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

Optimal. Leaf size=91 \[ -\frac{1}{3} x \left (a+b x^3\right )^{2/3}-\frac{2 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}+\frac{4 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0195956, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {388, 239} \[ -\frac{1}{3} x \left (a+b x^3\right )^{2/3}-\frac{2 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}+\frac{4 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)/(a + b*x^3)^(1/3),x]

[Out]

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

Mathematica [A]  time = 0.0746889, size = 134, normalized size = 1.47 \[ \frac{2 a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-3 \sqrt [3]{b} x \left (a+b x^3\right )^{2/3}-4 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+4 \sqrt{3} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.229, size = 0, normalized size = 0. \begin{align*} \int{(-b{x}^{3}+a){\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.71535, size = 950, normalized size = 10.44 \begin{align*} \left [\frac{6 \, \sqrt{\frac{1}{3}} a 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 ) - 3 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b x - 4 \, a \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 ) + 2 \, a \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 )}{9 \, b}, -\frac{12 \, \sqrt{\frac{1}{3}} a 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 ) + 3 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b x + 4 \, a \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 ) - 2 \, a \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 )}{9 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/9*(6*sqrt(1/3)*a*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) - 3*(b*x^3
+ a)^(2/3)*b*x - 4*a*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 2*a*(-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))/b, -1/9*(12*sqrt(1/3)*a*b*sqrt(-(-b)^(1/3)/b)*ar
ctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 3*(b*x^3 + a)^(2/3)*b*x + 4*a*(-
b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - 2*a*(-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))/b]

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Sympy [C]  time = 3.37955, size = 76, normalized size = 0.84 \begin{align*} \frac{a^{\frac{2}{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{b x^{3} e^{i \pi }}{a}} \right )}}{3 \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{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \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)**(1/3),x)

[Out]

a**(2/3)*x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) - b*x**4*gamma(4/3)*h
yper((1/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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