3.5.84 \(\int \frac {1}{x (a+b x^3)^{4/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=271 \[ \frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} c}-\frac {\log (x)}{2 a^{4/3} c}+\frac {d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}-\frac {d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}}-\frac {d^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c (b c-a d)^{4/3}}+\frac {b}{a \sqrt [3]{a+b x^3} (b c-a d)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.32, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {446, 85, 156, 55, 617, 204, 31, 56} \begin {gather*} \frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} c}-\frac {\log (x)}{2 a^{4/3} c}+\frac {d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}-\frac {d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}}-\frac {d^{4/3} \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 (b c-a d)^{4/3}}+\frac {b}{a \sqrt [3]{a+b x^3} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

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

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 85

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[(f*(e + f*x)^(p +
1))/((p + 1)*(b*e - a*f)*(d*e - c*f)), x] + Dist[1/((b*e - a*f)*(d*e - c*f)), Int[((b*d*e - b*c*f - a*d*f - b
*d*f*x)*(e + f*x)^(p + 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]

Rule 156

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

________________________________________________________________________________________

Mathematica [C]  time = 0.05, size = 86, normalized size = 0.32 \begin {gather*} \frac {a d \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {d \left (b x^3+a\right )}{a d-b c}\right )+(b c-a d) \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {b x^3}{a}+1\right )}{a c \sqrt [3]{a+b x^3} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

(a*d*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)] + (b*c - a*d)*Hypergeometric2F1[-1/3, 1,
2/3, 1 + (b*x^3)/a])/(a*c*(b*c - a*d)*(a + b*x^3)^(1/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.85, size = 360, normalized size = 1.33 \begin {gather*} \frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{a}\right )}{3 a^{4/3} c}-\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 a^{4/3} 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} a^{4/3} c}-\frac {d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 c (b c-a d)^{4/3}}+\frac {d^{4/3} \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 (b c-a d)^{4/3}}-\frac {d^{4/3} \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 (b c-a d)^{4/3}}-\frac {b}{a \sqrt [3]{a+b x^3} (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.80, size = 975, normalized size = 3.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.79, size = 389, normalized size = 1.44 \begin {gather*} -\frac {d^{2} \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^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\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^{2} c^{3} - 2 \, \sqrt {3} a b c^{2} d + \sqrt {3} a^{2} 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^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} + \frac {b}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a b c - a^{2} d\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 {4}{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 {4}{3}} c} + \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {4}{3}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {4}{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)^(4/3)/(d*x^3+c),x)

[Out]

int(1/x/(b*x^3+a)^(4/3)/(d*x^3+c),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{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)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x), x)

________________________________________________________________________________________

mupad [B]  time = 5.34, size = 3804, normalized size = 14.04

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^3)^(4/3)*(c + d*x^3)),x)

[Out]

