3.212 \(\int \frac{(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx\)

Optimal. Leaf size=115 \[ \frac{i e^{-\frac{a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c d f^2}+\frac{(f h-e i) (a+b \log (c (e+f x)))^{p+1}}{b d f^2 (p+1)} \]

[Out]

((f*h - e*i)*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^2*(1 + p)) + (i*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/
b)]*(a + b*Log[c*(e + f*x)])^p)/(c*d*E^(a/b)*f^2*(-((a + b*Log[c*(e + f*x)])/b))^p)

________________________________________________________________________________________

Rubi [A]  time = 0.28188, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {2411, 12, 2353, 2299, 2181, 2302, 30} \[ \frac{i e^{-\frac{a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c d f^2}+\frac{(f h-e i) (a+b \log (c (e+f x)))^{p+1}}{b d f^2 (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[((h + i*x)*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x),x]

[Out]

((f*h - e*i)*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^2*(1 + p)) + (i*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/
b)]*(a + b*Log[c*(e + f*x)])^p)/(c*d*E^(a/b)*f^2*(-((a + b*Log[c*(e + f*x)])/b))^p)

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/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 2302

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(h+212 x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-212 e+f h}{f}+\frac{212 x}{f}\right ) (a+b \log (c x))^p}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-212 e+f h}{f}+\frac{212 x}{f}\right ) (a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{212 (a+b \log (c x))^p}{f}+\frac{(-212 e+f h) (a+b \log (c x))^p}{f x}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac{212 \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^2}-\frac{(212 e-f h) \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac{212 \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c d f^2}-\frac{(212 e-f h) \operatorname{Subst}\left (\int x^p \, dx,x,a+b \log (c (e+f x))\right )}{b d f^2}\\ &=-\frac{(212 e-f h) (a+b \log (c (e+f x)))^{1+p}}{b d f^2 (1+p)}+\frac{212 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log (c (e+f x))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^2}\\ \end{align*}

Mathematica [A]  time = 0.2233, size = 106, normalized size = 0.92 \[ \frac{(a+b \log (c (e+f x)))^p \left (\frac{i e^{-\frac{a}{b}} \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c}+\frac{(f h-e i) (a+b \log (c (e+f x)))}{b (p+1)}\right )}{d f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((h + i*x)*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x),x]

[Out]

((a + b*Log[c*(e + f*x)])^p*(((f*h - e*i)*(a + b*Log[c*(e + f*x)]))/(b*(1 + p)) + (i*Gamma[1 + p, -((a + b*Log
[c*(e + f*x)])/b)])/(c*E^(a/b)*(-((a + b*Log[c*(e + f*x)])/b))^p)))/(d*f^2)

________________________________________________________________________________________

Maple [F]  time = 0.487, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ix+h \right ) \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) ^{p}}{dfx+de}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x+h)*(a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e),x)

[Out]

int((i*x+h)*(a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} i \int \frac{{\left (b \log \left (f x + e\right ) + b \log \left (c\right ) + a\right )}^{p} x}{d f x + d e}\,{d x} + \frac{{\left (b c \log \left (c f x + c e\right ) + a c\right )}{\left (b \log \left (c f x + c e\right ) + a\right )}^{p} h}{b c d f{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

i*integrate((b*log(f*x + e) + b*log(c) + a)^p*x/(d*f*x + d*e), x) + (b*c*log(c*f*x + c*e) + a*c)*(b*log(c*f*x
+ c*e) + a)^p*h/(b*c*d*f*(p + 1))

________________________________________________________________________________________

Fricas [A]  time = 1.82375, size = 270, normalized size = 2.35 \begin{align*} \frac{{\left (b i p + b i\right )} e^{\left (-\frac{b p \log \left (-\frac{1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac{b \log \left (c f x + c e\right ) + a}{b}\right ) +{\left (a c f h - a c e i +{\left (b c f h - b c e i\right )} \log \left (c f x + c e\right )\right )}{\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{b c d f^{2} p + b c d f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*i*p + b*i)*e^(-(b*p*log(-1/b) + a)/b)*gamma(p + 1, -(b*log(c*f*x + c*e) + a)/b) + (a*c*f*h - a*c*e*i + (b*
c*f*h - b*c*e*i)*log(c*f*x + c*e))*(b*log(c*f*x + c*e) + a)^p)/(b*c*d*f^2*p + b*c*d*f^2)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)*(a+b*ln(c*(f*x+e)))**p/(d*f*x+d*e),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((i*x + h)*(b*log((f*x + e)*c) + a)^p/(d*f*x + d*e), x)