3.5.60 \(\int \frac {1}{x \sqrt [3]{a+b x^3} (c+d x^3)} \, dx\)

Optimal. Leaf size=244 \[ -\frac {\sqrt [3]{d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{b c-a d}}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{b c-a d}}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{b c-a d}}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a} c}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} c}-\frac {\log (x)}{2 \sqrt [3]{a} c} \]

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Rubi [A]  time = 0.21, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {446, 86, 55, 617, 204, 31, 56} \begin {gather*} -\frac {\sqrt [3]{d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{b c-a d}}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{b c-a d}}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{b c-a d}}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a} c}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} c}-\frac {\log (x)}{2 \sqrt [3]{a} c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(b*c - a*d)/b]

Rule 86

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

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 446

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

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]

Rubi steps

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

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Mathematica [C]  time = 0.18, size = 140, normalized size = 0.57 \begin {gather*} \frac {3 \sqrt [3]{a} d \left (a+b x^3\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {d \left (b x^3+a\right )}{a d-b c}\right )-(b c-a d) \left (3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )-3 \log (x)\right )}{6 \sqrt [3]{a} c (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^(1/3)*d*(a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)] - (b*c - a*d)*(2
*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*Log[x] + 3*Log[a^(1/3) - (a + b*x^3)^(1/3)]))
/(6*a^(1/3)*c*(-(b*c) + a*d))

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IntegrateAlgebraic [A]  time = 0.49, size = 332, normalized size = 1.36 \begin {gather*} -\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{6 \sqrt [3]{a} c}-\frac {\sqrt [3]{d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{6 c \sqrt [3]{b c-a d}}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 c \sqrt [3]{b c-a d}}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{b c-a d}}\right )}{\sqrt {3} c \sqrt [3]{b c-a d}}+\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{a}\right )}{3 \sqrt [3]{a} c}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 628, normalized size = 2.57 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) - 2 \, \sqrt {3} a \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - a \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \left (\frac {d}{b c - a d}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} d + {\left (b c - a d\right )} \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}}\right ) + 2 \, a \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} \log \left ({\left (b c - a d\right )} \left (\frac {d}{b c - a d}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} d\right ) - a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{6 \, a c}, -\frac {2 \, \sqrt {3} a \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + a \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \left (\frac {d}{b c - a d}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} d + {\left (b c - a d\right )} \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}}\right ) - 2 \, a \left (\frac {d}{b c - a d}\right )^{\frac {1}{3}} \log \left ({\left (b c - a d\right )} \left (\frac {d}{b c - a d}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} d\right ) - 6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) + a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{6 \, a c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(1/3)*a*sqrt(-1/a^(2/3))*log((2*b*x^3 + 3*sqrt(1/3)*(2*(b*x^3 + a)^(2/3)*a^(2/3) - (b*x^3 + a)^(1/
3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x^3 + a)^(1/3)*a^(2/3) + 3*a)/x^3) - 2*sqrt(3)*a*(d/(b*c - a*d))^(1/3)
*arctan(2/3*sqrt(3)*(b*x^3 + a)^(1/3)*(d/(b*c - a*d))^(1/3) - 1/3*sqrt(3)) - a*(d/(b*c - a*d))^(1/3)*log(-(b*x
^3 + a)^(1/3)*(b*c - a*d)*(d/(b*c - a*d))^(2/3) + (b*x^3 + a)^(2/3)*d + (b*c - a*d)*(d/(b*c - a*d))^(1/3)) + 2
*a*(d/(b*c - a*d))^(1/3)*log((b*c - a*d)*(d/(b*c - a*d))^(2/3) + (b*x^3 + a)^(1/3)*d) - a^(2/3)*log((b*x^3 + a
)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*a^(2/3)*log((b*x^3 + a)^(1/3) - a^(1/3)))/(a*c), -1/6*(2*sq
rt(3)*a*(d/(b*c - a*d))^(1/3)*arctan(2/3*sqrt(3)*(b*x^3 + a)^(1/3)*(d/(b*c - a*d))^(1/3) - 1/3*sqrt(3)) + a*(d
/(b*c - a*d))^(1/3)*log(-(b*x^3 + a)^(1/3)*(b*c - a*d)*(d/(b*c - a*d))^(2/3) + (b*x^3 + a)^(2/3)*d + (b*c - a*
d)*(d/(b*c - a*d))^(1/3)) - 2*a*(d/(b*c - a*d))^(1/3)*log((b*c - a*d)*(d/(b*c - a*d))^(2/3) + (b*x^3 + a)^(1/3
)*d) - 6*sqrt(1/3)*a^(2/3)*arctan(sqrt(1/3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3)) + a^(2/3)*log((b*x^3 + a)
^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) - 2*a^(2/3)*log((b*x^3 + a)^(1/3) - a^(1/3)))/(a*c)]