log(9*a^7*b^14*c^11*d^4 - ((a + b*x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 297*a^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^1
1*d^5 - 4455*a^10*b^12*c^10*d^6 + 8937*a^11*b^11*c^9*d^7 - 12663*a^12*b^10*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 -
9855*a^14*b^8*c^6*d^10 + 5400*a^15*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^12 + 486*a^17*b^5*c^3*d^13 - 54*a^18*b^4
*c^2*d^14) - (-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^
(2/3)*(243*a^10*b^15*c^15*d^3 - 2916*a^11*b^14*c^14*d^4 + 15795*a^12*b^13*c^13*d^5 - 51030*a^13*b^12*c^12*d^6
+ 109350*a^14*b^11*c^11*d^7 - 163296*a^15*b^10*c^10*d^8 + 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c^8*d^10 +
69255*a^18*b^7*c^7*d^11 - 24300*a^19*b^6*c^6*d^12 + 5103*a^20*b^5*c^5*d^13 - 486*a^21*b^4*c^4*d^14))*(-d^4/(2
7*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3) - 90*a^8*b^13*c
^10*d^5 + 405*a^9*b^12*c^9*d^6 - 1071*a^10*b^11*c^8*d^7 + 1827*a^11*b^10*c^7*d^8 - 2079*a^12*b^9*c^6*d^9 + 157
5*a^13*b^8*c^5*d^10 - 765*a^14*b^7*c^4*d^11 + 216*a^15*b^6*c^3*d^12 - 27*a^16*b^5*c^2*d^13)*(-d^4/(27*b^4*c^7
+ 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3) + log(9*a^7*b^14*c^11*d^4
- ((a + b*x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 297*a^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*
c^10*d^6 + 8937*a^11*b^11*c^9*d^7 - 12663*a^12*b^10*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 9855*a^14*b^8*c^6*d^10
+ 5400*a^15*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^12 + 486*a^17*b^5*c^3*d^13 - 54*a^18*b^4*c^2*d^14) - (1/(27*a^4
*c^3))^(2/3)*(243*a^10*b^15*c^15*d^3 - 2916*a^11*b^14*c^14*d^4 + 15795*a^12*b^13*c^13*d^5 - 51030*a^13*b^12*c^
12*d^6 + 109350*a^14*b^11*c^11*d^7 - 163296*a^15*b^10*c^10*d^8 + 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c^8
*d^10 + 69255*a^18*b^7*c^7*d^11 - 24300*a^19*b^6*c^6*d^12 + 5103*a^20*b^5*c^5*d^13 - 486*a^21*b^4*c^4*d^14))*(
1/(27*a^4*c^3))^(1/3) - 90*a^8*b^13*c^10*d^5 + 405*a^9*b^12*c^9*d^6 - 1071*a^10*b^11*c^8*d^7 + 1827*a^11*b^10*
c^7*d^8 - 2079*a^12*b^9*c^6*d^9 + 1575*a^13*b^8*c^5*d^10 - 765*a^14*b^7*c^4*d^11 + 216*a^15*b^6*c^3*d^12 - 27*
a^16*b^5*c^2*d^13)*(1/(27*a^4*c^3))^(1/3) + (log(((3^(1/2)*1i - 1)*(-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^
3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3)*((a + b*x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 297*a^8
*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*c^10*d^6 + 8937*a^11*b^11*c^9*d^7 - 12663*a^12*b^10*c
^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 9855*a^14*b^8*c^6*d^10 + 5400*a^15*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^12 + 4
86*a^17*b^5*c^3*d^13 - 54*a^18*b^4*c^2*d^14) - ((3^(1/2)*1i - 1)^2*(-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^
3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(2/3)*(243*a^10*b^15*c^15*d^3 - 2916*a^11*b^14*c^14*d^4
+ 15795*a^12*b^13*c^13*d^5 - 51030*a^13*b^12*c^12*d^6 + 109350*a^14*b^11*c^11*d^7 - 163296*a^15*b^10*c^10*d^8
+ 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c^8*d^10 + 69255*a^18*b^7*c^7*d^11 - 24300*a^19*b^6*c^6*d^12 + 510
3*a^20*b^5*c^5*d^13 - 486*a^21*b^4*c^4*d^14))/4))/2 - 9*a^7*b^14*c^11*d^4 + 90*a^8*b^13*c^10*d^5 - 405*a^9*b^1
2*c^9*d^6 + 1071*a^10*b^11*c^8*d^7 - 1827*a^11*b^10*c^7*d^8 + 2079*a^12*b^9*c^6*d^9 - 1575*a^13*b^8*c^5*d^10 +
765*a^14*b^7*c^4*d^11 - 216*a^15*b^6*c^3*d^12 + 27*a^16*b^5*c^2*d^13)*(3^(1/2)*1i - 1)*(-d^4/(27*b^4*c^7 + 27
*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3))/2 - (log(((3^(1/2)*1i + 1)*(
-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3)*((a + b*
