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]

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

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Rubi [A]  time = 0.49, antiderivative size = 419, normalized size of antiderivative = 1.00, 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 31

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

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*}

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Mathematica [A]  time = 0.63, 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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fricas [B]  time = 85.30, size = 897, normalized size = 2.14 \[ \frac {3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{4} + 2 \, \sqrt {3} {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, \sqrt {3} {\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - {\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, {\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{9 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{6} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{3}\right )}} \]

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]

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

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giac [A]  time = 0.22, size = 664, normalized size = 1.58 \[ -\frac {2 \, {\left (b^{3} c - 4 \, a b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, {\left (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}\right )}} - \frac {2 \, {\left (4 \, b c d^{2} - a d^{3}\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{9 \, {\left (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}\right )}} + \frac {2 \, {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b d\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 )}{3 \, {\left (\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}\right )}} + \frac {2 \, {\left (4 \, \left (-c d^{2}\right )^{\frac {1}{3}} b c d - \left (-c d^{2}\right )^{\frac {1}{3}} a d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (\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}\right )}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, {\left (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}\right )}} + \frac {{\left (4 \, \left (-c d^{2}\right )^{\frac {1}{3}} b c d - \left (-c d^{2}\right )^{\frac {1}{3}} a d^{2}\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \, {\left (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}\right )}} + \frac {b^{2} c d x^{4} + a b d^{2} x^{4} + b^{2} c^{2} x + a^{2} d^{2} x}{3 \, {\left (b d x^{6} + b c x^{3} + a d x^{3} + a c\right )} {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )}} \]

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))

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maple [A]  time = 0.06, size = 606, normalized size = 1.45 \[ \frac {a \,d^{3} x}{3 \left (a d -b c \right )^{3} \left (d \,x^{3}+c \right ) c}-\frac {b^{3} c x}{3 \left (a d -b c \right )^{3} \left (b \,x^{3}+a \right ) a}+\frac {b^{2} d x}{3 \left (a d -b c \right )^{3} \left (b \,x^{3}+a \right )}-\frac {b \,d^{2} x}{3 \left (a d -b c \right )^{3} \left (d \,x^{3}+c \right )}+\frac {2 \sqrt {3}\, a \,d^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}} c}+\frac {2 a \,d^{2} \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}} c}-\frac {a \,d^{2} \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}} c}-\frac {2 \sqrt {3}\, b^{2} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {2 b^{2} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {b^{2} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {8 \sqrt {3}\, b d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {8 \sqrt {3}\, b d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {8 b d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {8 b d \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {4 b d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {4 b d \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \]

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/(a/b)^(2/3)*ln(x+(a/b)
^(1/3))*d-2/9*b^2/(a*d-b*c)^3/a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-4/9*b/(a*d-b*c)^3/(a/b)^(2/3)*ln(x^2-(a/b)^(1/
3)*x+(a/b)^(2/3))*d+1/9*b^2/(a*d-b*c)^3/a/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*c+8/9*b/(a*d-b*c)^3/(a
/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-2/9*b^2/(a*d-b*c)^3/a/(a/b)^(2/3)*3^(1/2)*arctan(1
/3*3^(1/2)*(2/(a/b)^(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 [B]  time = 1.26, size = 784, normalized size = 1.87 \[ \frac {2 \, \sqrt {3} {\left (b^{2} c - 4 \, a b d\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 \, {\left (a b^{3} c^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a^{2} b^{2} c^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a^{3} b c d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} {\left (4 \, b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{9 \, {\left (b^{3} c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - 3 \, a b^{2} c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}} + 3 \, a^{2} b c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a^{3} c d^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {{\left (b^{2} c - 4 \, a b d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a b^{3} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{2} b^{2} c^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b c d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{4} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {{\left (4 \, b c d - a d^{2}\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \, {\left (b^{3} c^{4} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 3 \, a b^{2} c^{3} d \left (\frac {c}{d}\right )^{\frac {2}{3}} + 3 \, a^{2} b c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a^{3} c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {2 \, {\left (b^{2} c - 4 \, a b d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{3} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{2} b^{2} c^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b c d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{4} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {2 \, {\left (4 \, b c d - a d^{2}\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \, {\left (b^{3} c^{4} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 3 \, a b^{2} c^{3} d \left (\frac {c}{d}\right )^{\frac {2}{3}} + 3 \, a^{2} b c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a^{3} c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{4} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{3 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{6} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{3}\right )}} \]

