3.170 \(\int (f+g x)^3 (a+b \log (c (d+e x)^n))^n \, dx\)

Optimal. Leaf size=474 \[ \frac{g^2 3^{-n} e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^4}+\frac{3 g 2^{-n-1} e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^4}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e^4}+\frac{g^3 4^{-n-1} e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^4} \]

[Out]

(4^(-1 - n)*g^3*(d + e*x)^4*Gamma[1 + n, (-4*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^n)/
(e^4*E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n) + (g^2*(e*f - d*g)*(d + e*
x)^3*Gamma[1 + n, (-3*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^n)/(3^n*e^4*E^((3*a)/(b*n)
)*(c*(d + e*x)^n)^(3/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n) + (3*2^(-1 - n)*g*(e*f - d*g)^2*(d + e*x)^2*G
amma[1 + n, (-2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^n)/(e^4*E^((2*a)/(b*n))*(c*(d +
e*x)^n)^(2/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n) + ((e*f - d*g)^3*(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^4*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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Rubi [A]  time = 0.548706, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2401, 2389, 2300, 2181, 2390, 2310} \[ \frac{g^2 3^{-n} e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^4}+\frac{3 g 2^{-n-1} e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^4}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e^4}+\frac{g^3 4^{-n-1} e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^4} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(a + b*Log[c*(d + e*x)^n])^n,x]

[Out]

(4^(-1 - n)*g^3*(d + e*x)^4*Gamma[1 + n, (-4*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^n)/
(e^4*E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n) + (g^2*(e*f - d*g)*(d + e*
x)^3*Gamma[1 + n, (-3*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^n)/(3^n*e^4*E^((3*a)/(b*n)
)*(c*(d + e*x)^n)^(3/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n) + (3*2^(-1 - n)*g*(e*f - d*g)^2*(d + e*x)^2*G
amma[1 + n, (-2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^n)/(e^4*E^((2*a)/(b*n))*(c*(d +
e*x)^n)^(2/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n) + ((e*f - d*g)^3*(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^4*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d
+ e*x)^n])/(b*n)))^n)

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

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 2390

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

Rule 2310

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

Rubi steps

\begin{align*} \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx &=\int \left (\frac{(e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^n}{e^3}+\frac{3 g (e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^n}{e^3}+\frac{3 g^2 (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^n}{e^3}+\frac{g^3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^n}{e^3}\right ) \, dx\\ &=\frac{g^3 \int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx}{e^3}+\frac{\left (3 g^2 (e f-d g)\right ) \int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx}{e^3}+\frac{\left (3 g (e f-d g)^2\right ) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx}{e^3}+\frac{(e f-d g)^3 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx}{e^3}\\ &=\frac{g^3 \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e^4}+\frac{(e f-d g)^3 \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e^4}\\ &=\frac{\left (g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int e^{\frac{4 x}{n}} (a+b x)^n \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{\frac{3 x}{n}} (a+b x)^n \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{\frac{2 x}{n}} (a+b x)^n \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left ((e f-d g)^3 (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^4 n}\\ &=\frac{4^{-1-n} e^{-\frac{4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \Gamma \left (1+n,-\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\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^4}+\frac{3^{-n} e^{-\frac{3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \Gamma \left (1+n,-\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\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^4}+\frac{3\ 2^{-1-n} e^{-\frac{2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \Gamma \left (1+n,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\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^4}+\frac{e^{-\frac{a}{b n}} (e f-d g)^3 (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^4}\\ \end{align*}

Mathematica [A]  time = 1.74115, size = 343, normalized size = 0.72 \[ \frac{3^{-n} 4^{-n-1} e^{-\frac{4 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \left (2^{n+1} e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (3^n e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (2^{n+1} e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )+3 g (d+e x) \text{Gamma}\left (n+1,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )+g^2 2^{n+1} (d+e x)^2 \text{Gamma}\left (n+1,-\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )+g^3 3^n (d+e x)^3 \text{Gamma}\left (n+1,-\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )}{e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(a + b*Log[c*(d + e*x)^n])^n,x]

[Out]

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

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Maple [F]  time = 0.938, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{3} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(a+b*ln(c*(e*x+d)^n))^n,x)

[Out]

int((g*x+f)^3*(a+b*ln(c*(e*x+d)^n))^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((g*x+f)^3*(a+b*log(c*(e*x+d)^n))^n,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*log(c*(e*x+d)^n))^n,x, algorithm="fricas")

[Out]

integral((g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3)*(b*log((e*x + d)^n*c) + a)^n, x)

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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((g*x+f)**3*(a+b*ln(c*(e*x+d)**n))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x + f\right )}^{3}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*log(c*(e*x+d)^n))^n,x, algorithm="giac")

[Out]

integrate((g*x + f)^3*(b*log((e*x + d)^n*c) + a)^n, x)