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

Optimal. Leaf size=180 \[ \frac {1}{27} \left (-2 a^2+3 a b+3 a c-9 b c\right ) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {1}{27} \left (2 \sqrt {3} a^2-3 \sqrt {3} a b-3 \sqrt {3} a c+9 \sqrt {3} b c\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{a+x^3}+x}\right )+\frac {1}{54} \left (2 a^2-3 a b-3 a c+9 b c\right ) \log \left (x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}+x^2\right )+\frac {1}{18} \left (a+x^3\right )^{2/3} \left (-4 a x+6 b x+6 c x+3 x^4\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 124, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {528, 388, 239} \begin {gather*} -\frac {1}{18} \left (2 a^2-3 a (b+c)+9 b c\right ) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {\left (2 a^2-3 a (b+c)+9 b c\right ) \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{18} x \left (a+x^3\right )^{2/3} (4 a-3 b-6 c)+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*((4*a - 3*b - 6*c)*x*(a + x^3)^(2/3)) + (x*(a + x^3)^(2/3)*(b + x^3))/6 + ((2*a^2 + 9*b*c - 3*a*(b + c))
*ArcTan[(1 + (2*x)/(a + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - ((2*a^2 + 9*b*c - 3*a*(b + c))*Log[-x + (a + x^3)^
(1/3)])/18

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 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 ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx &=\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{6} \int \frac {-b (a-6 c)-(4 a-3 b-6 c) x^3}{\sqrt [3]{a+x^3}} \, dx\\ &=-\frac {1}{18} (4 a-3 b-6 c) x \left (a+x^3\right )^{2/3}+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} \left (2 a^2+9 b c-3 a (b+c)\right ) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx\\ &=-\frac {1}{18} (4 a-3 b-6 c) x \left (a+x^3\right )^{2/3}+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {\left (2 a^2+9 b c-3 a (b+c)\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{18} \left (2 a^2+9 b c-3 a (b+c)\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 259, normalized size = 1.44 \begin {gather*} \frac {1}{54} \left (\left (-4 a^2+6 a (b+c)-18 b c\right ) \log \left (1-\frac {x}{\sqrt [3]{a+x^3}}\right )+2 \sqrt {3} \left (2 a^2-3 a (b+c)+9 b c\right ) \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )+2 a^2 \log \left (\frac {x}{\sqrt [3]{a+x^3}}+\frac {x^2}{\left (a+x^3\right )^{2/3}}+1\right )+9 b c \log \left (\frac {x}{\sqrt [3]{a+x^3}}+\frac {x^2}{\left (a+x^3\right )^{2/3}}+1\right )+18 b x \left (a+x^3\right )^{2/3}-3 a b \log \left (\frac {x}{\sqrt [3]{a+x^3}}+\frac {x^2}{\left (a+x^3\right )^{2/3}}+1\right )+18 c x \left (a+x^3\right )^{2/3}-3 a c \log \left (\frac {x}{\sqrt [3]{a+x^3}}+\frac {x^2}{\left (a+x^3\right )^{2/3}}+1\right )-12 a x \left (a+x^3\right )^{2/3}+9 x^4 \left (a+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*a*x*(a + x^3)^(2/3) + 18*b*x*(a + x^3)^(2/3) + 18*c*x*(a + x^3)^(2/3) + 9*x^4*(a + x^3)^(2/3) + 2*Sqrt[3]
*(2*a^2 + 9*b*c - 3*a*(b + c))*ArcTan[(1 + (2*x)/(a + x^3)^(1/3))/Sqrt[3]] + (-4*a^2 - 18*b*c + 6*a*(b + c))*L
og[1 - x/(a + x^3)^(1/3)] + 2*a^2*Log[1 + x^2/(a + x^3)^(2/3) + x/(a + x^3)^(1/3)] - 3*a*b*Log[1 + x^2/(a + x^
3)^(2/3) + x/(a + x^3)^(1/3)] - 3*a*c*Log[1 + x^2/(a + x^3)^(2/3) + x/(a + x^3)^(1/3)] + 9*b*c*Log[1 + x^2/(a
+ x^3)^(2/3) + x/(a + x^3)^(1/3)])/54

