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

Optimal. Leaf size=288 \[ \frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)} \]

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Rubi [A]  time = 0.14, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {482, 292, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, 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 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 482

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

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

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Mathematica [A]  time = 0.14, size = 224, normalized size = 0.78 \begin {gather*} \frac {-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{c}}-\frac {2 \sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{c}}-\frac {2 \sqrt {3} \sqrt [3]{d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 a d-6 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 201, normalized size = 0.70 \begin {gather*} \frac {2 \, \sqrt {3} \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} \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 ) + \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 ) + \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 \, \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (-\frac {d}{c}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c - a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.25, size = 290, normalized size = 1.01 \begin {gather*} -\frac {b \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 b c - a^{2} d\right )}} + \frac {d \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^{2} - a c 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 b^{2} c - \sqrt {3} a^{2} b 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^{2} d - \sqrt {3} a c d^{2}} + \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 b^{2} c - a^{2} b 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^{2} d - a c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 222, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.29, size = 265, normalized size = 0.92 \begin {gather*} \frac {\sqrt {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 )}{3 \, {\left (b c - a d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {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 )}{3 \, {\left (b c - a d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {\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 \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {\log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.42, size = 982, normalized size = 3.41 \begin {gather*} \ln \left (b\,x+a^3\,d^2\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}+a\,b^2\,c^2\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}-2\,a^2\,b\,c\,d\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )\,{\left (\frac {b}{27\,a^4\,d^3-81\,a^3\,b\,c\,d^2+81\,a^2\,b^2\,c^2\,d-27\,a\,b^3\,c^3}\right )}^{1/3}+\ln \left (d\,x+b^2\,c^3\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}+a^2\,c\,d^2\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}-2\,a\,b\,c^2\,d\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )\,{\left (\frac {d}{-27\,a^3\,c\,d^3+81\,a^2\,b\,c^2\,d^2-81\,a\,b^2\,c^3\,d+27\,b^3\,c^4}\right )}^{1/3}+\frac {\ln \left (b^4\,d^4\,x-\frac {b\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,a\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b}{27\,a^4\,d^3-81\,a^3\,b\,c\,d^2+81\,a^2\,b^2\,c^2\,d-27\,a\,b^3\,c^3}\right )}^{1/3}}{2}-\frac {\ln \left (b^4\,d^4\,x+\frac {b\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,a\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b}{27\,a^4\,d^3-81\,a^3\,b\,c\,d^2+81\,a^2\,b^2\,c^2\,d-27\,a\,b^3\,c^3}\right )}^{1/3}}{2}+\frac {\ln \left (b^4\,d^4\,x+\frac {d\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,c\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d}{-27\,a^3\,c\,d^3+81\,a^2\,b\,c^2\,d^2-81\,a\,b^2\,c^3\,d+27\,b^3\,c^4}\right )}^{1/3}}{2}-\frac {\ln \left (b^4\,d^4\,x-\frac {d\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,c\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d}{-27\,a^3\,c\,d^3+81\,a^2\,b\,c^2\,d^2-81\,a\,b^2\,c^3\,d+27\,b^3\,c^4}\right )}^{1/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x^3)*(c + d*x^3)),x)

[Out]

