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

Optimal. Leaf size=212 \[ \frac {1}{162} \left (a+x^3\right )^{2/3} \left (28 a^2 x-108 a b x-21 a x^4+162 b^2 x+81 b x^4+18 x^7\right )+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {1}{243} \left (-14 \sqrt {3} a^3+54 \sqrt {3} a^2 b-81 \sqrt {3} a b^2+81 \sqrt {3} b^3\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{a+x^3}+x}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}+x^2\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 171, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {416, 528, 388, 239} \begin {gather*} \frac {1}{162} x \left (28 a^2-87 a b+99 b^2\right ) \left (a+x^3\right )^{2/3}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (\sqrt [3]{a+x^3}-x\right )-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {1}{54} x (7 a-15 b) \left (a+x^3\right )^{2/3} \left (b+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + x^3)^3/(a + x^3)^(1/3),x]

[Out]

((28*a^2 - 87*a*b + 99*b^2)*x*(a + x^3)^(2/3))/162 - ((7*a - 15*b)*x*(a + x^3)^(2/3)*(b + x^3))/54 + (x*(a + x
^3)^(2/3)*(b + x^3)^2)/9 - ((14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*ArcTan[(1 + (2*x)/(a + x^3)^(1/3))/Sqrt[3]
])/(81*Sqrt[3]) + ((14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*Log[-x + (a + x^3)^(1/3)])/162

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 388

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

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 528

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

Rubi steps

\begin {align*} \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx &=\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{9} \int \frac {\left (b+x^3\right ) \left (-((a-9 b) b)+(-7 a+15 b) x^3\right )}{\sqrt [3]{a+x^3}} \, dx\\ &=-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{54} \int \frac {b \left (7 a^2-21 a b+54 b^2\right )+\left (28 a^2-87 a b+99 b^2\right ) x^3}{\sqrt [3]{a+x^3}} \, dx\\ &=\frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{81} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx\\ &=\frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 5.15, size = 151, normalized size = 0.71 \begin {gather*} \frac {1}{486} \left (3 x \left (a+x^3\right )^{2/3} \left (28 a^2-3 a \left (36 b+7 x^3\right )+9 \left (18 b^2+9 b x^3+2 x^6\right )\right )+\left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \left (-2 \log \left (1-\frac {x}{\sqrt [3]{a+x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {x}{\sqrt [3]{a+x^3}}+\frac {x^2}{\left (a+x^3\right )^{2/3}}+1\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(a + x^3)^(2/3)*(28*a^2 - 3*a*(36*b + 7*x^3) + 9*(18*b^2 + 9*b*x^3 + 2*x^6)) + (-14*a^3 + 54*a^2*b - 81*a
*b^2 + 81*b^3)*(2*Sqrt[3]*ArcTan[(1 + (2*x)/(a + x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - x/(a + x^3)^(1/3)] + Log[1 +
x^2/(a + x^3)^(2/3) + x/(a + x^3)^(1/3)]))/486

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IntegrateAlgebraic [A]  time = 1.24, size = 212, normalized size = 1.00 \begin {gather*} \frac {1}{162} \left (a+x^3\right )^{2/3} \left (28 a^2 x-108 a b x+162 b^2 x-21 a x^4+81 b x^4+18 x^7\right )+\frac {1}{243} \left (-14 \sqrt {3} a^3+54 \sqrt {3} a^2 b-81 \sqrt {3} a b^2+81 \sqrt {3} b^3\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + x^3)^(2/3)*(28*a^2*x - 108*a*b*x + 162*b^2*x - 21*a*x^4 + 81*b*x^4 + 18*x^7))/162 + ((-14*Sqrt[3]*a^3 +
54*Sqrt[3]*a^2*b - 81*Sqrt[3]*a*b^2 + 81*Sqrt[3]*b^3)*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))])/243 + ((14*
a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*Log[-x + (a + x^3)^(1/3)])/243 + ((-14*a^3 + 54*a^2*b - 81*a*b^2 + 81*b^3)
*Log[x^2 + x*(a + x^3)^(1/3) + (a + x^3)^(2/3)])/486

