3.21.53 \(\int \frac {\sqrt [3]{b-a x^6} (b+a x^6)}{x^2 (-b+c x^3+a x^6)} \, dx\)

Optimal. Leaf size=147 \[ -\frac {1}{6} \sqrt [3]{c} \log \left (\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}+c^{2/3} x^2\right )+\frac {1}{3} \sqrt [3]{c} \log \left (\sqrt [3]{b-a x^6}-\sqrt [3]{c} x\right )+\frac {\sqrt [3]{c} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{2 \sqrt [3]{b-a x^6}+\sqrt [3]{c} x}\right )}{\sqrt {3}}+\frac {\sqrt [3]{b-a x^6}}{x} \]

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Rubi [C]  time = 3.84, antiderivative size = 1003, normalized size of antiderivative = 6.82, number of steps used = 38, number of rules used = 9, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6728, 365, 364, 1562, 465, 430, 429, 511, 510} \begin {gather*} -\frac {2 a^2 c \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^5}{5 \sqrt {c^2+4 a b} \left (c^2-\sqrt {c^2+4 a b} c+2 a b\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {c^2+4 a b}}\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^5}{5 \left (2 a b+c \left (c-\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (\frac {c}{\sqrt {c^2+4 a b}}+1\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^5}{5 \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^5}{5 \sqrt {c^2+4 a b} \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^2}{\sqrt {c^2+4 a b} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1-\frac {c}{\sqrt {c^2+4 a b}}\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^2}{2 \left (2 a b+c \left (c-\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {a \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^2}{\sqrt {c^2+4 a b} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (\frac {c}{\sqrt {c^2+4 a b}}+1\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^2}{2 \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{\sqrt [3]{1-\frac {a x^6}{b}} x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)),x]

[Out]

-((a*x^2*(b - a*x^6)^(1/3)*AppellF1[1/3, 1, -1/3, 4/3, (2*a^2*x^6)/(2*a*b + c^2 - c*Sqrt[4*a*b + c^2]), (a*x^6
)/b])/(Sqrt[4*a*b + c^2]*(1 - (a*x^6)/b)^(1/3))) - (a*c*(1 - c/Sqrt[4*a*b + c^2])*x^2*(b - a*x^6)^(1/3)*Appell
F1[1/3, 1, -1/3, 4/3, (2*a^2*x^6)/(2*a*b + c^2 - c*Sqrt[4*a*b + c^2]), (a*x^6)/b])/(2*(2*a*b + c*(c - Sqrt[4*a
*b + c^2]))*(1 - (a*x^6)/b)^(1/3)) + (a*x^2*(b - a*x^6)^(1/3)*AppellF1[1/3, 1, -1/3, 4/3, (2*a^2*x^6)/(2*a*b +
c*(c + Sqrt[4*a*b + c^2])), (a*x^6)/b])/(Sqrt[4*a*b + c^2]*(1 - (a*x^6)/b)^(1/3)) - (a*c*(1 + c/Sqrt[4*a*b +
c^2])*x^2*(b - a*x^6)^(1/3)*AppellF1[1/3, 1, -1/3, 4/3, (2*a^2*x^6)/(2*a*b + c*(c + Sqrt[4*a*b + c^2])), (a*x^
6)/b])/(2*(2*a*b + c*(c + Sqrt[4*a*b + c^2]))*(1 - (a*x^6)/b)^(1/3)) - (2*a^2*c*x^5*(b - a*x^6)^(1/3)*AppellF1
[5/6, 1, -1/3, 11/6, (2*a^2*x^6)/(2*a*b + c^2 - c*Sqrt[4*a*b + c^2]), (a*x^6)/b])/(5*Sqrt[4*a*b + c^2]*(2*a*b
+ c^2 - c*Sqrt[4*a*b + c^2])*(1 - (a*x^6)/b)^(1/3)) - (2*a^2*(1 - c/Sqrt[4*a*b + c^2])*x^5*(b - a*x^6)^(1/3)*A
ppellF1[5/6, 1, -1/3, 11/6, (2*a^2*x^6)/(2*a*b + c^2 - c*Sqrt[4*a*b + c^2]), (a*x^6)/b])/(5*(2*a*b + c*(c - Sq
rt[4*a*b + c^2]))*(1 - (a*x^6)/b)^(1/3)) + (2*a^2*c*x^5*(b - a*x^6)^(1/3)*AppellF1[5/6, 1, -1/3, 11/6, (2*a^2*
x^6)/(2*a*b + c*(c + Sqrt[4*a*b + c^2])), (a*x^6)/b])/(5*Sqrt[4*a*b + c^2]*(2*a*b + c*(c + Sqrt[4*a*b + c^2]))
*(1 - (a*x^6)/b)^(1/3)) - (2*a^2*(1 + c/Sqrt[4*a*b + c^2])*x^5*(b - a*x^6)^(1/3)*AppellF1[5/6, 1, -1/3, 11/6,
(2*a^2*x^6)/(2*a*b + c*(c + Sqrt[4*a*b + c^2])), (a*x^6)/b])/(5*(2*a*b + c*(c + Sqrt[4*a*b + c^2]))*(1 - (a*x^
6)/b)^(1/3)) + ((b - a*x^6)^(1/3)*Hypergeometric2F1[-1/3, -1/6, 5/6, (a*x^6)/b])/(x*(1 - (a*x^6)/b)^(1/3))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 465

