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

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

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {481, 200, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac {\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{d} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x^3)*(c + d*x^3)),x]

[Out]

(a^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(1/3)*(b*c - a*d)) - (c^(1/3)*ArcTan[(c
^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*d^(1/3)*(b*c - a*d)) - (a^(1/3)*Log[a^(1/3) + b^(1/3)*x])/(
3*b^(1/3)*(b*c - a*d)) + (c^(1/3)*Log[c^(1/3) + d^(1/3)*x])/(3*d^(1/3)*(b*c - a*d)) + (a^(1/3)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(1/3)*(b*c - a*d)) - (c^(1/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*
x^2])/(6*d^(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 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 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 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 {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac {a \int \frac {1}{a+b x^3} \, dx}{b c-a d}+\frac {c \int \frac {1}{c+d x^3} \, dx}{b c-a d}\\ &=-\frac {\sqrt [3]{a} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 (b c-a d)}-\frac {\sqrt [3]{a} \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}{3 (b c-a d)}+\frac {\sqrt [3]{c} \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 (b c-a d)}+\frac {\sqrt [3]{c} \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}{3 (b c-a d)}\\ &=-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac {\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}-\frac {a^{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]{a} \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]{b} (b c-a d)}+\frac {c^{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]{c} \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]{d} (b c-a d)}\\ &=-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac {\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)}-\frac {\sqrt [3]{a} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b} (b c-a d)}+\frac {\sqrt [3]{c} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d} (b c-a d)}\\ &=\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{d} (b c-a d)}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac {\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 224, normalized size = 0.78 \[ \frac {\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{d}}-\frac {2 \sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.94, size = 199, normalized size = 0.69 \[ -\frac {2 \, \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 2 \, \sqrt {3} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} d x \left (-\frac {c}{d}\right )^{\frac {2}{3}} - \sqrt {3} c}{3 \, c}\right ) - \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right ) + 2 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 2 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}{6 \, {\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 278, normalized size = 0.97 \[ \frac {a \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b c - a^{2} d\right )}} - \frac {c \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c - \sqrt {3} a b d} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d - \sqrt {3} a d^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c - a b d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d - a d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 246, normalized size = 0.85 \[ \frac {\sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {a \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {\sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {2}{3}} d}-\frac {c \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {2}{3}} d}+\frac {c \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.21, size = 317, normalized size = 1.10 \[ -\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} c \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {c \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} - \frac {a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {c \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 8.12, size = 1265, normalized size = 4.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x^3)*(c + d*x^3)),x)

[Out]

