3.32.27 \(\int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx\)
Optimal. Leaf size=752 \[ -\frac {3 \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (3 a^2 x^4-3 a^2 x^2+a x^3-a x-2 x^2+2\right )}{10 x^2}-\frac {\log \left (\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (\sqrt [3]{1-a} x^2-\sqrt [3]{1-a}\right )+1\right )}{2 \sqrt [3]{1-a}}+\frac {\log \left (\left (\sqrt [3]{a+1} x^2-\sqrt [3]{a+1}\right ) \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}-1\right )}{2 \sqrt [3]{a+1}}+\frac {\log \left (\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (\sqrt [3]{1-a}-\sqrt [3]{1-a} x^2\right )+\left ((1-a)^{2/3} x^4-2 (1-a)^{2/3} x^2+(1-a)^{2/3}\right ) \left (\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}\right )^{2/3}+1\right )}{4 \sqrt [3]{1-a}}-\frac {\log \left (\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (\sqrt [3]{a+1} x^2-\sqrt [3]{a+1}\right )+\left ((a+1)^{2/3} x^4-2 (a+1)^{2/3} x^2+(a+1)^{2/3}\right ) \left (\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}\right )^{2/3}+1\right )}{4 \sqrt [3]{a+1}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{1-a} \left (2 x^2-2\right ) \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1-a}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a+1} \left (2 x^2-2\right ) \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a+1}} \]
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Rubi [A] time = 0.54, antiderivative size = 609, normalized size of antiderivative = 0.81,
number of steps used = 12, number of rules used = 7, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used
= {6688, 6718, 912, 129, 155, 12, 91} \begin {gather*} -\frac {3 (5-3 a) \left (1-x^2\right ) (a x+1) \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}}}{20 x}+\frac {3 (3 a+5) \left (1-x^2\right ) (a x+1) \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}}}{20 x}-\frac {3 \left (1-x^2\right ) (a x+1) \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}}}{5 x^2}+\frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log (1-x)}{4 \sqrt [3]{a+1} \sqrt [3]{x}}+\frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log (x+1)}{4 \sqrt [3]{a-1} \sqrt [3]{x}}-\frac {3 \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a-1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a-1} \sqrt [3]{x}}-\frac {3 \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a+1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a+1} \sqrt [3]{x}}-\frac {\sqrt {3} \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a-1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a-1} \sqrt [3]{x}}-\frac {\sqrt {3} \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a+1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a+1} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)/x^3,x]
[Out]
(-3*(1 + a*x)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2))/(5*x^2) - (3*(5 - 3*a)*(1 + a*x)*(-(x/((1 + a*x)
*(1 - x^2)^3)))^(1/3)*(1 - x^2))/(20*x) + (3*(5 + 3*a)*(1 + a*x)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2
))/(20*x) - (Sqrt[3]*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*ArcTan[1/Sqrt[3] + (2*(1 +
a*x)^(1/3))/(Sqrt[3]*(-1 + a)^(1/3)*x^(1/3))])/(2*(-1 + a)^(1/3)*x^(1/3)) - (Sqrt[3]*(1 + a*x)^(1/3)*(-(x/((1
+ a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*ArcTan[1/Sqrt[3] + (2*(1 + a*x)^(1/3))/(Sqrt[3]*(1 + a)^(1/3)*x^(1/3))]
)/(2*(1 + a)^(1/3)*x^(1/3)) + ((1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[1 - x])/(4*(
1 + a)^(1/3)*x^(1/3)) + ((1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[1 + x])/(4*(-1 + a
)^(1/3)*x^(1/3)) - (3*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[-x^(1/3) + (1 + a*x)^
(1/3)/(-1 + a)^(1/3)])/(4*(-1 + a)^(1/3)*x^(1/3)) - (3*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1
- x^2)*Log[-x^(1/3) + (1 + a*x)^(1/3)/(1 + a)^(1/3)])/(4*(1 + a)^(1/3)*x^(1/3))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 91
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Rule 129
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && S
umSimplerQ[p, 1])))
Rule 155
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1])))
Rule 912
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] && !IntegerQ[m] && !IntegerQ[n]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6718
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracP
art[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !FreeQ[z, x]
