3.26 \(\int \frac{1}{(a+b x^3)^2 (c+d x^3)^2} \, dx\)

Optimal. Leaf size=419 \[ -\frac{b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}+\frac{2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}-\frac{2 b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} (b c-a d)^3}-\frac{d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac{2 d^{5/3} (4 b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^3}+\frac{b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}+\frac{d x (a d+b c)}{3 a c \left (c+d x^3\right ) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.493458, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {414, 527, 522, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}+\frac{2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}-\frac{2 b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} (b c-a d)^3}-\frac{d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac{2 d^{5/3} (4 b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^3}+\frac{b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}+\frac{d x (a d+b c)}{3 a c \left (c+d x^3\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^3)^2*(c + d*x^3)^2),x]

[Out]

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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

Mathematica [A]  time = 0.639885, size = 381, normalized size = 0.91 \[ \frac{1}{9} \left (\frac{b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3} (a d-b c)^3}+\frac{2 b^{5/3} (4 a d-b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3} (a d-b c)^3}+\frac{2 \sqrt{3} b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3} (a d-b c)^3}+\frac{3 b^2 x}{a \left (a+b x^3\right ) (b c-a d)^2}+\frac{d^{5/3} (a d-4 b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3} (b c-a d)^3}+\frac{2 \sqrt{3} d^{5/3} (a d-4 b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{5/3} (b c-a d)^3}+\frac{3 d^2 x}{c \left (c+d x^3\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^3)^2*(c + d*x^3)^2),x]

[Out]

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

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Maple [A]  time = 0.014, size = 606, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^2/(d*x^3+c)^2,x)

[Out]

1/3*b^2/(a*d-b*c)^3*x/(b*x^3+a)*d-1/3*b^3/(a*d-b*c)^3/a*x/(b*x^3+a)*c+8/9*b/(a*d-b*c)^3/(1/b*a)^(2/3)*ln(x+(1/
b*a)^(1/3))*d-2/9*b^2/(a*d-b*c)^3/a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c-4/9*b/(a*d-b*c)^3/(1/b*a)^(2/3)*ln(x^2
-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+1/9*b^2/(a*d-b*c)^3/a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+
8/9*b/(a*d-b*c)^3/(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*b^2/(a*d-b*c)^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+1/3*d^3/(a*d-b*c)^3/c*x/(d*x^3+c)*a-1/3*d^2/(a*d
-b*c)^3*x/(d*x^3+c)*b+2/9*d^2/(a*d-b*c)^3/c/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a-8/9*d/(a*d-b*c)^3/(c/d)^(2/3)*ln(x
+(c/d)^(1/3))*b-1/9*d^2/(a*d-b*c)^3/c/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*a+4/9*d/(a*d-b*c)^3/(c/d)^
(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*b+2/9*d^2/(a*d-b*c)^3/c/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d
)^(1/3)*x-1))*a-8/9*d/(a*d-b*c)^3/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*b

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [A]  time = 1.65298, size = 896, normalized size = 2.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^2/(d*x^3+c)^2,x, algorithm="giac")

[Out]

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