3.27.60 \(\int \frac {(-4 b+a x^3) \sqrt [3]{b+a x^3}}{x^5 (-2 b+a x^3)} \, dx\)

Optimal. Leaf size=238 \[ -\frac {a^{4/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{a x^3+b}-3 \sqrt [3]{a} x\right )}{2 \sqrt [3]{2} 3^{2/3} b}-\frac {a^{4/3} \tan ^{-1}\left (\frac {3^{5/6} \sqrt [3]{a} x}{2 \sqrt [3]{2} \sqrt [3]{a x^3+b}+\sqrt [3]{3} \sqrt [3]{a} x}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}+\frac {a^{4/3} \log \left (3 a^{2/3} x^2+\sqrt [3]{2} 3^{2/3} \sqrt [3]{a} x \sqrt [3]{a x^3+b}+2^{2/3} \sqrt [3]{3} \left (a x^3+b\right )^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b}+\frac {\left (-2 a x^3-b\right ) \sqrt [3]{a x^3+b}}{2 b x^4} \]

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Rubi [A]  time = 0.35, antiderivative size = 226, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {580, 583, 12, 494, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {a^{4/3} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{a x^3+b}}\right )}{2 \sqrt [3]{2} 3^{2/3} b}-\frac {a^{4/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a} x}{\sqrt [6]{3} \sqrt [3]{a x^3+b}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}+\frac {a^{4/3} \log \left (\frac {3^{2/3} a^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac {\sqrt [3]{6} \sqrt [3]{a} x}{\sqrt [3]{a x^3+b}}+2^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b}-\frac {a \sqrt [3]{a x^3+b}}{b x}-\frac {\sqrt [3]{a x^3+b}}{2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4*b + a*x^3)*(b + a*x^3)^(1/3))/(x^5*(-2*b + a*x^3)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 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 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {\int \frac {16 a b^2+4 a^2 b x^3}{x^2 \left (-2 b+a x^3\right ) \left (b+a x^3\right )^{2/3}} \, dx}{8 b}\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {\int \frac {24 a^2 b^3 x}{\left (-2 b+a x^3\right ) \left (b+a x^3\right )^{2/3}} \, dx}{16 b^3}\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {1}{2} \left (3 a^2\right ) \int \frac {x}{\left (-2 b+a x^3\right ) \left (b+a x^3\right )^{2/3}} \, dx\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {x}{-2 b+3 a b x^3} \, dx,x,\frac {x}{\sqrt [3]{b+a x^3}}\right )\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {a^{5/3} \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{2} \sqrt [3]{b}+\sqrt [3]{3} \sqrt [3]{a} \sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{6} b^{2/3}}+\frac {a^{5/3} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{2} \sqrt [3]{b}+\sqrt [3]{3} \sqrt [3]{a} \sqrt [3]{b} x}{2^{2/3} b^{2/3}+\sqrt [3]{6} \sqrt [3]{a} b^{2/3} x+3^{2/3} a^{2/3} b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{6} b^{2/3}}\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {a^{4/3} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{6} \sqrt [3]{a} b^{2/3}+2\ 3^{2/3} a^{2/3} b^{2/3} x}{2^{2/3} b^{2/3}+\sqrt [3]{6} \sqrt [3]{a} b^{2/3} x+3^{2/3} a^{2/3} b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{b+a x^3}}\right )}{4 \sqrt [3]{2} 3^{2/3} b}-\frac {\left (3^{2/3} a^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} b^{2/3}+\sqrt [3]{6} \sqrt [3]{a} b^{2/3} x+3^{2/3} a^{2/3} b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{b+a x^3}}\right )}{4 \sqrt [3]{b}}\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {a^{4/3} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (2^{2/3}+\frac {3^{2/3} a^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{6} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}\right )}{4 \sqrt [3]{2} 3^{2/3} b}+\frac {\left (\sqrt [3]{\frac {3}{2}} a^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}\right )}{2 b}\\ &=-\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {a^{4/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}-\frac {a^{4/3} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (2^{2/3}+\frac {3^{2/3} a^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{6} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}\right )}{4 \sqrt [3]{2} 3^{2/3} b}\\ \end {align*}

