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

Optimal. Leaf size=139 \[ \frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{38 a^3 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{81 \sqrt [3]{b}}+\frac{76 a^3 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{81 \sqrt{3} \sqrt [3]{b}}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3} \]

[Out]

(38*a^2*x*(a + b*x^3)^(2/3))/81 - (8*a*x*(a + b*x^3)^(5/3))/27 - (x*(a - b*x^3)*(a + b*x^3)^(5/3))/9 + (76*a^3
*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]*b^(1/3)) - (38*a^3*Log[-(b^(1/3)*x) + (a +
b*x^3)^(1/3)])/(81*b^(1/3))

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Rubi [A]  time = 0.0566985, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {416, 388, 195, 239} \[ \frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{38 a^3 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{81 \sqrt [3]{b}}+\frac{76 a^3 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{81 \sqrt{3} \sqrt [3]{b}}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(38*a^2*x*(a + b*x^3)^(2/3))/81 - (8*a*x*(a + b*x^3)^(5/3))/27 - (x*(a - b*x^3)*(a + b*x^3)^(5/3))/9 + (76*a^3
*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]*b^(1/3)) - (38*a^3*Log[-(b^(1/3)*x) + (a +
b*x^3)^(1/3)])/(81*b^(1/3))

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 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 )^2 \left (a+b x^3\right )^{2/3} \, dx &=-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{\int \left (a+b x^3\right )^{2/3} \left (10 a^2 b-16 a b^2 x^3\right ) \, dx}{9 b}\\ &=-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{1}{27} \left (38 a^2\right ) \int \left (a+b x^3\right )^{2/3} \, dx\\ &=\frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{1}{81} \left (76 a^3\right ) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\\ &=\frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{76 a^3 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{81 \sqrt{3} \sqrt [3]{b}}-\frac{38 a^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{81 \sqrt [3]{b}}\\ \end{align*}

Mathematica [A]  time = 0.137231, size = 151, normalized size = 1.09 \[ \frac{1}{243} \left (3 \left (a+b x^3\right )^{2/3} \left (5 a^2 x-24 a b x^4+9 b^2 x^7\right )+\frac{38 a^3 \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{\sqrt [3]{b}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x^3)^(2/3)*(5*a^2*x - 24*a*b*x^4 + 9*b^2*x^7) + (38*a^3*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*
x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b
^(1/3)*x)/(a + b*x^3)^(1/3)]))/b^(1/3))/243

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Maple [F]  time = 0.233, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{3}+a \right ) ^{2} \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)^2*(b*x^3+a)^(2/3),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.63047, size = 1076, normalized size = 7.74 \begin{align*} \left [\frac{114 \, \sqrt{\frac{1}{3}} a^{3} 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 ) - 76 \, a^{3} \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 ) + 38 \, a^{3} \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 (9 \, b^{3} x^{7} - 24 \, a b^{2} x^{4} + 5 \, a^{2} b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{243 \, b}, -\frac{228 \, \sqrt{\frac{1}{3}} a^{3} 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 ) + 76 \, a^{3} \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 ) - 38 \, a^{3} \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 (9 \, b^{3} x^{7} - 24 \, a b^{2} x^{4} + 5 \, a^{2} b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{243 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/243*(114*sqrt(1/3)*a^3*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) - 76
*a^3*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 38*a^3*(-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*(9*b^3*x^7 - 24*a*b^2*x^4 + 5*a^2*b*x)*(b*x^3 + a)^(2/3))/b
, -1/243*(228*sqrt(1/3)*a^3*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) + 76*a^3*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - 38*a^3*(-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*(9*b^3*x^7 - 24*a*b^2*x^4 + 5*a^2*b*x
)*(b*x^3 + a)^(2/3))/b]

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Sympy [C]  time = 4.77737, size = 126, normalized size = 0.91 \begin{align*} \frac{a^{\frac{8}{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{2 a^{\frac{5}{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 )} + \frac{a^{\frac{2}{3}} b^{2} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(8/3)*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) - 2*a**(5/3)*b*x**4*
gamma(4/3)*hyper((-2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + a**(2/3)*b**2*x**7*gamma(7/3)
*hyper((-2/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3))

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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 )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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