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

Optimal. Leaf size=193 \[ \frac {\left (6561 b^2-728 a^6\right ) \log \left (\sqrt [3]{a x^2+x^3}-x\right )}{6561}+\frac {\left (728 a^6-6561 b^2\right ) \log \left (x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}+x^2\right )}{13122}+\frac {\left (728 \sqrt {3} a^6-6561 \sqrt {3} b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{a x^2+x^3}+x}\right )}{6561}+\frac {\left (a x^2+x^3\right )^{2/3} \left (-7280 a^5+5460 a^4 x-4680 a^3 x^2+4212 a^2 x^3-3888 a x^4+3645 x^5\right )}{21870 x} \]

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Rubi [B]  time = 0.42, antiderivative size = 416, normalized size of antiderivative = 2.16, number of steps used = 12, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2053, 2011, 59, 2024} \begin {gather*} -\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {728 a^6 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(182*a^4*(a*x^2 + x^3)^(2/3))/729 - (728*a^5*(a*x^2 + x^3)^(2/3))/(2187*x) - (52*a^3*x*(a*x^2 + x^3)^(2/3))/24
3 + (26*a^2*x^2*(a*x^2 + x^3)^(2/3))/135 - (8*a*x^3*(a*x^2 + x^3)^(2/3))/45 + (x^4*(a*x^2 + x^3)^(2/3))/6 - (7
28*a^6*x^(2/3)*(a + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2187*Sqrt[3]*(a*x^2 + x
^3)^(1/3)) + (Sqrt[3]*b^2*x^(2/3)*(a + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(a*x^
2 + x^3)^(1/3) - (364*a^6*x^(2/3)*(a + x)^(1/3)*Log[x])/(6561*(a*x^2 + x^3)^(1/3)) + (b^2*x^(2/3)*(a + x)^(1/3
)*Log[x])/(2*(a*x^2 + x^3)^(1/3)) - (364*a^6*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)])/(2187*(a*x
^2 + x^3)^(1/3)) + (3*b^2*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)])/(2*(a*x^2 + x^3)^(1/3))

Rule 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2024

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

Rule 2053

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx &=\int \left (-\frac {b^2}{\sqrt [3]{a x^2+x^3}}+\frac {x^6}{\sqrt [3]{a x^2+x^3}}\right ) \, dx\\ &=-\left (b^2 \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx\right )+\int \frac {x^6}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}-\frac {1}{9} (8 a) \int \frac {x^5}{\sqrt [3]{a x^2+x^3}} \, dx-\frac {\left (b^2 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{\sqrt [3]{a x^2+x^3}}\\ &=-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {1}{135} \left (104 a^2\right ) \int \frac {x^4}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {1}{81} \left (52 a^3\right ) \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {1}{729} \left (364 a^4\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (728 a^5\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx}{2187}\\ &=\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (728 a^6\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx}{6561}\\ &=\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (728 a^6 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{6561 \sqrt [3]{a x^2+x^3}}\\ &=\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^6 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 192, normalized size = 0.99 \begin {gather*} \frac {3 x \sqrt [3]{\frac {a+x}{a}} \left (a^6 \, _2F_1\left (-\frac {17}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-6 a^6 \, _2F_1\left (-\frac {14}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+15 a^6 \, _2F_1\left (-\frac {11}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-20 a^6 \, _2F_1\left (-\frac {8}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+15 a^6 \, _2F_1\left (-\frac {5}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-6 a^6 \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+a^6 \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-b^2 \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )\right )}{\sqrt [3]{x^2 (a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*((a + x)/a)^(1/3)*(a^6*Hypergeometric2F1[-17/3, 1/3, 4/3, -(x/a)] - 6*a^6*Hypergeometric2F1[-14/3, 1/3, 4
/3, -(x/a)] + 15*a^6*Hypergeometric2F1[-11/3, 1/3, 4/3, -(x/a)] - 20*a^6*Hypergeometric2F1[-8/3, 1/3, 4/3, -(x
/a)] + 15*a^6*Hypergeometric2F1[-5/3, 1/3, 4/3, -(x/a)] - 6*a^6*Hypergeometric2F1[-2/3, 1/3, 4/3, -(x/a)] + a^
6*Hypergeometric2F1[1/3, 1/3, 4/3, -(x/a)] - b^2*Hypergeometric2F1[1/3, 1/3, 4/3, -(x/a)]))/(x^2*(a + x))^(1/3
)