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx &=\int \frac {\sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}}}{x^3} \, dx\\ &=\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{x^{8/3} \sqrt [3]{1+a x} \left (-1+x^2\right )} \, dx}{\sqrt [3]{x}}\\ &=\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \left (-\frac {1}{2 (1-x) x^{8/3} \sqrt [3]{1+a x}}-\frac {1}{2 x^{8/3} (1+x) \sqrt [3]{1+a x}}\right ) \, dx}{\sqrt [3]{x}}\\ &=-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{(1-x) x^{8/3} \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{x^{8/3} (1+x) \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}\\ &=-\frac {3 (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{5 x^2}+\frac {\left (3 \sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {\frac {1}{3} (-5+3 a)-a x}{(1-x) x^{5/3} \sqrt [3]{1+a x}} \, dx}{10 \sqrt [3]{x}}+\frac {\left (3 \sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {\frac {1}{3} (5+3 a)+a x}{x^{5/3} (1+x) \sqrt [3]{1+a x}} \, dx}{10 \sqrt [3]{x}}\\ &=-\frac {3 (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{5 x^2}-\frac {3 (5-3 a) (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{20 x}+\frac {3 (5+3 a) (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{20 x}-\frac {\left (9 \sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {10}{9 (1-x) x^{2/3} \sqrt [3]{1+a x}} \, dx}{20 \sqrt [3]{x}}-\frac {\left (9 \sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {10}{9 x^{2/3} (1+x) \sqrt [3]{1+a x}} \, dx}{20 \sqrt [3]{x}}\\ &=-\frac {3 (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{5 x^2}-\frac {3 (5-3 a) (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{20 x}+\frac {3 (5+3 a) (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{20 x}-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{(1-x) x^{2/3} \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{x^{2/3} (1+x) \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}\\ &=-\frac {3 (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{5 x^2}-\frac {3 (5-3 a) (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{20 x}+\frac {3 (5+3 a) (1+a x) \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right )}{20 x}-\frac {\sqrt {3} \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+a x}}{\sqrt {3} \sqrt [3]{-1+a} \sqrt [3]{x}}\right )}{2 \sqrt [3]{-1+a} \sqrt [3]{x}}-\frac {\sqrt {3} \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+a x}}{\sqrt {3} \sqrt [3]{1+a} \sqrt [3]{x}}\right )}{2 \sqrt [3]{1+a} \sqrt [3]{x}}+\frac {\sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log (1-x)}{4 \sqrt [3]{1+a} \sqrt [3]{x}}+\frac {\sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log (1+x)}{4 \sqrt [3]{-1+a} \sqrt [3]{x}}-\frac {3 \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+a x}}{\sqrt [3]{-1+a}}\right )}{4 \sqrt [3]{-1+a} \sqrt [3]{x}}-\frac {3 \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+a x}}{\sqrt [3]{1+a}}\right )}{4 \sqrt [3]{1+a} \sqrt [3]{x}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 95, normalized size = 0.13 \begin {gather*} -\frac {3 \left (x^2-1\right ) \sqrt [3]{\frac {x}{\left (x^2-1\right )^3 (a x+1)}} \left (3 a^2 x^2+5 x^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(a-1) x}{a x+1}\right )+5 x^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(a+1) x}{a x+1}\right )+a x-2\right )}{10 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)/x^3,x]
[Out]
(-3*(x/((1 + a*x)*(-1 + x^2)^3))^(1/3)*(-1 + x^2)*(-2 + a*x + 3*a^2*x^2 + 5*x^2*Hypergeometric2F1[1/3, 1, 4/3,
((-1 + a)*x)/(1 + a*x)] + 5*x^2*Hypergeometric2F1[1/3, 1, 4/3, ((1 + a)*x)/(1 + a*x)]))/(10*x^2)
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IntegrateAlgebraic [A] time = 3.74, size = 752, normalized size = 1.00 \begin {gather*} -\frac {3 \left (2-a x-2 x^2-3 a^2 x^2+a x^3+3 a^2 x^4\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{10 x^2}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{1-a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1-a}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{1+a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1+a}}-\frac {\log \left (1+\left (-\sqrt [3]{1-a}+\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1-a}}+\frac {\log \left (-1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1+a}}+\frac {\log \left (1+\left (\sqrt [3]{1-a}-\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1-a}}-\frac {\log \left (1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)/x^3,x]
