3.568 \(\int \frac{x^4}{(a+b x^3)^{2/3}} \, dx\)

Optimal. Leaf size=97 \[ \frac{a \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{5/3}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}+\frac{x^2 \sqrt [3]{a+b x^3}}{3 b} \]

[Out]

(x^2*(a + b*x^3)^(1/3))/(3*b) + (2*a*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(5/3)
) + (a*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/(3*b^(5/3))

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Rubi [A]  time = 0.0688983, antiderivative size = 148, normalized size of antiderivative = 1.53, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {321, 331, 292, 31, 634, 617, 204, 628} \[ \frac{2 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}-\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{9 b^{5/3}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}+\frac{x^2 \sqrt [3]{a+b x^3}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^3)^(2/3),x]

[Out]

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

Rule 321

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 292

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac{(2 a) \int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx}{3 b}\\ &=\frac{x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{x}{1-b x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 b}\\ &=\frac{x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{b} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3}}\\ &=\frac{x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac{2 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{4/3}}\\ &=\frac{x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac{2 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}-\frac{a \log \left (1+\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}}\right )}{9 b^{5/3}}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{5/3}}\\ &=\frac{x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac{2 a \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}+\frac{2 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}-\frac{a \log \left (1+\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}}\right )}{9 b^{5/3}}\\ \end{align*}

Mathematica [C]  time = 0.0097423, size = 53, normalized size = 0.55 \[ \frac{x^2 \left (-a \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right )+a+b x^3\right )}{3 b \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a + b*x^3 - a*Hypergeometric2F1[2/3, 1, 5/3, (b*x^3)/(a + b*x^3)]))/(3*b*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^3+a)^(2/3),x)

[Out]

int(x^4/(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(x^4/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.55896, size = 448, normalized size = 4.62 \begin{align*} \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{2} x^{2} - 2 \, \sqrt{3} a{\left (b^{2}\right )}^{\frac{1}{6}} b \arctan \left (\frac{{\left (\sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}\right )}{\left (b^{2}\right )}^{\frac{1}{6}}}{3 \, b^{2} x}\right ) + 2 \, a{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (-\frac{{\left (b^{2}\right )}^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) - a{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (\frac{{\left (b^{2}\right )}^{\frac{1}{3}} b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right )}{9 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

1/9*(3*(b*x^3 + a)^(1/3)*b^2*x^2 - 2*sqrt(3)*a*(b^2)^(1/6)*b*arctan(1/3*(sqrt(3)*(b^2)^(1/3)*b*x + 2*sqrt(3)*(
b*x^3 + a)^(1/3)*(b^2)^(2/3))*(b^2)^(1/6)/(b^2*x)) + 2*a*(b^2)^(2/3)*log(-((b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b
)/x) - a*(b^2)^(2/3)*log(((b^2)^(1/3)*b*x^2 + (b*x^3 + a)^(1/3)*(b^2)^(2/3)*x + (b*x^3 + a)^(2/3)*b)/x^2))/b^3

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Sympy [C]  time = 1.6814, size = 37, normalized size = 0.38 \begin{align*} \frac{x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{8}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**3+a)**(2/3),x)

[Out]

x**5*gamma(5/3)*hyper((2/3, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*gamma(8/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^3 + a)^(2/3), x)