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Mathematica [C]  time = 0.24, size = 108, normalized size = 0.45 \begin {gather*} \frac {\frac {3 a^2 x^6 \left (\frac {2 a x^3}{b}+2\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {3 a x^3}{a x^3-2 b}\right )}{\left (2-\frac {a x^3}{b}\right )^{2/3}}-4 \left (2 a^2 x^6+3 a b x^3+b^2\right )}{8 b x^4 \left (a x^3+b\right )^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-4*b + a*x^3)*(b + a*x^3)^(1/3))/(x^5*(-2*b + a*x^3)),x]

[Out]

(-4*(b^2 + 3*a*b*x^3 + 2*a^2*x^6) + (3*a^2*x^6*(2 + (2*a*x^3)/b)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (3*a*x
^3)/(-2*b + a*x^3)])/(2 - (a*x^3)/b)^(2/3))/(8*b*x^4*(b + a*x^3)^(2/3))

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IntegrateAlgebraic [A]  time = 0.57, size = 238, normalized size = 1.00 \begin {gather*} \frac {\left (-b-2 a x^3\right ) \sqrt [3]{b+a x^3}}{2 b x^4}-\frac {a^{4/3} \tan ^{-1}\left (\frac {3^{5/6} \sqrt [3]{a} x}{\sqrt [3]{3} \sqrt [3]{a} x+2 \sqrt [3]{2} \sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}-\frac {a^{4/3} \log \left (-3 \sqrt [3]{a} x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (3 a^{2/3} x^2+\sqrt [3]{2} 3^{2/3} \sqrt [3]{a} x \sqrt [3]{b+a x^3}+2^{2/3} \sqrt [3]{3} \left (b+a x^3\right )^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4*b + a*x^3)*(b + a*x^3)^(1/3))/(x^5*(-2*b + a*x^3)),x]

[Out]

((-b - 2*a*x^3)*(b + a*x^3)^(1/3))/(2*b*x^4) - (a^(4/3)*ArcTan[(3^(5/6)*a^(1/3)*x)/(3^(1/3)*a^(1/3)*x + 2*2^(1
/3)*(b + a*x^3)^(1/3))])/(2*2^(1/3)*3^(1/6)*b) - (a^(4/3)*Log[-3*a^(1/3)*x + 2^(1/3)*3^(2/3)*(b + a*x^3)^(1/3)
])/(2*2^(1/3)*3^(2/3)*b) + (a^(4/3)*Log[3*a^(2/3)*x^2 + 2^(1/3)*3^(2/3)*a^(1/3)*x*(b + a*x^3)^(1/3) + 2^(2/3)*
3^(1/3)*(b + a*x^3)^(2/3)])/(4*2^(1/3)*3^(2/3)*b)

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fricas [B]  time = 17.59, size = 418, normalized size = 1.76 \begin {gather*} -\frac {2 \cdot 18^{\frac {2}{3}} \sqrt {3} \left (-a\right )^{\frac {1}{3}} a x^{4} \arctan \left (\frac {4 \cdot 18^{\frac {2}{3}} \sqrt {3} {\left (4 \, a^{2} x^{7} - 7 \, a b x^{4} - 2 \, b^{2} x\right )} {\left (a x^{3} + b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} + 6 \cdot 18^{\frac {1}{3}} \sqrt {3} {\left (55 \, a^{2} x^{8} + 50 \, a b x^{5} + 4 \, b^{2} x^{2}\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} {\left (377 \, a^{3} x^{9} + 600 \, a^{2} b x^{6} + 204 \, a b^{2} x^{3} + 8 \, b^{3}\right )}}{3 \, {\left (487 \, a^{3} x^{9} + 480 \, a^{2} b x^{6} + 12 \, a b^{2} x^{3} - 8 \, b^{3}\right )}}\right ) - 2 \cdot 18^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} a x^{4} \log \left (-\frac {3 \cdot 18^{\frac {2}{3}} {\left (a x^{3} + b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a x^{2} + 18 \, {\left (a x^{3} + b\right )}^{\frac {2}{3}} a x + 18^{\frac {1}{3}} {\left (a x^{3} - 2 \, b\right )} \left (-a\right )^{\frac {2}{3}}}{18 \, {\left (a x^{3} - 2 \, b\right )}}\right ) + 18^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} a x^{4} \log \left (\frac {36 \cdot 18^{\frac {1}{3}} {\left (4 \, a x^{4} + b x\right )} {\left (a x^{3} + b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - 18^{\frac {2}{3}} {\left (55 \, a^{2} x^{6} + 50 \, a b x^{3} + 4 \, b^{2}\right )} \left (-a\right )^{\frac {1}{3}} + 54 \, {\left (7 \, a^{2} x^{5} + 4 \, a b x^{2}\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}}}{18 \, {\left (a^{2} x^{6} - 4 \, a b x^{3} + 4 \, b^{2}\right )}}\right ) + 108 \, {\left (2 \, a x^{3} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}}}{216 \, b x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3-4*b)*(a*x^3+b)^(1/3)/x^5/(a*x^3-2*b),x, algorithm="fricas")