[Out]
(-3*(2 - a*x - 2*x^2 - 3*a^2*x^2 + a*x^3 + 3*a^2*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 +
a*x^7))^(1/3))/(10*x^2) + (Sqrt[3]*ArcTan[1/Sqrt[3] - ((1 - a)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*
x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(2*(1 - a)^(1/3)) - (Sqrt[3]*ArcTan[1/Sqrt[3] + ((1 + a
)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(2*(1 +
a)^(1/3)) - Log[1 + (-(1 - a)^(1/3) + (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x
^6 + a*x^7))^(1/3)]/(2*(1 - a)^(1/3)) + Log[-1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3
*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)]/(2*(1 + a)^(1/3)) + Log[1 + ((1 - a)^(1/3) - (1 - a)^(1/3)*x^2
)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 - a)^(2/3) - 2*(1 - a)^(2/3)*x^
2 + (1 - a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 - a)^(1/3
)) - Log[1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x
^7))^(1/3) + ((1 + a)^(2/3) - 2*(1 + a)^(2/3)*x^2 + (1 + a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4
- 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 + a)^(1/3))
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fricas [A] time = 0.59, size = 3627, normalized size = 4.82
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)/x^3,x, algorithm="fricas")
[Out]
[1/20*(5*sqrt(3)*(a^2 - 1)*x^2*sqrt(-1/(a - 1)^(2/3))*log(-((3*a - 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a
- 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 - a)*x^5 + (a - 1)*
x^4 - 2*(a^2 - a)*x^3 - 2*(a - 1)*x^2 + (a^2 - a)*x + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x
+ 3*x^2 - 1))^(2/3) + (a*x + 1)*(a - 1)^(1/3))*sqrt(-1/(a - 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a - 1)^(1/
3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x + 1)) + 5*sqrt(3)*(a^2 - 1)*x
^2*sqrt(-1/(a + 1)^(2/3))*log(((3*a + 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^
5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 + a)*x^5 + (a + 1)*x^4 - 2*(a^2 + a)*x^3 - 2*(a
+ 1)*x^2 + (a^2 + a)*x + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x +
1)*(a + 1)^(1/3))*sqrt(-1/(a + 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a + 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^6
+ 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x - 1)) - 5*(a + 1)^(2/3)*(a - 1)*x^2*log((x^2 - 1)*(a + 1)^
(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a
+ 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) - 5*(a + 1)*(a -
1)^(2/3)*x^2*log((x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3)
+ ((a - 1)*x^4 - 2*(a - 1)*x^2 + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3)
+ (a - 1)^(1/3)) + 10*(a + 1)^(2/3)*(a - 1)*x^2*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3
- 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (a + 1)^(2/3)) + 10*(a + 1)*(a - 1)^(2/3)*x^2*log(((a - 1)*x^2 - a + 1)*(
x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (a - 1)^(2/3)) - 6*(3*(a^4 - a^2)*x^4 +
(a^3 - a)*x^3 - (3*a^4 - a^2 - 2)*x^2 + 2*a^2 - (a^3 - a)*x - 2)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4
- a*x + 3*x^2 - 1))^(1/3))/((a^2 - 1)*x^2), 1/20*(5*sqrt(3)*(a^2 - 1)*x^2*sqrt(-1/(a - 1)^(2/3))*log(-((3*a -
2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^
2 - 1))^(1/3) - 2*((a^2 - a)*x^5 + (a - 1)*x^4 - 2*(a^2 - a)*x^3 - 2*(a - 1)*x^2 + (a^2 - a)*x + a - 1)*(x/(a*
x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x + 1)*(a - 1)^(1/3))*sqrt(-1/(a - 1)^(2/
3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a - 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^
(1/3) + 1)/(x + 1)) - 5*(a + 1)^(2/3)*(a - 1)*x^2*log((x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*
x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*
x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) - 5*(a + 1)*(a - 1)^(2/3)*x^2*log((x^2 - 1)*(a - 1)^(2/
3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a - 1)*x^4 - 2*(a - 1)*x^2 + a -
1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a - 1)^(1/3)) + 10*(a + 1)^(2/3)*(
a - 1)*x^2*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (
a + 1)^(2/3)) + 10*(a + 1)*(a - 1)^(2/3)*x^2*log(((a - 1)*x^2 - a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3
*x^4 - a*x + 3*x^2 - 1))^(1/3) - (a - 1)^(2/3)) - 10*sqrt(3)*(a^2 - 1)*x^2*arctan(1/3*sqrt(3)*(2*(x^2 - 1)*(a
