3.509 \(\int x^2 (a+b \log (c (d+\frac{e}{x^{2/3}})^n)) \, dx\)

Optimal. Leaf size=121 \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{2 b e^2 n x^{5/3}}{15 d^2}-\frac{2 b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{2 b e^3 n x}{9 d^3}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 d^{9/2}}+\frac{2 b e n x^{7/3}}{21 d} \]

[Out]

(-2*b*e^4*n*x^(1/3))/(3*d^4) + (2*b*e^3*n*x)/(9*d^3) - (2*b*e^2*n*x^(5/3))/(15*d^2) + (2*b*e*n*x^(7/3))/(21*d)
+ (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*d^(9/2)) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/3

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Rubi [A]  time = 0.0718126, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {2455, 263, 341, 302, 205} \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{2 b e^2 n x^{5/3}}{15 d^2}-\frac{2 b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{2 b e^3 n x}{9 d^3}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 d^{9/2}}+\frac{2 b e n x^{7/3}}{21 d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*(d + e/x^(2/3))^n]),x]

[Out]

(-2*b*e^4*n*x^(1/3))/(3*d^4) + (2*b*e^3*n*x)/(9*d^3) - (2*b*e^2*n*x^(5/3))/(15*d^2) + (2*b*e*n*x^(7/3))/(21*d)
+ (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*d^(9/2)) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/3

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 263

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

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

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

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+\frac{1}{9} (2 b e n) \int \frac{x^{4/3}}{d+\frac{e}{x^{2/3}}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+\frac{1}{9} (2 b e n) \int \frac{x^2}{e+d x^{2/3}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \frac{x^8}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \left (-\frac{e^3}{d^4}+\frac{e^2 x^2}{d^3}-\frac{e x^4}{d^2}+\frac{x^6}{d}+\frac{e^4}{d^4 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{2 b e^3 n x}{9 d^3}-\frac{2 b e^2 n x^{5/3}}{15 d^2}+\frac{2 b e n x^{7/3}}{21 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{2 b e^3 n x}{9 d^3}-\frac{2 b e^2 n x^{5/3}}{15 d^2}+\frac{2 b e n x^{7/3}}{21 d}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )\\ \end{align*}

Mathematica [C]  time = 0.0153182, size = 65, normalized size = 0.54 \[ \frac{a x^3}{3}+\frac{1}{3} b x^3 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )+\frac{2 b e n x^{7/3} \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};-\frac{e}{d x^{2/3}}\right )}{21 d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*(d + e/x^(2/3))^n]),x]

[Out]

(a*x^3)/3 + (2*b*e*n*x^(7/3)*Hypergeometric2F1[-7/2, 1, -5/2, -(e/(d*x^(2/3)))])/(21*d) + (b*x^3*Log[c*(d + e/
x^(2/3))^n])/3

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Maple [F]  time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*(d+e/x^(2/3))^n)),x)

[Out]

int(x^2*(a+b*ln(c*(d+e/x^(2/3))^n)),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^2*(a+b*log(c*(d+e/x^(2/3))^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.07102, size = 971, normalized size = 8.02 \begin{align*} \left [\frac{105 \, b d^{4} x^{3} \log \left (c\right ) + 105 \, a d^{4} x^{3} - 42 \, b d^{2} e^{2} n x^{\frac{5}{3}} + 105 \, b e^{4} n \sqrt{-\frac{e}{d}} \log \left (\frac{d^{3} x^{2} - 2 \, d^{2} e x \sqrt{-\frac{e}{d}} - e^{3} + 2 \,{\left (d^{3} x \sqrt{-\frac{e}{d}} + d e^{2}\right )} x^{\frac{2}{3}} - 2 \,{\left (d^{2} e x - d e^{2} \sqrt{-\frac{e}{d}}\right )} x^{\frac{1}{3}}}{d^{3} x^{2} + e^{3}}\right ) + 70 \, b d e^{3} n x + 105 \, b d^{4} n \log \left (d x^{\frac{2}{3}} + e\right ) - 210 \, b d^{4} n \log \left (x^{\frac{1}{3}}\right ) + 105 \,{\left (b d^{4} n x^{3} - b d^{4} n\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right ) + 30 \,{\left (b d^{3} e n x^{2} - 7 \, b e^{4} n\right )} x^{\frac{1}{3}}}{315 \, d^{4}}, \frac{105 \, b d^{4} x^{3} \log \left (c\right ) + 105 \, a d^{4} x^{3} - 42 \, b d^{2} e^{2} n x^{\frac{5}{3}} + 210 \, b e^{4} n \sqrt{\frac{e}{d}} \arctan \left (\frac{d x^{\frac{1}{3}} \sqrt{\frac{e}{d}}}{e}\right ) + 70 \, b d e^{3} n x + 105 \, b d^{4} n \log \left (d x^{\frac{2}{3}} + e\right ) - 210 \, b d^{4} n \log \left (x^{\frac{1}{3}}\right ) + 105 \,{\left (b d^{4} n x^{3} - b d^{4} n\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right ) + 30 \,{\left (b d^{3} e n x^{2} - 7 \, b e^{4} n\right )} x^{\frac{1}{3}}}{315 \, d^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(2/3))^n)),x, algorithm="fricas")