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

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

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

Rule 1562

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Dist[(f*x)^m
/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x
], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx &=\int \left (-\frac {\sqrt [3]{b-a x^6}}{x^2}+\frac {x \left (c+2 a x^3\right ) \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6}\right ) \, dx\\ &=-\int \frac {\sqrt [3]{b-a x^6}}{x^2} \, dx+\int \frac {x \left (c+2 a x^3\right ) \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6} \, dx\\ &=-\frac {\sqrt [3]{b-a x^6} \int \frac {\sqrt [3]{1-\frac {a x^6}{b}}}{x^2} \, dx}{\sqrt [3]{1-\frac {a x^6}{b}}}+\int \left (\frac {c x \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6}+\frac {2 a x^4 \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+(2 a) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6} \, dx+c \int \frac {x \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6} \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+(2 a) \int \left (\frac {\left (-c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c-\sqrt {4 a b+c^2}+2 a x^3\right )}+\frac {\left (c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c+\sqrt {4 a b+c^2}+2 a x^3\right )}\right ) \, dx+c \int \left (\frac {2 a x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c-\sqrt {4 a b+c^2}+2 a x^3\right )}-\frac {2 a x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c+\sqrt {4 a b+c^2}+2 a x^3\right )}\right ) \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {(2 a c) \int \frac {x \sqrt [3]{b-a x^6}}{c-\sqrt {4 a b+c^2}+2 a x^3} \, dx}{\sqrt {4 a b+c^2}}-\frac {(2 a c) \int \frac {x \sqrt [3]{b-a x^6}}{c+\sqrt {4 a b+c^2}+2 a x^3} \, dx}{\sqrt {4 a b+c^2}}+\left (2 a \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{c-\sqrt {4 a b+c^2}+2 a x^3} \, dx+\left (2 a \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{c+\sqrt {4 a b+c^2}+2 a x^3} \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {(2 a c) \int \left (\frac {\left (c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx}{\sqrt {4 a b+c^2}}+\frac {(2 a c) \int \left (\frac {\left (-c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx}{\sqrt {4 a b+c^2}}+\left (2 a \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \left (\frac {\left (-c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx+\left (2 a \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \left (\frac {\left (c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {\left (2 a^2 c\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}+\frac {\left (2 a^2 c\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}+\left (2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx+\left (a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx+\left (2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx-\left (a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6} \, dx+\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right )^2\right ) \int \frac {x \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}+\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right )^2\right ) \int \frac {x \sqrt [3]{b-a x^6}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {1}{2} \left (a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )-\frac {1}{2} \left (a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )+\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right )^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2}}+\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right )^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2}}-\frac {\left (2 a^2 c \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (2 a^2 c \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt [3]{1-\frac {a x^6}{b}}}\\ &=-\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c^2-c \sqrt {4 a b+c^2}\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {\left (a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right )^2 \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right )^2 \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}\\ &=-\frac {a x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{2 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {a x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{2 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c^2-c \sqrt {4 a b+c^2}\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}\\ \end {align*}

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Mathematica [F]  time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)),x]

[Out]

Integrate[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)), x]

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IntegrateAlgebraic [A]  time = 2.10, size = 147, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{b-a x^6}}{x}+\frac {\sqrt [3]{c} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{\sqrt [3]{c} x+2 \sqrt [3]{b-a x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{c} \log \left (-\sqrt [3]{c} x+\sqrt [3]{b-a x^6}\right )-\frac {1}{6} \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)),x]

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*x^6+b)^(1/3)*(a*x^6+b)/x^2/(a*x^6+c*x^3-b),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + b\right )} {\left (-a x^{6} + b\right )}^{\frac {1}{3}}}{{\left (a x^{6} + c x^{3} - b\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*x^6+b)^(1/3)*(a*x^6+b)/x^2/(a*x^6+c*x^3-b),x, algorithm="giac")

[Out]

integrate((a*x^6 + b)*(-a*x^6 + b)^(1/3)/((a*x^6 + c*x^3 - b)*x^2), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-a \,x^{6}+b \right )^{\frac {1}{3}} \left (a \,x^{6}+b \right )}{x^{2} \left (a \,x^{6}+x^{3} c -b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*x^6+b)^(1/3)*(a*x^6+b)/x^2/(a*x^6+c*x^3-b),x)

[Out]

int((-a*x^6+b)^(1/3)*(a*x^6+b)/x^2/(a*x^6+c*x^3-b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + b\right )} {\left (-a x^{6} + b\right )}^{\frac {1}{3}}}{{\left (a x^{6} + c x^{3} - b\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*x^6+b)^(1/3)*(a*x^6+b)/x^2/(a*x^6+c*x^3-b),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)*(-a*x^6 + b)^(1/3)/((a*x^6 + c*x^3 - b)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a\,x^6+b\right )\,{\left (b-a\,x^6\right )}^{1/3}}{x^2\,\left (a\,x^6+c\,x^3-b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + a*x^6)*(b - a*x^6)^(1/3))/(x^2*(a*x^6 - b + c*x^3)),x)

[Out]

int(((b + a*x^6)*(b - a*x^6)^(1/3))/(x^2*(a*x^6 - b + c*x^3)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*x**6+b)**(1/3)*(a*x**6+b)/x**2/(a*x**6+c*x**3-b),x)

[Out]

Timed out

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