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

Optimal. Leaf size=65 \[ a x+b x \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-\frac{2 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}+\frac{2 b e n \sqrt [3]{x}}{d} \]

[Out]

(2*b*e*n*x^(1/3))/d + a*x - (2*b*e^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/d^(3/2) + b*x*Log[c*(d + e/x^(2/
3))^n]

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Rubi [A]  time = 0.0417612, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2448, 263, 243, 321, 205} \[ a x+b x \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-\frac{2 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}+\frac{2 b e n \sqrt [3]{x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e/x^(2/3))^n],x]

[Out]

(2*b*e*n*x^(1/3))/d + a*x - (2*b*e^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/d^(3/2) + b*x*Log[c*(d + e/x^(2/
3))^n]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

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 243

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

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

Mathematica [C]  time = 0.0133864, size = 53, normalized size = 0.82 \[ a x+b x \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )+\frac{2 b e n \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{e}{d x^{2/3}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.204, size = 168, normalized size = 2.6 \begin{align*} ax+xb\ln \left ( c \left ({ \left ( e+d{x}^{{\frac{2}{3}}} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) +{\frac{2\,b{e}^{2}n}{3\,d}\arctan \left ({\frac{x{d}^{2}}{e}{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+2\,{\frac{enb\sqrt [3]{x}}{d}}-{\frac{2\,b{e}^{2}n}{3\,d}\arctan \left ({ \left ( 2\,d\sqrt [3]{x}+\sqrt{3}\sqrt{d}\sqrt{e} \right ){\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{2\,b{e}^{2}n}{3\,d}\arctan \left ({ \left ( \sqrt{3}\sqrt{d}\sqrt{e}-2\,d\sqrt [3]{x} \right ){\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{4\,b{e}^{2}n}{3\,d}\arctan \left ({d\sqrt [3]{x}{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+x*b*ln(c*((e+d*x^(2/3))/x^(2/3))^n)+2/3*b*e^2*n/d/(d*e)^(1/2)*arctan(x*d^2/e/(d*e)^(1/2))+2*b*e*n*x^(1/3)/
d-2/3*b*e^2*n/d/(d*e)^(1/2)*arctan((2*d*x^(1/3)+3^(1/2)*d^(1/2)*e^(1/2))/(d*e)^(1/2))+2/3*b*e^2*n/d/(d*e)^(1/2
)*arctan((3^(1/2)*d^(1/2)*e^(1/2)-2*d*x^(1/3))/(d*e)^(1/2))-4/3*b*e^2*n/d/(d*e)^(1/2)*arctan(d*x^(1/3)/(d*e)^(
1/2))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.99349, size = 664, normalized size = 10.22 \begin{align*} \left [\frac{b e 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 ) + b d n \log \left (d x^{\frac{2}{3}} + e\right ) + b d x \log \left (c\right ) - 2 \, b d n \log \left (x^{\frac{1}{3}}\right ) + 2 \, b e n x^{\frac{1}{3}} + a d x +{\left (b d n x - b d n\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )}{d}, -\frac{2 \, b e n \sqrt{\frac{e}{d}} \arctan \left (\frac{d x^{\frac{1}{3}} \sqrt{\frac{e}{d}}}{e}\right ) - b d n \log \left (d x^{\frac{2}{3}} + e\right ) - b d x \log \left (c\right ) + 2 \, b d n \log \left (x^{\frac{1}{3}}\right ) - 2 \, b e n x^{\frac{1}{3}} - a d x -{\left (b d n x - b d n\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )}{d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(b*e*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)) + b*d*n*log(d*x^(2/3) + e) + b*d*x*log(c) - 2*b*d*n*log(x^(1/
3)) + 2*b*e*n*x^(1/3) + a*d*x + (b*d*n*x - b*d*n)*log((d*x + e*x^(1/3))/x))/d, -(2*b*e*n*sqrt(e/d)*arctan(d*x^
(1/3)*sqrt(e/d)/e) - b*d*n*log(d*x^(2/3) + e) - b*d*x*log(c) + 2*b*d*n*log(x^(1/3)) - 2*b*e*n*x^(1/3) - a*d*x
- (b*d*n*x - b*d*n)*log((d*x + e*x^(1/3))/x))/d]

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Sympy [A]  time = 161.897, size = 61, normalized size = 0.94 \begin{align*} a x + b \left (\frac{2 e n \left (\frac{3 \sqrt [3]{x}}{d} - \frac{3 e \operatorname{atan}{\left (\frac{\sqrt [3]{x}}{\sqrt{\frac{e}{d}}} \right )}}{d^{2} \sqrt{\frac{e}{d}}}\right )}{3} + x \log{\left (c \left (d + \frac{e}{x^{\frac{2}{3}}}\right )^{n} \right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*(2*e*n*(3*x**(1/3)/d - 3*e*atan(x**(1/3)/sqrt(e/d))/(d**2*sqrt(e/d)))/3 + x*log(c*(d + e/x**(2/3))**n)
)

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Giac [A]  time = 1.32507, size = 77, normalized size = 1.18 \begin{align*} -{\left ({\left (2 \,{\left (\frac{\arctan \left (\sqrt{d} x^{\frac{1}{3}} e^{\left (-\frac{1}{2}\right )}\right ) e^{\frac{1}{2}}}{d^{\frac{3}{2}}} - \frac{x^{\frac{1}{3}}}{d}\right )} e - x \log \left (d + \frac{e}{x^{\frac{2}{3}}}\right )\right )} n - x \log \left (c\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((2*(arctan(sqrt(d)*x^(1/3)*e^(-1/2))*e^(1/2)/d^(3/2) - x^(1/3)/d)*e - x*log(d + e/x^(2/3)))*n - x*log(c))*b
+ a*x