[Out]

-1/216*(2*18^(2/3)*sqrt(3)*(-a)^(1/3)*a*x^4*arctan(1/3*(4*18^(2/3)*sqrt(3)*(4*a^2*x^7 - 7*a*b*x^4 - 2*b^2*x)*(
a*x^3 + b)^(2/3)*(-a)^(1/3) + 6*18^(1/3)*sqrt(3)*(55*a^2*x^8 + 50*a*b*x^5 + 4*b^2*x^2)*(a*x^3 + b)^(1/3)*(-a)^
(2/3) + sqrt(3)*(377*a^3*x^9 + 600*a^2*b*x^6 + 204*a*b^2*x^3 + 8*b^3))/(487*a^3*x^9 + 480*a^2*b*x^6 + 12*a*b^2
*x^3 - 8*b^3)) - 2*18^(2/3)*(-a)^(1/3)*a*x^4*log(-1/18*(3*18^(2/3)*(a*x^3 + b)^(1/3)*(-a)^(1/3)*a*x^2 + 18*(a*
x^3 + b)^(2/3)*a*x + 18^(1/3)*(a*x^3 - 2*b)*(-a)^(2/3))/(a*x^3 - 2*b)) + 18^(2/3)*(-a)^(1/3)*a*x^4*log(1/18*(3
6*18^(1/3)*(4*a*x^4 + b*x)*(a*x^3 + b)^(2/3)*(-a)^(2/3) - 18^(2/3)*(55*a^2*x^6 + 50*a*b*x^3 + 4*b^2)*(-a)^(1/3
) + 54*(7*a^2*x^5 + 4*a*b*x^2)*(a*x^3 + b)^(1/3))/(a^2*x^6 - 4*a*b*x^3 + 4*b^2)) + 108*(2*a*x^3 + b)*(a*x^3 +
b)^(1/3))/(b*x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} + b\right )}^{\frac {1}{3}} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} - 2 \, b\right )} x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3-4*b)*(a*x^3+b)^(1/3)/x^5/(a*x^3-2*b),x, algorithm="giac")

[Out]

integrate((a*x^3 + b)^(1/3)*(a*x^3 - 4*b)/((a*x^3 - 2*b)*x^5), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}-4 b \right ) \left (a \,x^{3}+b \right )^{\frac {1}{3}}}{x^{5} \left (a \,x^{3}-2 b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3-4*b)*(a*x^3+b)^(1/3)/x^5/(a*x^3-2*b),x)

[Out]

int((a*x^3-4*b)*(a*x^3+b)^(1/3)/x^5/(a*x^3-2*b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} + b\right )}^{\frac {1}{3}} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} - 2 \, b\right )} x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3-4*b)*(a*x^3+b)^(1/3)/x^5/(a*x^3-2*b),x, algorithm="maxima")

[Out]

integrate((a*x^3 + b)^(1/3)*(a*x^3 - 4*b)/((a*x^3 - 2*b)*x^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^3+b\right )}^{1/3}\,\left (4\,b-a\,x^3\right )}{x^5\,\left (2\,b-a\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + a*x^3)^(1/3)*(4*b - a*x^3))/(x^5*(2*b - a*x^3)),x)

[Out]

int(((b + a*x^3)^(1/3)*(4*b - a*x^3))/(x^5*(2*b - a*x^3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{3} - 4 b\right ) \sqrt [3]{a x^{3} + b}}{x^{5} \left (a x^{3} - 2 b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3-4*b)*(a*x**3+b)**(1/3)/x**5/(a*x**3-2*b),x)

[Out]

Integral((a*x**3 - 4*b)*(a*x**3 + b)**(1/3)/(x**5*(a*x**3 - 2*b)), x)

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