3.210 \(\int \frac{(h+i x)^3 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx\)

Optimal. Leaf size=305 \[ \frac{3 i^2 2^{-p-1} e^{-\frac{2 a}{b}} (f h-e i) (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 (a+b \log (c (e+f x)))}{b}\right )}{c^2 d f^4}+\frac{i^3 3^{-p-1} e^{-\frac{3 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 (a+b \log (c (e+f x)))}{b}\right )}{c^3 d f^4}+\frac{3 i e^{-\frac{a}{b}} (f h-e i)^2 (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c d f^4}+\frac{(f h-e i)^3 (a+b \log (c (e+f x)))^{p+1}}{b d f^4 (p+1)} \]

[Out]

((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^4*(1 + p)) + (3^(-1 - p)*i^3*Gamma[1 + p, (-3*(a + b*L
og[c*(e + f*x)]))/b]*(a + b*Log[c*(e + f*x)])^p)/(c^3*d*E^((3*a)/b)*f^4*(-((a + b*Log[c*(e + f*x)])/b))^p) + (
3*2^(-1 - p)*i^2*(f*h - e*i)*Gamma[1 + p, (-2*(a + b*Log[c*(e + f*x)]))/b]*(a + b*Log[c*(e + f*x)])^p)/(c^2*d*
E^((2*a)/b)*f^4*(-((a + b*Log[c*(e + f*x)])/b))^p) + (3*i*(f*h - e*i)^2*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^4*(-((a + b*Log[c*(e + f*x)])/b))^p)

________________________________________________________________________________________

Rubi [A]  time = 0.663943, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2411, 12, 2353, 2299, 2181, 2302, 30, 2309} \[ \frac{3 i^2 2^{-p-1} e^{-\frac{2 a}{b}} (f h-e i) (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 (a+b \log (c (e+f x)))}{b}\right )}{c^2 d f^4}+\frac{i^3 3^{-p-1} e^{-\frac{3 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 (a+b \log (c (e+f x)))}{b}\right )}{c^3 d f^4}+\frac{3 i e^{-\frac{a}{b}} (f h-e i)^2 (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c d f^4}+\frac{(f h-e i)^3 (a+b \log (c (e+f x)))^{p+1}}{b d f^4 (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[((h + i*x)^3*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x),x]

[Out]

((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^4*(1 + p)) + (3^(-1 - p)*i^3*Gamma[1 + p, (-3*(a + b*L
og[c*(e + f*x)]))/b]*(a + b*Log[c*(e + f*x)])^p)/(c^3*d*E^((3*a)/b)*f^4*(-((a + b*Log[c*(e + f*x)])/b))^p) + (
3*2^(-1 - p)*i^2*(f*h - e*i)*Gamma[1 + p, (-2*(a + b*Log[c*(e + f*x)]))/b]*(a + b*Log[c*(e + f*x)])^p)/(c^2*d*
E^((2*a)/b)*f^4*(-((a + b*Log[c*(e + f*x)])/b))^p) + (3*i*(f*h - e*i)^2*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^4*(-((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]