[Out]

[1/315*(105*b*d^4*x^3*log(c) + 105*a*d^4*x^3 - 42*b*d^2*e^2*n*x^(5/3) + 105*b*e^4*n*sqrt(-e/d)*log((d^3*x^2 -
2*d^2*e*x*sqrt(-e/d) - e^3 + 2*(d^3*x*sqrt(-e/d) + d*e^2)*x^(2/3) - 2*(d^2*e*x - d*e^2*sqrt(-e/d))*x^(1/3))/(d
^3*x^2 + e^3)) + 70*b*d*e^3*n*x + 105*b*d^4*n*log(d*x^(2/3) + e) - 210*b*d^4*n*log(x^(1/3)) + 105*(b*d^4*n*x^3
- b*d^4*n)*log((d*x + e*x^(1/3))/x) + 30*(b*d^3*e*n*x^2 - 7*b*e^4*n)*x^(1/3))/d^4, 1/315*(105*b*d^4*x^3*log(c
) + 105*a*d^4*x^3 - 42*b*d^2*e^2*n*x^(5/3) + 210*b*e^4*n*sqrt(e/d)*arctan(d*x^(1/3)*sqrt(e/d)/e) + 70*b*d*e^3*
n*x + 105*b*d^4*n*log(d*x^(2/3) + e) - 210*b*d^4*n*log(x^(1/3)) + 105*(b*d^4*n*x^3 - b*d^4*n)*log((d*x + e*x^(
1/3))/x) + 30*(b*d^3*e*n*x^2 - 7*b*e^4*n)*x^(1/3))/d^4]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*(d+e/x**(2/3))**n)),x)

[Out]

Timed out

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Giac [A]  time = 1.36602, size = 131, normalized size = 1.08 \begin{align*} \frac{1}{3} \, b x^{3} \log \left (c\right ) + \frac{1}{3} \, a x^{3} + \frac{1}{315} \,{\left (105 \, x^{3} \log \left (d + \frac{e}{x^{\frac{2}{3}}}\right ) + 2 \,{\left (\frac{105 \, \arctan \left (\sqrt{d} x^{\frac{1}{3}} e^{\left (-\frac{1}{2}\right )}\right ) e^{\frac{7}{2}}}{d^{\frac{9}{2}}} + \frac{15 \, d^{6} x^{\frac{7}{3}} - 21 \, d^{5} x^{\frac{5}{3}} e + 35 \, d^{4} x e^{2} - 105 \, d^{3} x^{\frac{1}{3}} e^{3}}{d^{7}}\right )} e\right )} b n \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(2/3))^n)),x, algorithm="giac")

[Out]

1/3*b*x^3*log(c) + 1/3*a*x^3 + 1/315*(105*x^3*log(d + e/x^(2/3)) + 2*(105*arctan(sqrt(d)*x^(1/3)*e^(-1/2))*e^(
7/2)/d^(9/2) + (15*d^6*x^(7/3) - 21*d^5*x^(5/3)*e + 35*d^4*x*e^2 - 105*d^3*x^(1/3)*e^3)/d^7)*e)*b*n