3.27 \(\int (a-b x^3) (a+b x^3)^{2/3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{7 a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 \sqrt [3]{b}}+\frac{7 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} \sqrt [3]{b}}+\frac{7}{18} a x \left (a+b x^3\right )^{2/3}-\frac{1}{6} x \left (a+b x^3\right )^{5/3} \]

[Out]

(7*a*x*(a + b*x^3)^(2/3))/18 - (x*(a + b*x^3)^(5/3))/6 + (7*a^2*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/S
qrt[3]])/(9*Sqrt[3]*b^(1/3)) - (7*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(18*b^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0322832, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {388, 195, 239} \[ -\frac{7 a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 \sqrt [3]{b}}+\frac{7 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} \sqrt [3]{b}}+\frac{7}{18} a x \left (a+b x^3\right )^{2/3}-\frac{1}{6} x \left (a+b x^3\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)*(a + b*x^3)^(2/3),x]

[Out]

(7*a*x*(a + b*x^3)^(2/3))/18 - (x*(a + b*x^3)^(5/3))/6 + (7*a^2*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/S
qrt[3]])/(9*Sqrt[3]*b^(1/3)) - (7*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(18*b^(1/3))

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rubi steps

\begin{align*} \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx &=-\frac{1}{6} x \left (a+b x^3\right )^{5/3}+\frac{1}{6} (7 a) \int \left (a+b x^3\right )^{2/3} \, dx\\ &=\frac{7}{18} a x \left (a+b x^3\right )^{2/3}-\frac{1}{6} x \left (a+b x^3\right )^{5/3}+\frac{1}{9} \left (7 a^2\right ) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\\ &=\frac{7}{18} a x \left (a+b x^3\right )^{2/3}-\frac{1}{6} x \left (a+b x^3\right )^{5/3}+\frac{7 a^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{9 \sqrt{3} \sqrt [3]{b}}-\frac{7 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 \sqrt [3]{b}}\\ \end{align*}

Mathematica [C]  time = 0.066334, size = 62, normalized size = 0.55 \[ \frac{1}{6} x \left (a+b x^3\right )^{2/3} \left (\frac{7 a \, _2F_1\left (-\frac{2}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (\frac{b x^3}{a}+1\right )^{2/3}}-a-b x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x^3)^(2/3)*(-a - b*x^3 + (7*a*Hypergeometric2F1[-2/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(2/3)
))/6

________________________________________________________________________________________

Maple [F]  time = 0.295, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{3}+a \right ) \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.66543, size = 1022, normalized size = 9.12 \begin{align*} \left [\frac{21 \, \sqrt{\frac{1}{3}} a^{2} b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} b x^{3} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + 2 \, a\right ) - 14 \, a^{2} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 7 \, a^{2} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 3 \,{\left (3 \, b^{2} x^{4} - 4 \, a b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{54 \, b}, -\frac{42 \, \sqrt{\frac{1}{3}} a^{2} b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} x - 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}{x}\right ) + 14 \, a^{2} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - 7 \, a^{2} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 3 \,{\left (3 \, b^{2} x^{4} - 4 \, a b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{54 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(21*sqrt(1/3)*a^2*b*sqrt((-b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((
-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) - 14*a
^2*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 7*a^2*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(
1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) - 3*(3*b^2*x^4 - 4*a*b*x)*(b*x^3 + a)^(2/3))/b, -1/54*(42*sqrt(1/3
)*a^2*b*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 14
*a^2*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - 7*a^2*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)
^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) + 3*(3*b^2*x^4 - 4*a*b*x)*(b*x^3 + a)^(2/3))/b]

________________________________________________________________________________________

Sympy [C]  time = 5.08027, size = 80, normalized size = 0.71 \begin{align*} \frac{a^{\frac{5}{3}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} - \frac{a^{\frac{2}{3}} b x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(5/3)*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) - a**(2/3)*b*x**4*ga
mma(4/3)*hyper((-2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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