3.516 \(\int (a+b \log (c (d (e+f x)^p)^q))^n \, dx\)

Optimal. Leaf size=131 \[ \frac{(e+f x) e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.151447, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2300, 2181, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^n,x]

[Out]

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

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]

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 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 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

Mathematica [A]  time = 0.121365, size = 131, normalized size = 1. \[ \frac{(e+f x) e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.278, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.02874, size = 192, normalized size = 1.47 \begin{align*} \frac{e^{\left (-\frac{b n p q \log \left (-\frac{1}{b p q}\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \Gamma \left (n + 1, -\frac{b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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