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giac [A]  time = 0.74, size = 326, normalized size = 1.34 \begin {gather*} \frac {d \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{2} d - \sqrt {3} a c d^{2}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{2} d - a c d^{2}\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {1}{3}} c} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {1}{3}} c} + \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {1}{3}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right ) x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.44, size = 702, normalized size = 2.88 \begin {gather*} \ln \left (b^5\,d^4\,{\left (b\,x^3+a\right )}^{1/3}-\frac {d\,\left (27\,b^4\,c^2\,d^3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )-243\,a\,b^4\,c^4\,d^3\,{\left (\frac {d}{27\,b\,c^4-27\,a\,c^3\,d}\right )}^{2/3}\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )\right )}{27\,b\,c^4-27\,a\,c^3\,d}\right )\,{\left (\frac {d}{27\,b\,c^4-27\,a\,c^3\,d}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}-a\,c^2\,{\left (\frac {1}{a\,c^3}\right )}^{2/3}\right )\,{\left (\frac {1}{27\,a\,c^3}\right )}^{1/3}+\frac {\ln \left (b^5\,d^4\,{\left (b\,x^3+a\right )}^{1/3}-\frac {d\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^4\,c^2\,d^3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )-\frac {243\,a\,b^4\,c^4\,d^3\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {d}{27\,b\,c^4-27\,a\,c^3\,d}\right )}^{2/3}\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{4}\right )}{8\,\left (27\,b\,c^4-27\,a\,c^3\,d\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d}{27\,b\,c^4-27\,a\,c^3\,d}\right )}^{1/3}}{2}-\frac {\ln \left (b^5\,d^4\,{\left (b\,x^3+a\right )}^{1/3}+\frac {d\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^4\,c^2\,d^3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )-\frac {243\,a\,b^4\,c^4\,d^3\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {d}{27\,b\,c^4-27\,a\,c^3\,d}\right )}^{2/3}\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{4}\right )}{8\,\left (27\,b\,c^4-27\,a\,c^3\,d\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d}{27\,b\,c^4-27\,a\,c^3\,d}\right )}^{1/3}}{2}-\ln \left (\sqrt {3}\,a\,c^2\,{\left (\frac {1}{a\,c^3}\right )}^{2/3}+{\left (b\,x^3+a\right )}^{1/3}\,2{}\mathrm {i}+a\,c^2\,{\left (\frac {1}{a\,c^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a\,c^3}\right )}^{1/3}+\ln \left (-\sqrt {3}\,a\,c^2\,{\left (\frac {1}{a\,c^3}\right )}^{2/3}+{\left (b\,x^3+a\right )}^{1/3}\,2{}\mathrm {i}+a\,c^2\,{\left (\frac {1}{a\,c^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a\,c^3}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b^5*d^4*(a + b*x^3)^(1/3) - (d*(27*b^4*c^2*d^3*(a + b*x^3)^(1/3)*(2*a^2*d^2 + b^2*c^2 - 2*a*b*c*d) - 243*a
*b^4*c^4*d^3*(d/(27*b*c^4 - 27*a*c^3*d))^(2/3)*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)))/(27*b*c^4 - 27*a*c^3*d))*(d
/(27*b*c^4 - 27*a*c^3*d))^(1/3) + log((a + b*x^3)^(1/3) - a*c^2*(1/(a*c^3))^(2/3))*(1/(27*a*c^3))^(1/3) + (log
(b^5*d^4*(a + b*x^3)^(1/3) - (d*(3^(1/2)*1i - 1)^3*(27*b^4*c^2*d^3*(a + b*x^3)^(1/3)*(2*a^2*d^2 + b^2*c^2 - 2*
a*b*c*d) - (243*a*b^4*c^4*d^3*(3^(1/2)*1i - 1)^2*(d/(27*b*c^4 - 27*a*c^3*d))^(2/3)*(2*a^2*d^2 + b^2*c^2 - 3*a*
b*c*d))/4))/(8*(27*b*c^4 - 27*a*c^3*d)))*(3^(1/2)*1i - 1)*(d/(27*b*c^4 - 27*a*c^3*d))^(1/3))/2 - (log(b^5*d^4*
(a + b*x^3)^(1/3) + (d*(3^(1/2)*1i + 1)^3*(27*b^4*c^2*d^3*(a + b*x^3)^(1/3)*(2*a^2*d^2 + b^2*c^2 - 2*a*b*c*d)
- (243*a*b^4*c^4*d^3*(3^(1/2)*1i + 1)^2*(d/(27*b*c^4 - 27*a*c^3*d))^(2/3)*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/4
))/(8*(27*b*c^4 - 27*a*c^3*d)))*(3^(1/2)*1i + 1)*(d/(27*b*c^4 - 27*a*c^3*d))^(1/3))/2 - log((a + b*x^3)^(1/3)*
2i + a*c^2*(1/(a*c^3))^(2/3)*1i + 3^(1/2)*a*c^2*(1/(a*c^3))^(2/3))*((3^(1/2)*1i)/2 + 1/2)*(1/(27*a*c^3))^(1/3)
+ log((a + b*x^3)^(1/3)*2i + a*c^2*(1/(a*c^3))^(2/3)*1i - 3^(1/2)*a*c^2*(1/(a*c^3))^(2/3))*((3^(1/2)*1i)/2 -
1/2)*(1/(27*a*c^3))^(1/3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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