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fricas [A]  time = 0.96, size = 190, normalized size = 0.90 \begin {gather*} \frac {1}{243} \, \sqrt {3} {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{243} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{486} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (\frac {x^{2} + {\left (x^{3} + a\right )}^{\frac {1}{3}} x + {\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{162} \, {\left (18 \, x^{7} - 3 \, {\left (7 \, a - 27 \, b\right )} x^{4} + 2 \, {\left (14 \, a^{2} - 54 \, a b + 81 \, b^{2}\right )} x\right )} {\left (x^{3} + a\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/243*sqrt(3)*(14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + a)^(1/3))/x) +
1/243*(14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*log(-(x - (x^3 + a)^(1/3))/x) - 1/486*(14*a^3 - 54*a^2*b + 81*a*
b^2 - 81*b^3)*log((x^2 + (x^3 + a)^(1/3)*x + (x^3 + a)^(2/3))/x^2) + 1/162*(18*x^7 - 3*(7*a - 27*b)*x^4 + 2*(1
4*a^2 - 54*a*b + 81*b^2)*x)*(x^3 + a)^(2/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.60, size = 480, normalized size = 2.26 \begin {gather*} \frac {14}{243} \, \sqrt {3} a^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {1}{6} \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right )\right )} b^{3} - \frac {7}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {14}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{6} \, {\left (2 \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {6 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a}{x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}}\right )} b^{2} - \frac {1}{18} \, {\left (4 \, \sqrt {3} a^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - 2 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 4 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {3 \, {\left (\frac {7 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{2}}{x^{2}} - \frac {4 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1}\right )} b + \frac {\frac {67 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{x^{2}} - \frac {77 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{3}}{x^{5}} + \frac {28 \, {\left (x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}}{162 \, {\left (\frac {3 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} + a\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} + a\right )}^{3}}{x^{9}} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14/243*sqrt(3)*a^3*arctan(1/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - 1/6*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 +
a)^(1/3)/x + 1)) - log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 2*log((x^3 + a)^(1/3)/x - 1))*b^3 - 7/2
43*a^3*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 14/243*a^3*log((x^3 + a)^(1/3)/x - 1) + 1/6*(2*sqrt(
3)*a*arctan(1/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - a*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 2*a*
log((x^3 + a)^(1/3)/x - 1) + 6*(x^3 + a)^(2/3)*a/(x^2*((x^3 + a)/x^3 - 1)))*b^2 - 1/18*(4*sqrt(3)*a^2*arctan(1
/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - 2*a^2*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 4*a^2*log((x^
3 + a)^(1/3)/x - 1) + 3*(7*(x^3 + a)^(2/3)*a^2/x^2 - 4*(x^3 + a)^(5/3)*a^2/x^5)/(2*(x^3 + a)/x^3 - (x^3 + a)^2
/x^6 - 1))*b + 1/162*(67*(x^3 + a)^(2/3)*a^3/x^2 - 77*(x^3 + a)^(5/3)*a^3/x^5 + 28*(x^3 + a)^(8/3)*a^3/x^8)/(3
*(x^3 + a)/x^3 - 3*(x^3 + a)^2/x^6 + (x^3 + a)^3/x^9 - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + x^3)^3/(a + x^3)^(1/3),x)

[Out]

int((b + x^3)^3/(a + x^3)^(1/3), x)

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sympy [C]  time = 3.99, size = 151, normalized size = 0.71 \begin {gather*} \frac {b^{3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {b^{2} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} + \frac {x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {13}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(4/3)) + b**2*x**4*gamma(
4/3)*hyper((1/3, 4/3), (7/3,), x**3*exp_polar(I*pi)/a)/(a**(1/3)*gamma(7/3)) + b*x**7*gamma(7/3)*hyper((1/3, 7
/3), (10/3,), x**3*exp_polar(I*pi)/a)/(a**(1/3)*gamma(10/3)) + x**10*gamma(10/3)*hyper((1/3, 10/3), (13/3,), x
**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(13/3))

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