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

Optimal. Leaf size=299 \[ -\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{4/3} (b c-a d)}-\frac{1}{a c x} \]

[Out]

-(1/(a*c*x)) + (b^(4/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*(b*c - a*d)) - (d^
(4/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(4/3)*(b*c - a*d)) + (b^(4/3)*Log[a^(1/3)
+ b^(1/3)*x])/(3*a^(4/3)*(b*c - a*d)) - (d^(4/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(4/3)*(b*c - a*d)) - (b^(4/3)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*(b*c - a*d)) + (d^(4/3)*Log[c^(2/3) - c^(1/3)*d^(1/
3)*x + d^(2/3)*x^2])/(6*c^(4/3)*(b*c - a*d))

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Rubi [A]  time = 0.271776, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {480, 584, 292, 31, 634, 617, 204, 628} \[ -\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{4/3} (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a*c*x)) + (b^(4/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*(b*c - a*d)) - (d^
(4/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(4/3)*(b*c - a*d)) + (b^(4/3)*Log[a^(1/3)
+ b^(1/3)*x])/(3*a^(4/3)*(b*c - a*d)) - (d^(4/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(4/3)*(b*c - a*d)) - (b^(4/3)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*(b*c - a*d)) + (d^(4/3)*Log[c^(2/3) - c^(1/3)*d^(1/
3)*x + d^(2/3)*x^2])/(6*c^(4/3)*(b*c - a*d))

Rule 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 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{1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac{1}{a c x}+\frac{\int \frac{x \left (-b c-a d-b d x^3\right )}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{a c}\\ &=-\frac{1}{a c x}+\frac{\int \left (-\frac{b^2 c x}{(b c-a d) \left (a+b x^3\right )}-\frac{a d^2 x}{(-b c+a d) \left (c+d x^3\right )}\right ) \, dx}{a c}\\ &=-\frac{1}{a c x}-\frac{b^2 \int \frac{x}{a+b x^3} \, dx}{a (b c-a d)}+\frac{d^2 \int \frac{x}{c+d x^3} \, dx}{c (b c-a d)}\\ &=-\frac{1}{a c x}+\frac{b^{5/3} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3} (b c-a d)}-\frac{b^{5/3} \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}{3 a^{4/3} (b c-a d)}-\frac{d^{5/3} \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{4/3} (b c-a d)}+\frac{d^{5/3} \int \frac{\sqrt [3]{c}+\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{4/3} (b c-a d)}\\ &=-\frac{1}{a c x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{b^{4/3} \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}{6 a^{4/3} (b c-a d)}-\frac{b^{5/3} \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a (b c-a d)}+\frac{d^{4/3} \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}{6 c^{4/3} (b c-a d)}+\frac{d^{5/3} \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c (b c-a d)}\\ &=-\frac{1}{a c x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac{b^{4/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3} (b c-a d)}+\frac{d^{4/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{4/3} (b c-a d)}\\ &=-\frac{1}{a c x}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} (b c-a d)}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{4/3} (b c-a d)}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.136827, size = 244, normalized size = 0.82 \[ \frac{\frac{b^{4/3} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac{2 b^{4/3} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac{2 \sqrt{3} b^{4/3} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}+\frac{6 b}{a}-\frac{d^{4/3} x \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{4/3}}+\frac{2 d^{4/3} x \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{4/3}}+\frac{2 \sqrt{3} d^{4/3} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{4/3}}-\frac{6 d}{c}}{6 a d x-6 b c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*b)/a - (6*d)/c - (2*Sqrt[3]*b^(4/3)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) + (2*Sqrt[3]*d^
(4/3)*x*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(4/3) - (2*b^(4/3)*x*Log[a^(1/3) + b^(1/3)*x])/a^(4/3)
+ (2*d^(4/3)*x*Log[c^(1/3) + d^(1/3)*x])/c^(4/3) + (b^(4/3)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
a^(4/3) - (d^(4/3)*x*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(4/3))/(-6*b*c*x + 6*a*d*x)