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IntegrateAlgebraic [A]  time = 0.71, size = 180, normalized size = 1.00 \begin {gather*} \frac {1}{18} \left (a+x^3\right )^{2/3} \left (-4 a x+6 b x+6 c x+3 x^4\right )+\frac {1}{27} \left (2 \sqrt {3} a^2-3 \sqrt {3} a b-3 \sqrt {3} a c+9 \sqrt {3} b c\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{27} \left (-2 a^2+3 a b+3 a c-9 b c\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{54} \left (2 a^2-3 a b-3 a c+9 b c\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)*(c + x^3))/(a + x^3)^(1/3),x]

[Out]

((a + x^3)^(2/3)*(-4*a*x + 6*b*x + 6*c*x + 3*x^4))/18 + ((2*Sqrt[3]*a^2 - 3*Sqrt[3]*a*b - 3*Sqrt[3]*a*c + 9*Sq
rt[3]*b*c)*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))])/27 + ((-2*a^2 + 3*a*b + 3*a*c - 9*b*c)*Log[-x + (a + x
^3)^(1/3)])/27 + ((2*a^2 - 3*a*b - 3*a*c + 9*b*c)*Log[x^2 + x*(a + x^3)^(1/3) + (a + x^3)^(2/3)])/54

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fricas [A]  time = 0.66, size = 158, normalized size = 0.88 \begin {gather*} -\frac {1}{27} \, \sqrt {3} {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{27} \, {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{54} \, {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\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}{18} \, {\left (3 \, x^{4} - 2 \, {\left (2 \, a - 3 \, b - 3 \, c\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)*(x^3+c)/(x^3+a)^(1/3),x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*(2*a^2 - 3*a*b - 3*(a - 3*b)*c)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + a)^(1/3))/x) - 1/27*(2*
a^2 - 3*a*b - 3*(a - 3*b)*c)*log(-(x - (x^3 + a)^(1/3))/x) + 1/54*(2*a^2 - 3*a*b - 3*(a - 3*b)*c)*log((x^2 + (
x^3 + a)^(1/3)*x + (x^3 + a)^(2/3))/x^2) + 1/18*(3*x^4 - 2*(2*a - 3*b - 3*c)*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 )} {\left (x^{3} + c\right )}}{{\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)*(x^3+c)/(x^3+a)^(1/3),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.43, size = 412, normalized size = 2.29 \begin {gather*} -\frac {2}{27} \, \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 ) - \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 c + \frac {1}{27} \, 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 ) - \frac {2}{27} \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{18} \, {\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 + \frac {1}{18} \, {\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 )} c - \frac {\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}}}{18 \, {\left (\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*sqrt(3)*a^2*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*c + 1/27
*a^2*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) - 2/27*a^2*log((x^3 + a)^(1/3)/x - 1) + 1/18*(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 + 1/18*(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)))*c - 1/18*(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)

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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 (x^3+c\right )}{{\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)*(c + x^3))/(a + x^3)^(1/3),x)

[Out]

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

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sympy [C]  time = 3.68, size = 153, normalized size = 0.85 \begin {gather*} \frac {b c 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 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 )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {c 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 )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {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 )}}{3 \sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*c*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*x**4*gamma(4/3)
*hyper((1/3, 4/3), (7/3,), x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(7/3)) + c*x**4*gamma(4/3)*hyper((1/3, 4/3
), (7/3,), x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(7/3)) + x**7*gamma(7/3)*hyper((1/3, 7/3), (10/3,), x**3*e
xp_polar(I*pi)/a)/(3*a**(1/3)*gamma(10/3))

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