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]

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

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mupad [B]  time = 24.31, size = 3637, normalized size = 8.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a^2*d^2 + b^2*c^2))/(3*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^4*(a*d + b*c))/(3*a*c*(a^2*d^2 + b^2
*c^2 - 2*a*b*c*d)))/(a*c + x^3*(a*d + b*c) + b*d*x^6) + log((2*((4*((54*b^3*d^3*x*(a*d - b*c)^2*(a^3*d^3 + b^3
*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) + 54*a*b^3*c*d^3*(a*d + b*c)*(a*d - b*c)^4*((b^5*(4*a*d - b*c)^3)
/(a^5*(a*d - b*c)^9))^(1/3))*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(2/3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6
*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*c^2*d^4 - 11*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(3*a^3
*c^3*(a*d - b*c)^4))*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(1/3))/9 - (16*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c
^6 + 268*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2*c^2*d^4 - 49*a*b^5*c^5*d - 49*a^5*b*c*d^5))/(27*a
^3*c^3*(a*d - b*c)^8))*(-(8*b^8*c^3 - 512*a^3*b^5*d^3 + 384*a^2*b^6*c*d^2 - 96*a*b^7*c^2*d)/(729*a^14*d^9 - 72
9*a^5*b^9*c^9 + 6561*a^6*b^8*c^8*d - 26244*a^7*b^7*c^7*d^2 + 61236*a^8*b^6*c^6*d^3 - 91854*a^9*b^5*c^5*d^4 + 9
1854*a^10*b^4*c^4*d^5 - 61236*a^11*b^3*c^3*d^6 + 26244*a^12*b^2*c^2*d^7 - 6561*a^13*b*c*d^8))^(1/3) + log((2*(
(4*((54*b^3*d^3*x*(a*d - b*c)^2*(a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) + 54*a*b^3*c*d^3*(a
*d + b*c)*(a*d - b*c)^4*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)^9))^(1/3))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d -
b*c)^9))^(2/3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*c^2*
d^4 - 11*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*c^3*(a*d - b*c)^4))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)^9))
^(1/3))/9 - (16*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c^6 + 268*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2*c^2
*d^4 - 49*a*b^5*c^5*d - 49*a^5*b*c*d^5))/(27*a^3*c^3*(a*d - b*c)^8))*(-(8*a^3*d^8 - 512*b^3*c^3*d^5 + 384*a*b^
2*c^2*d^6 - 96*a^2*b*c*d^7)/(729*b^9*c^14 - 729*a^9*c^5*d^9 + 6561*a^8*b*c^6*d^8 + 26244*a^2*b^7*c^12*d^2 - 61
236*a^3*b^6*c^11*d^3 + 91854*a^4*b^5*c^10*d^4 - 91854*a^5*b^4*c^9*d^5 + 61236*a^6*b^3*c^8*d^6 - 26244*a^7*b^2*
c^7*d^7 - 6561*a*b^8*c^13*d))^(1/3) + (log(((3^(1/2)*1i - 1)*(((3^(1/2)*1i - 1)^2*((54*b^3*d^3*x*(a*d - b*c)^2
*(a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) + 27*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a*d + b*c)*(a*d
- b*c)^4*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(1/3))*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(2/3)
)/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*c^2*d^4 - 11*a*b^5
*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*c^3*(a*d - b*c)^4))*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(1/3))/9 - (1
6*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c^6 + 268*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2*c^2*d^4 - 49*a*b^
5*c^5*d - 49*a^5*b*c*d^5))/(27*a^3*c^3*(a*d - b*c)^8))*(3^(1/2)*1i - 1)*(-(8*b^8*c^3 - 512*a^3*b^5*d^3 + 384*a
^2*b^6*c*d^2 - 96*a*b^7*c^2*d)/(729*a^14*d^9 - 729*a^5*b^9*c^9 + 6561*a^6*b^8*c^8*d - 26244*a^7*b^7*c^7*d^2 +