log(x + a*d*(a/(b*(a*d - b*c)^3))^(1/3) - b*c*(a/(b*(a*d - b*c)^3))^(1/3))*(-a/(27*b^4*c^3 - 27*a^3*b*d^3 + 81
*a^2*b^2*c*d^2 - 81*a*b^3*c^2*d))^(1/3) + log(x - a*d*(-c/(d*(a*d - b*c)^3))^(1/3) + b*c*(-c/(d*(a*d - b*c)^3)
)^(1/3))*(-c/(27*a^3*d^4 - 27*b^3*c^3*d + 81*a*b^2*c^2*d^2 - 81*a^2*b*c*d^3))^(1/3) + (log(((3^(1/2)*1i - 1)*(
a/(b*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 - (81*a*b^3*c*d^3*(3^(1/2)*1i
- 1)*(a*d + b*c)*(a*d - b*c)^4*(a/(b*(a*d - b*c)^3))^(1/3))/2)*(a/(b*(a*d - b*c)^3))^(2/3))/36 + 9*a*b^5*c^4*d
^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c^3*d^3 - 9*a^3*b^3*c^2*d^4))/6 - 3*a*b^2*c*d^2*x*(a^2*d^2 + b^2*c^2))*(3^(1/
2)*1i - 1)*(-a/(27*b^4*c^3 - 27*a^3*b*d^3 + 81*a^2*b^2*c*d^2 - 81*a*b^3*c^2*d))^(1/3))/2 - (log(((3^(1/2)*1i +
1)*(a/(b*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i + 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 + (81*a*b^3*c*d^3*(3^(1/2
)*1i + 1)*(a*d + b*c)*(a*d - b*c)^4*(a/(b*(a*d - b*c)^3))^(1/3))/2)*(a/(b*(a*d - b*c)^3))^(2/3))/36 + 9*a*b^5*
c^4*d^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c^3*d^3 - 9*a^3*b^3*c^2*d^4))/6 + 3*a*b^2*c*d^2*x*(a^2*d^2 + b^2*c^2))*(
3^(1/2)*1i + 1)*(-a/(27*b^4*c^3 - 27*a^3*b*d^3 + 81*a^2*b^2*c*d^2 - 81*a*b^3*c^2*d))^(1/3))/2 + (log(((3^(1/2)
*1i - 1)*(-c/(d*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 - (81*a*b^3*c*d^3*(
3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c)^4*(-c/(d*(a*d - b*c)^3))^(1/3))/2)*(-c/(d*(a*d - b*c)^3))^(2/3))/36 +
9*a*b^5*c^4*d^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c^3*d^3 - 9*a^3*b^3*c^2*d^4))/6 - 3*a*b^2*c*d^2*x*(a^2*d^2 + b^2
*c^2))*(3^(1/2)*1i - 1)*(-c/(27*a^3*d^4 - 27*b^3*c^3*d + 81*a*b^2*c^2*d^2 - 81*a^2*b*c*d^3))^(1/3))/2 - (log((
(3^(1/2)*1i + 1)*(-c/(d*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i + 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 + (81*a*b^3
*c*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*(a*d - b*c)^4*(-c/(d*(a*d - b*c)^3))^(1/3))/2)*(-c/(d*(a*d - b*c)^3))^(2/3
))/36 + 9*a*b^5*c^4*d^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c^3*d^3 - 9*a^3*b^3*c^2*d^4))/6 + 3*a*b^2*c*d^2*x*(a^2*d
^2 + b^2*c^2))*(3^(1/2)*1i + 1)*(-c/(27*a^3*d^4 - 27*b^3*c^3*d + 81*a*b^2*c^2*d^2 - 81*a^2*b*c*d^3))^(1/3))/2

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sympy [A]  time = 20.20, size = 342, normalized size = 1.19 \[ \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} d^{4} - 81 a^{2} b c d^{3} + 81 a b^{2} c^{2} d^{2} - 27 b^{3} c^{3} d\right ) + c, \left (t \mapsto t \log {\left (x + \frac {162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} + \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} b d^{3} - 81 a^{2} b^{2} c d^{2} + 81 a b^{3} c^{2} d - 27 b^{4} c^{3}\right ) - a, \left (t \mapsto t \log {\left (x + \frac {162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**3*(27*a**3*d**4 - 81*a**2*b*c*d**3 + 81*a*b**2*c**2*d**2 - 27*b**3*c**3*d) + c, Lambda(_t, _t*log(
x + (162*_t**4*a**4*b*d**5 - 648*_t**4*a**3*b**2*c*d**4 + 972*_t**4*a**2*b**3*c**2*d**3 - 648*_t**4*a*b**4*c**
3*d**2 + 162*_t**4*b**5*c**4*d - 3*_t*a**2*d**2 + 6*_t*a*b*c*d - 3*_t*b**2*c**2)/(a*d + b*c)))) + RootSum(_t**
3*(27*a**3*b*d**3 - 81*a**2*b**2*c*d**2 + 81*a*b**3*c**2*d - 27*b**4*c**3) - a, Lambda(_t, _t*log(x + (162*_t*
*4*a**4*b*d**5 - 648*_t**4*a**3*b**2*c*d**4 + 972*_t**4*a**2*b**3*c**2*d**3 - 648*_t**4*a*b**4*c**3*d**2 + 162
*_t**4*b**5*c**4*d - 3*_t*a**2*d**2 + 6*_t*a*b*c*d - 3*_t*b**2*c**2)/(a*d + b*c))))

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