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Maple [A]  time = 0.01, size = 257, normalized size = 0.9 \begin{align*} -{\frac{1}{acx}}+{\frac{d}{3\,c \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{d}{6\,c \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{3}}{3\,c \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{b}{3\,a \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b}{6\,a \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}}{3\,a \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/c/x+1/3*d/c/(a*d-b*c)/(1/d*c)^(1/3)*ln(x+(1/d*c)^(1/3))-1/6*d/c/(a*d-b*c)/(1/d*c)^(1/3)*ln(x^2-(1/d*c)^(1
/3)*x+(1/d*c)^(2/3))-1/3*d/c/(a*d-b*c)*3^(1/2)/(1/d*c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/d*c)^(1/3)*x-1))-1/3*b/a
/(a*d-b*c)/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6*b/a/(a*d-b*c)/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*b
/a/(a*d-b*c)*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(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(1/x^2/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89678, size = 585, normalized size = 1.96 \begin{align*} -\frac{2 \, \sqrt{3} b c x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 2 \, \sqrt{3} a d x \left (\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{d}{c}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) - b c x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - a d x \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (\frac{d}{c}\right )^{\frac{2}{3}} + c \left (\frac{d}{c}\right )^{\frac{1}{3}}\right ) + 2 \, b c x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 2 \, a d x \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d}{c}\right )^{\frac{2}{3}}\right ) + 6 \, b c - 6 \, a d}{6 \,{\left (a b c^{2} - a^{2} c d\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 39.3303, size = 661, normalized size = 2.21 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{7} d^{3} - 81 a^{6} b c d^{2} + 81 a^{5} b^{2} c^{2} d - 27 a^{4} b^{3} c^{3}\right ) + b^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 243 t^{5} a^{12} c^{4} d^{8} + 1215 t^{5} a^{11} b c^{5} d^{7} - 2430 t^{5} a^{10} b^{2} c^{6} d^{6} + 2673 t^{5} a^{9} b^{3} c^{7} d^{5} - 2430 t^{5} a^{8} b^{4} c^{8} d^{4} + 2673 t^{5} a^{7} b^{5} c^{9} d^{3} - 2430 t^{5} a^{6} b^{6} c^{10} d^{2} + 1215 t^{5} a^{5} b^{7} c^{11} d - 243 t^{5} a^{4} b^{8} c^{12} + 9 t^{2} a^{9} d^{9} - 18 t^{2} a^{8} b c d^{8} + 9 t^{2} a^{7} b^{2} c^{2} d^{7} + 9 t^{2} a^{2} b^{7} c^{7} d^{2} - 18 t^{2} a b^{8} c^{8} d + 9 t^{2} b^{9} c^{9}}{a^{4} b^{3} d^{7} + b^{7} c^{4} d^{3}} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} c^{4} d^{3} - 81 a^{2} b c^{5} d^{2} + 81 a b^{2} c^{6} d - 27 b^{3} c^{7}\right ) - d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 243 t^{5} a^{12} c^{4} d^{8} + 1215 t^{5} a^{11} b c^{5} d^{7} - 2430 t^{5} a^{10} b^{2} c^{6} d^{6} + 2673 t^{5} a^{9} b^{3} c^{7} d^{5} - 2430 t^{5} a^{8} b^{4} c^{8} d^{4} + 2673 t^{5} a^{7} b^{5} c^{9} d^{3} - 2430 t^{5} a^{6} b^{6} c^{10} d^{2} + 1215 t^{5} a^{5} b^{7} c^{11} d - 243 t^{5} a^{4} b^{8} c^{12} + 9 t^{2} a^{9} d^{9} - 18 t^{2} a^{8} b c d^{8} + 9 t^{2} a^{7} b^{2} c^{2} d^{7} + 9 t^{2} a^{2} b^{7} c^{7} d^{2} - 18 t^{2} a b^{8} c^{8} d + 9 t^{2} b^{9} c^{9}}{a^{4} b^{3} d^{7} + b^{7} c^{4} d^{3}} \right )} \right )\right )} - \frac{1}{a c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**3*(27*a**7*d**3 - 81*a**6*b*c*d**2 + 81*a**5*b**2*c**2*d - 27*a**4*b**3*c**3) + b**4, Lambda(_t, _
t*log(x + (-243*_t**5*a**12*c**4*d**8 + 1215*_t**5*a**11*b*c**5*d**7 - 2430*_t**5*a**10*b**2*c**6*d**6 + 2673*
_t**5*a**9*b**3*c**7*d**5 - 2430*_t**5*a**8*b**4*c**8*d**4 + 2673*_t**5*a**7*b**5*c**9*d**3 - 2430*_t**5*a**6*
b**6*c**10*d**2 + 1215*_t**5*a**5*b**7*c**11*d - 243*_t**5*a**4*b**8*c**12 + 9*_t**2*a**9*d**9 - 18*_t**2*a**8
*b*c*d**8 + 9*_t**2*a**7*b**2*c**2*d**7 + 9*_t**2*a**2*b**7*c**7*d**2 - 18*_t**2*a*b**8*c**8*d + 9*_t**2*b**9*
c**9)/(a**4*b**3*d**7 + b**7*c**4*d**3)))) + RootSum(_t**3*(27*a**3*c**4*d**3 - 81*a**2*b*c**5*d**2 + 81*a*b**
2*c**6*d - 27*b**3*c**7) - d**4, Lambda(_t, _t*log(x + (-243*_t**5*a**12*c**4*d**8 + 1215*_t**5*a**11*b*c**5*d
**7 - 2430*_t**5*a**10*b**2*c**6*d**6 + 2673*_t**5*a**9*b**3*c**7*d**5 - 2430*_t**5*a**8*b**4*c**8*d**4 + 2673
*_t**5*a**7*b**5*c**9*d**3 - 2430*_t**5*a**6*b**6*c**10*d**2 + 1215*_t**5*a**5*b**7*c**11*d - 243*_t**5*a**4*b
**8*c**12 + 9*_t**2*a**9*d**9 - 18*_t**2*a**8*b*c*d**8 + 9*_t**2*a**7*b**2*c**2*d**7 + 9*_t**2*a**2*b**7*c**7*
d**2 - 18*_t**2*a*b**8*c**8*d + 9*_t**2*b**9*c**9)/(a**4*b**3*d**7 + b**7*c**4*d**3)))) - 1/(a*c*x)

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Giac [A]  time = 1.15925, size = 412, normalized size = 1.38 \begin{align*} \frac{b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \left (-\frac{c}{d}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{3} - a c^{2} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \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 )}{\sqrt{3} a^{2} b c - \sqrt{3} a^{3} d} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \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 )}{\sqrt{3} b c^{3} - \sqrt{3} a c^{2} d} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{2} b c - a^{3} d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{1}{a c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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