3.173 \(\int (a+b \log (c (d+e x)^n))^n \, dx\)

Optimal. Leaf size=103 \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]

[Out]

(e*x+d)*GAMMA(1+n,(-a-b*ln(c*(e*x+d)^n))/b/n)*(a+b*ln(c*(e*x+d)^n))^n/e/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))/(((-a
-b*ln(c*(e*x+d)^n))/b/n)^n)

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Rubi [A]  time = 0.06, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2389, 2300, 2181} \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^n,x]

[Out]

((d + e*x)*Gamma[1 + n, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(e*E^(a/(b*n))*(c*(
d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n)

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2389

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

Rubi steps

\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e}\\ &=\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{\frac {x}{n}} (a+b x)^n \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+n,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n}}{e}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 103, normalized size = 1.00 \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*Gamma[1 + n, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(e*E^(a/(b*n))*(c*(
d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n)

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fricas [A]  time = 0.45, size = 60, normalized size = 0.58 \[ \frac {e^{\left (-\frac {b n^{2} \log \left (-\frac {1}{b n}\right ) + b \log \relax (c) + a}{b n}\right )} \Gamma \left (n + 1, -\frac {b n \log \left (e x + d\right ) + b \log \relax (c) + a}{b n}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log((e*x + d)^n*c) + a)^n, x)

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maple [F]  time = 1.24, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^n,x)

[Out]

int((a + b*log(c*(d + e*x)^n))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*(d + e*x)**n))**n, x)

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