Rule 2309

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

Rubi steps

\begin{align*} \int \frac{(h+210 x)^3 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-210 e+f h}{f}+\frac{210 x}{f}\right )^3 (a+b \log (c x))^p}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-210 e+f h}{f}+\frac{210 x}{f}\right )^3 (a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{630 (210 e-f h)^2 (a+b \log (c x))^p}{f^3}-\frac{(210 e-f h)^3 (a+b \log (c x))^p}{f^3 x}-\frac{132300 (210 e-f h) x (a+b \log (c x))^p}{f^3}+\frac{9261000 x^2 (a+b \log (c x))^p}{f^3}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac{9261000 \operatorname{Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^4}-\frac{(132300 (210 e-f h)) \operatorname{Subst}\left (\int x (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^4}+\frac{\left (630 (210 e-f h)^2\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^4}-\frac{(210 e-f h)^3 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f^4}\\ &=\frac{9261000 \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c^3 d f^4}-\frac{(132300 (210 e-f h)) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c^2 d f^4}+\frac{\left (630 (210 e-f h)^2\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c d f^4}-\frac{(210 e-f h)^3 \operatorname{Subst}\left (\int x^p \, dx,x,a+b \log (c (e+f x))\right )}{b d f^4}\\ &=-\frac{(210 e-f h)^3 (a+b \log (c (e+f x)))^{1+p}}{b d f^4 (1+p)}+\frac{343000\ 3^{2-p} e^{-\frac{3 a}{b}} \Gamma \left (1+p,-\frac{3 (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^3 d f^4}-\frac{33075\ 2^{1-p} e^{-\frac{2 a}{b}} (210 e-f h) \Gamma \left (1+p,-\frac{2 (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^2 d f^4}+\frac{630 e^{-\frac{a}{b}} (210 e-f h)^2 \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^4}\\ \end{align*}

Mathematica [A]  time = 1.30436, size = 247, normalized size = 0.81 \[ \frac{6^{-p-1} e^{-\frac{3 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \left (c 3^{p+1} e^{a/b} (f h-e i) \left (c 2^{p+1} e^{a/b} (f h-e i) \left (3 b i (p+1) \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )-b c e^{a/b} (f h-e i) \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{p+1}\right )+3 b i^2 (p+1) \text{Gamma}\left (p+1,-\frac{2 (a+b \log (c (e+f x)))}{b}\right )\right )+b i^3 2^{p+1} (p+1) \text{Gamma}\left (p+1,-\frac{3 (a+b \log (c (e+f x)))}{b}\right )\right )}{b c^3 d f^4 (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((h + i*x)^3*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x),x]

[Out]

(6^(-1 - p)*(a + b*Log[c*(e + f*x)])^p*(2^(1 + p)*b*i^3*(1 + p)*Gamma[1 + p, (-3*(a + b*Log[c*(e + f*x)]))/b]
+ 3^(1 + p)*c*E^(a/b)*(f*h - e*i)*(3*b*i^2*(1 + p)*Gamma[1 + p, (-2*(a + b*Log[c*(e + f*x)]))/b] + 2^(1 + p)*c
*E^(a/b)*(f*h - e*i)*(3*b*i*(1 + p)*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/b)] - b*c*E^(a/b)*(f*h - e*i)*(-((
a + b*Log[c*(e + f*x)])/b))^(1 + p)))))/(b*c^3*d*E^((3*a)/b)*f^4*(1 + p)*(-((a + b*Log[c*(e + f*x)])/b))^p)

________________________________________________________________________________________

Maple [F]  time = 0.678, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ix+h \right ) ^{3} \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)^3*(a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e),x)

[Out]

int((i*x+h)^3*(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*} \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^{3}}{b c d f{\left (p + 1\right )}} + \int \frac{{\left (i^{3} x^{3} + 3 \, h i^{2} x^{2} + 3 \, h^{2} i x\right )}{\left (b \log \left (f x + e\right ) + b \log \left (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)^3*(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^3/(b*c*d*f*(p + 1)) + integrate((i^3*x^3 + 3*h*i^2*x
^2 + 3*h^2*i*x)*(b*log(f*x + e) + b*log(c) + a)^p/(d*f*x + d*e), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (i^{3} x^{3} + 3 \, h i^{2} x^{2} + 3 \, h^{2} i x + h^{3}\right )}{\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{d f x + d e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^3*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),x, algorithm="fricas")

[Out]

integral((i^3*x^3 + 3*h*i^2*x^2 + 3*h^2*i*x + h^3)*(b*log(c*f*x + c*e) + a)^p/(d*f*x + d*e), x)

________________________________________________________________________________________

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)**3*(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 )}^{3}{\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)^3*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),x, algorithm="giac")

[Out]

integrate((i*x + h)^3*(b*log((f*x + e)*c) + a)^p/(d*f*x + d*e), x)