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IntegrateAlgebraic [A]  time = 0.78, size = 193, normalized size = 1.00 \begin {gather*} \frac {\left (a x^2+x^3\right )^{2/3} \left (-7280 a^5+5460 a^4 x-4680 a^3 x^2+4212 a^2 x^3-3888 a x^4+3645 x^5\right )}{21870 x}+\frac {\left (728 \sqrt {3} a^6-6561 \sqrt {3} b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a x^2+x^3}}\right )}{6561}+\frac {\left (-728 a^6+6561 b^2\right ) \log \left (-x+\sqrt [3]{a x^2+x^3}\right )}{6561}+\frac {\left (728 a^6-6561 b^2\right ) \log \left (x^2+x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}\right )}{13122} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*x^2 + x^3)^(2/3)*(-7280*a^5 + 5460*a^4*x - 4680*a^3*x^2 + 4212*a^2*x^3 - 3888*a*x^4 + 3645*x^5))/(21870*x)
+ ((728*Sqrt[3]*a^6 - 6561*Sqrt[3]*b^2)*ArcTan[(Sqrt[3]*x)/(x + 2*(a*x^2 + x^3)^(1/3))])/6561 + ((-728*a^6 +
6561*b^2)*Log[-x + (a*x^2 + x^3)^(1/3)])/6561 + ((728*a^6 - 6561*b^2)*Log[x^2 + x*(a*x^2 + x^3)^(1/3) + (a*x^2
+ x^3)^(2/3)])/13122

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fricas [A]  time = 0.52, size = 185, normalized size = 0.96 \begin {gather*} -\frac {10 \, \sqrt {3} {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 10 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \log \left (-\frac {x - {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \log \left (\frac {x^{2} + {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}} x + {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (7280 \, a^{5} - 5460 \, a^{4} x + 4680 \, a^{3} x^{2} - 4212 \, a^{2} x^{3} + 3888 \, a x^{4} - 3645 \, x^{5}\right )} {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{65610 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/65610*(10*sqrt(3)*(728*a^6 - 6561*b^2)*x*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(a*x^2 + x^3)^(1/3))/x) + 10*(72
8*a^6 - 6561*b^2)*x*log(-(x - (a*x^2 + x^3)^(1/3))/x) - 5*(728*a^6 - 6561*b^2)*x*log((x^2 + (a*x^2 + x^3)^(1/3
)*x + (a*x^2 + x^3)^(2/3))/x^2) + 3*(7280*a^5 - 5460*a^4*x + 4680*a^3*x^2 - 4212*a^2*x^3 + 3888*a*x^4 - 3645*x
^5)*(a*x^2 + x^3)^(2/3))/x

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giac [A]  time = 0.61, size = 197, normalized size = 1.02 \begin {gather*} -\frac {10 \, \sqrt {3} {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 5 \, {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \log \left ({\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \log \left ({\left | {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) + \frac {3 \, {\left (7280 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} - 41860 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} + 99320 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} - 123812 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 84592 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 29165 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )} x^{6}}{a^{6}}}{65610 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/65610*(10*sqrt(3)*(728*a^7 - 6561*a*b^2)*arctan(1/3*sqrt(3)*(2*(a/x + 1)^(1/3) + 1)) - 5*(728*a^7 - 6561*a*
b^2)*log((a/x + 1)^(2/3) + (a/x + 1)^(1/3) + 1) + 10*(728*a^7 - 6561*a*b^2)*log(abs((a/x + 1)^(1/3) - 1)) + 3*
(7280*a^7*(a/x + 1)^(17/3) - 41860*a^7*(a/x + 1)^(14/3) + 99320*a^7*(a/x + 1)^(11/3) - 123812*a^7*(a/x + 1)^(8
/3) + 84592*a^7*(a/x + 1)^(5/3) - 29165*a^7*(a/x + 1)^(2/3))*x^6/a^6)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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