x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 297*a^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*c^10*d^6 +
8937*a^11*b^11*c^9*d^7 - 12663*a^12*b^10*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 9855*a^14*b^8*c^6*d^10 + 5400*a^15
*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^12 + 486*a^17*b^5*c^3*d^13 - 54*a^18*b^4*c^2*d^14) - ((3^(1/2)*1i + 1)^2*(
-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(2/3)*(243*a^1
0*b^15*c^15*d^3 - 2916*a^11*b^14*c^14*d^4 + 15795*a^12*b^13*c^13*d^5 - 51030*a^13*b^12*c^12*d^6 + 109350*a^14*
b^11*c^11*d^7 - 163296*a^15*b^10*c^10*d^8 + 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c^8*d^10 + 69255*a^18*b^
7*c^7*d^11 - 24300*a^19*b^6*c^6*d^12 + 5103*a^20*b^5*c^5*d^13 - 486*a^21*b^4*c^4*d^14))/4))/2 + 9*a^7*b^14*c^1
1*d^4 - 90*a^8*b^13*c^10*d^5 + 405*a^9*b^12*c^9*d^6 - 1071*a^10*b^11*c^8*d^7 + 1827*a^11*b^10*c^7*d^8 - 2079*a
^12*b^9*c^6*d^9 + 1575*a^13*b^8*c^5*d^10 - 765*a^14*b^7*c^4*d^11 + 216*a^15*b^6*c^3*d^12 - 27*a^16*b^5*c^2*d^1
3)*(3^(1/2)*1i + 1)*(-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c
^6*d))^(1/3))/2 - b/((a + b*x^3)^(1/3)*(a^2*d - a*b*c)) + log(((a + b*x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 297*a
^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*c^10*d^6 + 8937*a^11*b^11*c^9*d^7 - 12663*a^12*b^10
*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 9855*a^14*b^8*c^6*d^10 + 5400*a^15*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^12 +
486*a^17*b^5*c^3*d^13 - 54*a^18*b^4*c^2*d^14) - ((3^(1/2)*1i)/2 - 1/2)^2*(1/(27*a^4*c^3))^(2/3)*(243*a^10*b^1
5*c^15*d^3 - 2916*a^11*b^14*c^14*d^4 + 15795*a^12*b^13*c^13*d^5 - 51030*a^13*b^12*c^12*d^6 + 109350*a^14*b^11*
c^11*d^7 - 163296*a^15*b^10*c^10*d^8 + 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c^8*d^10 + 69255*a^18*b^7*c^7
*d^11 - 24300*a^19*b^6*c^6*d^12 + 5103*a^20*b^5*c^5*d^13 - 486*a^21*b^4*c^4*d^14))*((3^(1/2)*1i)/2 - 1/2)*(1/(
27*a^4*c^3))^(1/3) - 9*a^7*b^14*c^11*d^4 + 90*a^8*b^13*c^10*d^5 - 405*a^9*b^12*c^9*d^6 + 1071*a^10*b^11*c^8*d^
7 - 1827*a^11*b^10*c^7*d^8 + 2079*a^12*b^9*c^6*d^9 - 1575*a^13*b^8*c^5*d^10 + 765*a^14*b^7*c^4*d^11 - 216*a^15
*b^6*c^3*d^12 + 27*a^16*b^5*c^2*d^13)*((3^(1/2)*1i)/2 - 1/2)*(1/(27*a^4*c^3))^(1/3) - log(((a + b*x^3)^(1/3)*(
27*a^7*b^15*c^13*d^3 - 297*a^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*c^10*d^6 + 8937*a^11*b^
11*c^9*d^7 - 12663*a^12*b^10*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 9855*a^14*b^8*c^6*d^10 + 5400*a^15*b^7*c^5*d^1
1 - 2052*a^16*b^6*c^4*d^12 + 486*a^17*b^5*c^3*d^13 - 54*a^18*b^4*c^2*d^14) - ((3^(1/2)*1i)/2 + 1/2)^2*(1/(27*a
^4*c^3))^(2/3)*(243*a^10*b^15*c^15*d^3 - 2916*a^11*b^14*c^14*d^4 + 15795*a^12*b^13*c^13*d^5 - 51030*a^13*b^12*
c^12*d^6 + 109350*a^14*b^11*c^11*d^7 - 163296*a^15*b^10*c^10*d^8 + 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c
^8*d^10 + 69255*a^18*b^7*c^7*d^11 - 24300*a^19*b^6*c^6*d^12 + 5103*a^20*b^5*c^5*d^13 - 486*a^21*b^4*c^4*d^14))
*((3^(1/2)*1i)/2 + 1/2)*(1/(27*a^4*c^3))^(1/3) + 9*a^7*b^14*c^11*d^4 - 90*a^8*b^13*c^10*d^5 + 405*a^9*b^12*c^9
*d^6 - 1071*a^10*b^11*c^8*d^7 + 1827*a^11*b^10*c^7*d^8 - 2079*a^12*b^9*c^6*d^9 + 1575*a^13*b^8*c^5*d^10 - 765*
a^14*b^7*c^4*d^11 + 216*a^15*b^6*c^3*d^12 - 27*a^16*b^5*c^2*d^13)*((3^(1/2)*1i)/2 + 1/2)*(1/(27*a^4*c^3))^(1/3
)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a + b x^{3}\right )^{\frac {4}{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)**(4/3)/(d*x**3+c),x)

[Out]

Integral(1/(x*(a + b*x**3)**(4/3)*(c + d*x**3)), x)

________________________________________________________________________________________