3.3.12 \(\int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx\) [212]

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

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 = {2458, 12, 2395, 2336, 2212, 2339, 30} \begin {gather*} \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)} \end {gather*}

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 12

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

Rule 30

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

Rule 2212

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

Rule 2336

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 2339

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 2395

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 2458

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]

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 106, normalized size = 0.92 \begin {gather*} \frac {(a+b \log (c (e+f x)))^p \left (\frac {(f h-e i) (a+b \log (c (e+f x)))}{b (1+p)}+\frac {e^{-\frac {a}{b}} i \Gamma \left (1+p,-\frac {a+b \log (c (e+f x))}{b}\right ) \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c}\right )}{d f^2} \end {gather*}

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)

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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (i x +h \right ) \left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{d f x +e d}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [A]
time = 0.10, size = 121, normalized size = 1.05 \begin {gather*} \frac {{\left (i \, b p + i \, b\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 - i \, a c e + {\left (b c f h - i \, b c e\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 {gather*}

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]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {h \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx + \int \frac {i x \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx}{d} \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{d\,e+d\,f\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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