\(\int (a+b \log (c (d+e x^{2/3})^n)) \, dx\) [466]

Optimal result

Integrand size = 18, antiderivative size = 72 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {2 b d n \sqrt [3]{x}}{e}+a x-\frac {2 b n x}{3}-\frac {2 b d^{3/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2498, 348, 308, 211} \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=a x-\frac {2 b d^{3/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )+\frac {2 b d n \sqrt [3]{x}}{e}-\frac {2 b n x}{3} \]

[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 211

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

Rule 308

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 348

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 2498

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]

Rubi steps \begin{align*} \text {integral}& = 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) \text {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) \text {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 ) \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {2 b d n \sqrt [3]{x}}{e}+a x-\frac {2 b n x}{3}-\frac {2 b d^{3/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right ) \]

[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]

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86

method	result	size
default	\(a x +b x \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )-\frac {2 b n x}{3}+\frac {2 b d n \,x^{\frac {1}{3}}}{e}-\frac {2 b n \,d^{2} \arctan \left (\frac {x^{\frac {1}{3}} e}{\sqrt {d e}}\right )}{e \sqrt {d e}}\)	\(62\)
parts	\(a x +b x \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )-\frac {2 b n x}{3}+\frac {2 b d n \,x^{\frac {1}{3}}}{e}-\frac {2 b n \,d^{2} \arctan \left (\frac {x^{\frac {1}{3}} e}{\sqrt {d e}}\right )}{e \sqrt {d e}}\)	\(62\)

[In]

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

[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)
)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.21 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\left [\frac {3 \, b e n x \log \left (e x^{\frac {2}{3}} + d\right ) + 3 \, b d n \sqrt {-\frac {d}{e}} \log \left (\frac {e^{3} x^{2} + 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - d^{3} - 2 \, {\left (e^{3} x \sqrt {-\frac {d}{e}} - d^{2} e\right )} x^{\frac {2}{3}} - 2 \, {\left (d e^{2} x + d^{2} e \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{e^{3} x^{2} + d^{3}}\right ) + 3 \, b e x \log \left (c\right ) + 6 \, b d n x^{\frac {1}{3}} - {\left (2 \, b e n - 3 \, a e\right )} x}{3 \, e}, \frac {3 \, b e n x \log \left (e x^{\frac {2}{3}} + d\right ) - 6 \, b d n \sqrt {\frac {d}{e}} \arctan \left (\frac {e x^{\frac {1}{3}} \sqrt {\frac {d}{e}}}{d}\right ) + 3 \, b e x \log \left (c\right ) + 6 \, b d n x^{\frac {1}{3}} - {\left (2 \, b e n - 3 \, a e\right )} x}{3 \, e}\right ] \]

[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]

Sympy [A] (verification not implemented)

Time = 1.90 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=a x + b \left (- \frac {2 e n \left (\frac {3 d^{2} \operatorname {atan}{\left (\frac {\sqrt [3]{x}}{\sqrt {\frac {d}{e}}} \right )}}{e^{3} \sqrt {\frac {d}{e}}} - \frac {3 d \sqrt [3]{x}}{e^{2}} + \frac {x}{e}\right )}{3} + x \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=-\frac {1}{3} \, {\left ({\left (2 \, e {\left (\frac {3 \, d^{2} \arctan \left (\frac {e x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {e^{2} x - 3 \, d e x^{\frac {1}{3}}}{e^{3}}\right )} - 3 \, x \log \left (e x^{\frac {2}{3}} + d\right )\right )} n - 3 \, x \log \left (c\right )\right )} b + a x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=a\,x+b\,x\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )-\frac {2\,b\,n\,x}{3}+\frac {2\,b\,d\,n\,x^{1/3}}{e}-\frac {2\,b\,d^{3/2}\,n\,\mathrm {atan}\left (\frac {\sqrt {e}\,x^{1/3}}{\sqrt {d}}\right )}{e^{3/2}} \]

[In]

int(a + b*log(c*(d + e*x^(2/3))^n),x)

[Out]

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