3.2700 \(\int \frac{x^m}{\sqrt [3]{a+b x^{3 (1+m)}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0473381, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {345, 239} \[ \frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{m+1}}{\sqrt [3]{a+b x^{3 (m+1)}}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} (m+1)}-\frac{\log \left (\sqrt [3]{b} x^{m+1}-\sqrt [3]{a+b x^{3 (m+1)}}\right )}{2 \sqrt [3]{b} (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

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

Mathematica [C]  time = 0.0350741, size = 67, normalized size = 0.69 \[ \frac{x^{m+1} \sqrt [3]{\frac{b x^{3 m+3}}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^{3 m+3}}{a}\right )}{(m+1) \sqrt [3]{a+b x^{3 m+3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt [3]{a+b{x}^{3+3\,m}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 3.91478, size = 117, normalized size = 1.21 \begin{align*} \frac{x x^{m} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{m}{3 \left (m + 1\right )} + 1 + \frac{1}{3 \left (m + 1\right )} \end{matrix}\middle |{\frac{b x^{3} x^{3 m} e^{i \pi }}{a}} \right )}}{3 a^{\frac{m}{3 \left (m + 1\right )}} a^{\frac{1}{3 \left (m + 1\right )}} m \Gamma \left (\frac{m}{3 \left (m + 1\right )} + 1 + \frac{1}{3 \left (m + 1\right )}\right ) + 3 a^{\frac{m}{3 \left (m + 1\right )}} a^{\frac{1}{3 \left (m + 1\right )}} \Gamma \left (\frac{m}{3 \left (m + 1\right )} + 1 + \frac{1}{3 \left (m + 1\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*x**m*gamma(1/3)*hyper((1/3, 1/3), (m/(3*(m + 1)) + 1 + 1/(3*(m + 1)),), b*x**3*x**(3*m)*exp_polar(I*pi)/a)/(
3*a**(m/(3*(m + 1)))*a**(1/(3*(m + 1)))*m*gamma(m/(3*(m + 1)) + 1 + 1/(3*(m + 1))) + 3*a**(m/(3*(m + 1)))*a**(
1/(3*(m + 1)))*gamma(m/(3*(m + 1)) + 1 + 1/(3*(m + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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