+ 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + (a + 1)^(1/3))/(a + 1)^(1/3
))/(a + 1)^(1/3) - 6*(3*(a^4 - a^2)*x^4 + (a^3 - a)*x^3 - (3*a^4 - a^2 - 2)*x^2 + 2*a^2 - (a^3 - a)*x - 2)*(x/
(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3))/((a^2 - 1)*x^2), 1/20*(5*sqrt(3)*(a^2 - 1)
*x^2*sqrt(-1/(a + 1)^(2/3))*log(((3*a + 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*
x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 + a)*x^5 + (a + 1)*x^4 - 2*(a^2 + a)*x^3 - 2*(
a + 1)*x^2 + (a^2 + a)*x + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x
+ 1)*(a + 1)^(1/3))*sqrt(-1/(a + 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a + 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^
6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x - 1)) - 5*(a + 1)^(2/3)*(a - 1)*x^2*log((x^2 - 1)*(a + 1
)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 +
a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) - 5*(a + 1)*(a
- 1)^(2/3)*x^2*log((x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/
3) + ((a - 1)*x^4 - 2*(a - 1)*x^2 + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/
3) + (a - 1)^(1/3)) + 10*(a + 1)^(2/3)*(a - 1)*x^2*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x
^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (a + 1)^(2/3)) + 10*(a + 1)*(a - 1)^(2/3)*x^2*log(((a - 1)*x^2 - a + 1)
*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (a - 1)^(2/3)) - 10*sqrt(3)*(a^2 - 1)
*x^2*arctan(1/3*sqrt(3)*(2*(x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 -
1))^(1/3) + (a - 1)^(1/3))/(a - 1)^(1/3))/(a - 1)^(1/3) - 6*(3*(a^4 - a^2)*x^4 + (a^3 - a)*x^3 - (3*a^4 - a^2
- 2)*x^2 + 2*a^2 - (a^3 - a)*x - 2)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3))/((
a^2 - 1)*x^2), -1/20*(5*(a + 1)^(2/3)*(a - 1)*x^2*log((x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*
x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*
x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) + 5*(a + 1)*(a - 1)^(2/3)*x^2*log((x^2 - 1)*(a - 1)^(2/
3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a - 1)*x^4 - 2*(a - 1)*x^2 + a -
1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a - 1)^(1/3)) - 10*(a + 1)^(2/3)*(
a - 1)*x^2*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (
a + 1)^(2/3)) - 10*(a + 1)*(a - 1)^(2/3)*x^2*log(((a - 1)*x^2 - a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3
*x^4 - a*x + 3*x^2 - 1))^(1/3) - (a - 1)^(2/3)) + 10*sqrt(3)*(a^2 - 1)*x^2*arctan(1/3*sqrt(3)*(2*(x^2 - 1)*(a
+ 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + (a + 1)^(1/3))/(a + 1)^(1/3
))/(a + 1)^(1/3) + 10*sqrt(3)*(a^2 - 1)*x^2*arctan(1/3*sqrt(3)*(2*(x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5
+ x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + (a - 1)^(1/3))/(a - 1)^(1/3))/(a - 1)^(1/3) + 6*(3*(a^4 -
a^2)*x^4 + (a^3 - a)*x^3 - (3*a^4 - a^2 - 2)*x^2 + 2*a^2 - (a^3 - a)*x - 2)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^
3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3))/((a^2 - 1)*x^2)]
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)/x^3,x, algorithm="giac")
[Out]
integrate((x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3)/x^3, x)
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {x}{a \,x^{7}-3 a \,x^{5}+x^{6}+3 a \,x^{3}-3 x^{4}-a x +3 x^{2}-1}\right )^{\frac {1}{3}}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)/x^3,x)
[Out]
int((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)/x^3,x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)/x^3,x, algorithm="maxima")
[Out]
integrate((x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3)/x^3, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (-\frac {x}{-a\,x^7-x^6+3\,a\,x^5+3\,x^4-3\,a\,x^3-3\,x^2+a\,x+1}\right )}^{1/3}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x/(a*x - 3*a*x^3 + 3*a*x^5 - a*x^7 - 3*x^2 + 3*x^4 - x^6 + 1))^(1/3)/x^3,x)
[Out]
int((-x/(a*x - 3*a*x^3 + 3*a*x^5 - a*x^7 - 3*x^2 + 3*x^4 - x^6 + 1))^(1/3)/x^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{\frac {x}{a x^{7} - 3 a x^{5} + 3 a x^{3} - a x + x^{6} - 3 x^{4} + 3 x^{2} - 1}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x**7-3*a*x**5+x**6+3*a*x**3-3*x**4-a*x+3*x**2-1))**(1/3)/x**3,x)
[Out]
Integral((x/(a*x**7 - 3*a*x**5 + 3*a*x**3 - a*x + x**6 - 3*x**4 + 3*x**2 - 1))**(1/3)/x**3, x)
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