61236*a^8*b^6*c^6*d^3 - 91854*a^9*b^5*c^5*d^4 + 91854*a^10*b^4*c^4*d^5 - 61236*a^11*b^3*c^3*d^6 + 26244*a^12*b
^2*c^2*d^7 - 6561*a^13*b*c*d^8))^(1/3))/2 - (log(((3^(1/2)*1i + 1)*(((3^(1/2)*1i + 1)^2*((54*b^3*d^3*x*(a*d -
b*c)^2*(a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) - 27*a*b^3*c*d^3*(3^(1/2)*1i + 1)*(a*d + b*c
)*(a*d - b*c)^4*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(1/3))*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))
^(2/3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*c^2*d^4 - 11
*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*c^3*(a*d - b*c)^4))*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(1/3))/
9 + (16*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c^6 + 268*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2*c^2*d^4 - 4
9*a*b^5*c^5*d - 49*a^5*b*c*d^5))/(27*a^3*c^3*(a*d - b*c)^8))*(3^(1/2)*1i + 1)*(-(8*b^8*c^3 - 512*a^3*b^5*d^3 +
384*a^2*b^6*c*d^2 - 96*a*b^7*c^2*d)/(729*a^14*d^9 - 729*a^5*b^9*c^9 + 6561*a^6*b^8*c^8*d - 26244*a^7*b^7*c^7*
d^2 + 61236*a^8*b^6*c^6*d^3 - 91854*a^9*b^5*c^5*d^4 + 91854*a^10*b^4*c^4*d^5 - 61236*a^11*b^3*c^3*d^6 + 26244*
a^12*b^2*c^2*d^7 - 6561*a^13*b*c*d^8))^(1/3))/2 + (log(((3^(1/2)*1i - 1)*(((3^(1/2)*1i - 1)^2*((54*b^3*d^3*x*(
a*d - b*c)^2*(a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) + 27*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a*d
+ b*c)*(a*d - b*c)^4*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)^9))^(1/3))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*
c)^9))^(2/3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*c^2*d^
4 - 11*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*c^3*(a*d - b*c)^4))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)^9))^(
1/3))/9 - (16*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c^6 + 268*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2*c^2*d
^4 - 49*a*b^5*c^5*d - 49*a^5*b*c*d^5))/(27*a^3*c^3*(a*d - b*c)^8))*(3^(1/2)*1i - 1)*(-(8*a^3*d^8 - 512*b^3*c^3
*d^5 + 384*a*b^2*c^2*d^6 - 96*a^2*b*c*d^7)/(729*b^9*c^14 - 729*a^9*c^5*d^9 + 6561*a^8*b*c^6*d^8 + 26244*a^2*b^
7*c^12*d^2 - 61236*a^3*b^6*c^11*d^3 + 91854*a^4*b^5*c^10*d^4 - 91854*a^5*b^4*c^9*d^5 + 61236*a^6*b^3*c^8*d^6 -
26244*a^7*b^2*c^7*d^7 - 6561*a*b^8*c^13*d))^(1/3))/2 - (log(((3^(1/2)*1i + 1)*(((3^(1/2)*1i + 1)^2*((54*b^3*d
^3*x*(a*d - b*c)^2*(a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) - 27*a*b^3*c*d^3*(3^(1/2)*1i + 1
)*(a*d + b*c)*(a*d - b*c)^4*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)^9))^(1/3))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*
d - b*c)^9))^(2/3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*
c^2*d^4 - 11*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*c^3*(a*d - b*c)^4))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)
^9))^(1/3))/9 + (16*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c^6 + 268*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2
*c^2*d^4 - 49*a*b^5*c^5*d - 49*a^5*b*c*d^5))/(27*a^3*c^3*(a*d - b*c)^8))*(3^(1/2)*1i + 1)*(-(8*a^3*d^8 - 512*b
^3*c^3*d^5 + 384*a*b^2*c^2*d^6 - 96*a^2*b*c*d^7)/(729*b^9*c^14 - 729*a^9*c^5*d^9 + 6561*a^8*b*c^6*d^8 + 26244*
a^2*b^7*c^12*d^2 - 61236*a^3*b^6*c^11*d^3 + 91854*a^4*b^5*c^10*d^4 - 91854*a^5*b^4*c^9*d^5 + 61236*a^6*b^3*c^8
*d^6 - 26244*a^7*b^2*c^7*d^7 - 6561*a*b^8*c^13*d))^(1/3))/2

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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