log(b*x + a^3*d^2*(b/(a*(a*d - b*c)^3))^(2/3) + a*b^2*c^2*(b/(a*(a*d - b*c)^3))^(2/3) - 2*a^2*b*c*d*(b/(a*(a*d
- b*c)^3))^(2/3))*(b/(27*a^4*d^3 - 27*a*b^3*c^3 + 81*a^2*b^2*c^2*d - 81*a^3*b*c*d^2))^(1/3) + log(d*x + b^2*c
^3*(-d/(c*(a*d - b*c)^3))^(2/3) + a^2*c*d^2*(-d/(c*(a*d - b*c)^3))^(2/3) - 2*a*b*c^2*d*(-d/(c*(a*d - b*c)^3))^
(2/3))*(d/(27*b^3*c^4 - 27*a^3*c*d^3 + 81*a^2*b*c^2*d^2 - 81*a*b^2*c^3*d))^(1/3) + (log(b^4*d^4*x - (b*(3^(1/2
)*1i - 1)^3*(27*b^3*d^3*x*(a^2*d^2 + b^2*c^2)*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(
a*d - b*c)^4*(b/(a*(a*d - b*c)^3))^(2/3))/4))/(216*a*(a*d - b*c)^3))*(3^(1/2)*1i - 1)*(b/(27*a^4*d^3 - 27*a*b^
3*c^3 + 81*a^2*b^2*c^2*d - 81*a^3*b*c*d^2))^(1/3))/2 - (log(b^4*d^4*x + (b*(3^(1/2)*1i + 1)^3*(27*b^3*d^3*x*(a
^2*d^2 + b^2*c^2)*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^4*(b/(a*(a*d - b*
c)^3))^(2/3))/4))/(216*a*(a*d - b*c)^3))*(3^(1/2)*1i + 1)*(b/(27*a^4*d^3 - 27*a*b^3*c^3 + 81*a^2*b^2*c^2*d - 8
1*a^3*b*c*d^2))^(1/3))/2 + (log(b^4*d^4*x + (d*(3^(1/2)*1i - 1)^3*(27*b^3*d^3*x*(a^2*d^2 + b^2*c^2)*(a*d - b*c
)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-d/(c*(a*d - b*c)^3))^(2/3))/4))/(216*c*(a
*d - b*c)^3))*(3^(1/2)*1i - 1)*(d/(27*b^3*c^4 - 27*a^3*c*d^3 + 81*a^2*b*c^2*d^2 - 81*a*b^2*c^3*d))^(1/3))/2 -
(log(b^4*d^4*x - (d*(3^(1/2)*1i + 1)^3*(27*b^3*d^3*x*(a^2*d^2 + b^2*c^2)*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1
/2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-d/(c*(a*d - b*c)^3))^(2/3))/4))/(216*c*(a*d - b*c)^3))*(3^(1/2)*1i +
1)*(d/(27*b^3*c^4 - 27*a^3*c*d^3 + 81*a^2*b*c^2*d^2 - 81*a*b^2*c^3*d))^(1/3))/2

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sympy [A]  time = 14.51, size = 515, normalized size = 1.79 \begin {gather*} \operatorname {RootSum} {\left (t^{3} \left (27 a^{4} d^{3} - 81 a^{3} b c d^{2} + 81 a^{2} b^{2} c^{2} d - 27 a b^{3} c^{3}\right ) - b, \left (t \mapsto t \log {\left (x + \frac {243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} + \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} c d^{3} - 81 a^{2} b c^{2} d^{2} + 81 a b^{2} c^{3} d - 27 b^{3} c^{4}\right ) + d, \left (t \mapsto t \log {\left (x + \frac {243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**3*(27*a**4*d**3 - 81*a**3*b*c*d**2 + 81*a**2*b**2*c**2*d - 27*a*b**3*c**3) - b, Lambda(_t, _t*log(
x + (243*_t**5*a**7*c*d**6 - 1458*_t**5*a**6*b*c**2*d**5 + 3645*_t**5*a**5*b**2*c**3*d**4 - 4860*_t**5*a**4*b*
*3*c**4*d**3 + 3645*_t**5*a**3*b**4*c**5*d**2 - 1458*_t**5*a**2*b**5*c**6*d + 243*_t**5*a*b**6*c**7 + 9*_t**2*
a**4*d**4 - 18*_t**2*a**3*b*c*d**3 + 18*_t**2*a**2*b**2*c**2*d**2 - 18*_t**2*a*b**3*c**3*d + 9*_t**2*b**4*c**4
)/(a*b*d**2 + b**2*c*d)))) + RootSum(_t**3*(27*a**3*c*d**3 - 81*a**2*b*c**2*d**2 + 81*a*b**2*c**3*d - 27*b**3*
c**4) + d, Lambda(_t, _t*log(x + (243*_t**5*a**7*c*d**6 - 1458*_t**5*a**6*b*c**2*d**5 + 3645*_t**5*a**5*b**2*c
**3*d**4 - 4860*_t**5*a**4*b**3*c**4*d**3 + 3645*_t**5*a**3*b**4*c**5*d**2 - 1458*_t**5*a**2*b**5*c**6*d + 243
*_t**5*a*b**6*c**7 + 9*_t**2*a**4*d**4 - 18*_t**2*a**3*b*c*d**3 + 18*_t**2*a**2*b**2*c**2*d**2 - 18*_t**2*a*b*
*3*c**3*d + 9*_t**2*b**4*c**4)/(a*b*d**2 + b**2*c*d))))

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