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

Optimal. Leaf size=72 \[ a x+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{e^{3/2}}+\frac{2 b d n \sqrt [3]{x}}{e}-\frac{2 b n x}{3} \]

[Out]

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

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Rubi [A]  time = 0.0530306, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2448, 341, 302, 205} \[ a x+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{e^{3/2}}+\frac{2 b d n \sqrt [3]{x}}{e}-\frac{2 b n x}{3} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e*x^(2/3))^n],x]

[Out]

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

Mathematica [A]  time = 0.0267923, size = 72, normalized size = 1. \[ a x+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{e^{3/2}}+\frac{2 b d n \sqrt [3]{x}}{e}-\frac{2 b n x}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.092, size = 62, normalized size = 0.9 \begin{align*} ax+bx\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) -{\frac{2\,bnx}{3}}+2\,{\frac{bdn\sqrt [3]{x}}{e}}-2\,{\frac{b{d}^{2}n}{e\sqrt{de}}\arctan \left ({\frac{e\sqrt [3]{x}}{\sqrt{de}}} \right ) } \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+b*x*ln(c*(d+e*x^(2/3))^n)-2/3*b*n*x+2*b*d*n*x^(1/3)/e-2*b/e*n*d^2/(d*e)^(1/2)*arctan(x^(1/3)*e/(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 [A]  time = 1.85239, size = 547, normalized size = 7.6 \begin{align*} \left [\frac{3 \, b e n x \log \left (e x^{\frac{2}{3}} + d\right ) + 3 \, b d n \sqrt{-\frac{d}{e}} \log \left (\frac{e^{3} x^{2} + 2 \, d e^{2} x \sqrt{-\frac{d}{e}} - d^{3} - 2 \,{\left (e^{3} x \sqrt{-\frac{d}{e}} - d^{2} e\right )} x^{\frac{2}{3}} - 2 \,{\left (d e^{2} x + d^{2} e \sqrt{-\frac{d}{e}}\right )} x^{\frac{1}{3}}}{e^{3} x^{2} + d^{3}}\right ) + 3 \, b e x \log \left (c\right ) + 6 \, b d n x^{\frac{1}{3}} -{\left (2 \, b e n - 3 \, a e\right )} x}{3 \, e}, \frac{3 \, b e n x \log \left (e x^{\frac{2}{3}} + d\right ) - 6 \, b d n \sqrt{\frac{d}{e}} \arctan \left (\frac{e x^{\frac{1}{3}} \sqrt{\frac{d}{e}}}{d}\right ) + 3 \, b e x \log \left (c\right ) + 6 \, b d n x^{\frac{1}{3}} -{\left (2 \, b e n - 3 \, a e\right )} x}{3 \, e}\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]

[1/3*(3*b*e*n*x*log(e*x^(2/3) + d) + 3*b*d*n*sqrt(-d/e)*log((e^3*x^2 + 2*d*e^2*x*sqrt(-d/e) - d^3 - 2*(e^3*x*s
qrt(-d/e) - d^2*e)*x^(2/3) - 2*(d*e^2*x + d^2*e*sqrt(-d/e))*x^(1/3))/(e^3*x^2 + d^3)) + 3*b*e*x*log(c) + 6*b*d
*n*x^(1/3) - (2*b*e*n - 3*a*e)*x)/e, 1/3*(3*b*e*n*x*log(e*x^(2/3) + d) - 6*b*d*n*sqrt(d/e)*arctan(e*x^(1/3)*sq
rt(d/e)/d) + 3*b*e*x*log(c) + 6*b*d*n*x^(1/3) - (2*b*e*n - 3*a*e)*x)/e]

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Sympy [A]  time = 13.4007, size = 133, normalized size = 1.85 \begin{align*} a x + b \left (- \frac{2 e n \left (\begin{cases} - \frac{3 i d^{\frac{3}{2}} \log{\left (- i \sqrt{d} \sqrt{\frac{1}{e}} + \sqrt [3]{x} \right )}}{2 e^{3} \sqrt{\frac{1}{e}}} + \frac{3 i d^{\frac{3}{2}} \log{\left (i \sqrt{d} \sqrt{\frac{1}{e}} + \sqrt [3]{x} \right )}}{2 e^{3} \sqrt{\frac{1}{e}}} - \frac{3 d \sqrt [3]{x}}{e^{2}} + \frac{x}{e} & \text{for}\: e \neq 0 \\\frac{3 x^{\frac{5}{3}}}{5 d} & \text{otherwise} \end{cases}\right )}{3} + x \log{\left (c \left (d + 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*Piecewise((-3*I*d**(3/2)*log(-I*sqrt(d)*sqrt(1/e) + x**(1/3))/(2*e**3*sqrt(1/e)) + 3*I*d**(3/2
)*log(I*sqrt(d)*sqrt(1/e) + x**(1/3))/(2*e**3*sqrt(1/e)) - 3*d*x**(1/3)/e**2 + x/e, Ne(e, 0)), (3*x**(5/3)/(5*
d), True))/3 + x*log(c*(d + e*x**(2/3))**n))

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Giac [A]  time = 1.28292, size = 92, normalized size = 1.28 \begin{align*} -\frac{1}{3} \,{\left ({\left (2 \,{\left (3 \, d^{\frac{3}{2}} \arctan \left (\frac{x^{\frac{1}{3}} e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )} -{\left (3 \, d x^{\frac{1}{3}} e - x e^{2}\right )} e^{\left (-3\right )}\right )} e - 3 \, x \log \left (x^{\frac{2}{3}} e + d\right )\right )} n - 3 \, 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]

-1/3*((2*(3*d^(3/2)*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(-5/2) - (3*d*x^(1/3)*e - x*e^2)*e^(-3))*e - 3*x*log(x^(
2/3)*e + d))*n - 3*x*log(c))*b + a*x