\frac{(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac{\left ((b c-a d)^2 (2 b c+7 a d)\right ) \int \frac{1}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^4}{4 b^2}+\frac{(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac{\left ((b c-a d)^2 (2 b c+7 a d)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^3}+\frac{\left ((b c-a d)^2 (2 b c+7 a d)\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{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^4}{4 b^2}+\frac{(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac{(b c-a d)^2 (2 b c+7 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{\left ((b c-a d)^2 (2 b c+7 a d)\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 ((b c-a d)^2 (2 b c+7 a d)\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{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^4}{4 b^2}+\frac{(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac{(b c-a d)^2 (2 b c+7 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (2 b c+7 a d) \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 ((b c-a d)^2 (2 b c+7 a d)\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{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^4}{4 b^2}+\frac{(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}-\frac{(b c-a d)^2 (2 b c+7 a d) \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{(b c-a d)^2 (2 b c+7 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (2 b c+7 a d) \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 = 0.15917, size = 227, normalized size = 0.97 \[ \frac{-\frac{2 (b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{4 (b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac{4 \sqrt{3} (b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+36 \sqrt [3]{b} d^2 x (3 b c-2 a d)+\frac{12 \sqrt [3]{b} x (b c-a d)^3}{a \left (a+b x^3\right )}+9 b^{4/3} d^3 x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*b^(1/3)*d^2*(3*b*c - 2*a*d)*x + 9*b^(4/3)*d^3*x^4 + (12*b^(1/3)*(b*c - a*d)^3*x)/(a*(a + b*x^3)) + (4*Sqrt
[3]*(b*c - a*d)^2*(2*b*c + 7*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (4*(b*c - a*d)
^2*(2*b*c + 7*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b*c - a*d)^2*(2*b*c + 7*a*d)*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 = 529, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^3/(b*x^3+a)^2,x)

[Out]

1/4*d^3*x^4/b^2-2*d^3/b^3*a*x+3*d^2/b^2*x*c-1/3/b^3*a^2*x/(b*x^3+a)*d^3+1/b^2*a*x/(b*x^3+a)*c*d^2-1/b*x/(b*x^3
+a)*c^2*d+1/3/a*x/(b*x^3+a)*c^3+7/9/b^4*a^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d^3-4/3/b^3*a/(1/b*a)^(2/3)*ln(x
+(1/b*a)^(1/3))*c*d^2+1/3/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c^2*d+2/9/b/a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3)
)*c^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))*d^3+2/3/b^3*a/(1/b*a)^(2/3)*ln(x^2-(1/b
*a)^(1/3)*x+(1/b*a)^(2/3))*c*d^2-1/6/b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^2*d-1/9/b/a/(1/
b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^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))*d^3-4/3/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))*c*d^2+1/3/b^2
/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c^2*d+2/9/b/a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/
3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c^3

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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((d*x^3+c)^3/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.71334, size = 2225, normalized size = 9.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^3/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.0055, size = 289, normalized size = 1.24 \begin{align*} - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\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} d^{9} + 1764 a^{8} b c d^{8} - 3465 a^{7} b^{2} c^{2} d^{7} + 2946 a^{6} b^{3} c^{3} d^{6} - 477 a^{5} b^{4} c^{4} d^{5} - 792 a^{4} b^{5} c^{5} d^{4} + 321 a^{3} b^{6} c^{6} d^{3} + 90 a^{2} b^{7} c^{7} d^{2} - 36 a b^{8} c^{8} d - 8 b^{9} c^{9}, \left ( t \mapsto t \log{\left (\frac{9 t a^{2} b^{3}}{7 a^{3} d^{3} - 12 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 2 b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x \left (2 a d^{3} - 3 b c d^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**3/(b*x**3+a)**2,x)

[Out]

-x*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/(3*a**2*b**3 + 3*a*b**4*x**3) + RootSum(729*_t*
*3*a**5*b**10 - 343*a**9*d**9 + 1764*a**8*b*c*d**8 - 3465*a**7*b**2*c**2*d**7 + 2946*a**6*b**3*c**3*d**6 - 477
*a**5*b**4*c**4*d**5 - 792*a**4*b**5*c**5*d**4 + 321*a**3*b**6*c**6*d**3 + 90*a**2*b**7*c**7*d**2 - 36*a*b**8*
c**8*d - 8*b**9*c**9, Lambda(_t, _t*log(9*_t*a**2*b**3/(7*a**3*d**3 - 12*a**2*b*c*d**2 + 3*a*b**2*c**2*d + 2*b
**3*c**3) + x))) + d**3*x**4/(4*b**2) - x*(2*a*d**3 - 3*b*c*d**2)/b**3

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Giac [A]  time = 1.11544, size = 495, normalized size = 2.12 \begin{align*} -\frac{{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\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^{3} + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{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^{2} b^{4}} + \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{3 \,{\left (b x^{3} + a\right )} a b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\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} d^{3} x^{4} + 12 \, b^{6} c d^{2} x - 8 \, a b^{5} d^{3} x}{4 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a^3*d^3)*(-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^3 + 3*(-a*b^2)^(1/3)*a*b^2*c^2*d - 12*(-a*b^2)^(1/3)*a^2*b*c*d^2 + 7*(
-a*b^2)^(1/3)*a^3*d^3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^4) + 1/3*(b^3*c^3*x - 3*a*
b^2*c^2*d*x + 3*a^2*b*c*d^2*x - a^3*d^3*x)/((b*x^3 + a)*a*b^3) + 1/18*(2*(-a*b^2)^(1/3)*b^3*c^3 + 3*(-a*b^2)^(
1/3)*a*b^2*c^2*d - 12*(-a*b^2)^(1/3)*a^2*b*c*d^2 + 7*(-a*b^2)^(1/3)*a^3*d^3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)
^(2/3))/(a^2*b^4) + 1/4*(b^6*d^3*x^4 + 12*b^6*c*d^2*x - 8*a*b^5*d^3*x)/b^8