3.101 \(\int \frac{f+g x}{(a+b \log (c (d+e x)^n))^3} \, dx\)

Optimal. Leaf size=261 \[ \frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac{2 g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}+\frac{(d+e x) (e f-d g)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.360439, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310, 2297} \[ \frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac{2 g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}+\frac{(d+e x) (e f-d g)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2400

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

Rule 2399

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, 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 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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]

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rubi steps

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

Mathematica [A]  time = 0.416702, size = 256, normalized size = 0.98 \[ -\frac{e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )-4 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+b n e^{\frac{2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} \left (a (d g+e f+2 e g x)+b (d g+e (f+2 g x)) \log \left (c (d+e x)^n\right )+b e n (f+g x)\right )\right )}{2 b^3 e^2 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 3.707, size = 0, normalized size = 0. \begin{align*} \int{\frac{gx+f}{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (2 \, a e^{2} g +{\left (e^{2} g n + 2 \, e^{2} g \log \left (c\right )\right )} b\right )} x^{2} +{\left (d e f + d^{2} g\right )} a +{\left (d e f n +{\left (d e f + d^{2} g\right )} \log \left (c\right )\right )} b +{\left ({\left (e^{2} f + 3 \, d e g\right )} a +{\left (e^{2} f n + d e g n +{\left (e^{2} f + 3 \, d e g\right )} \log \left (c\right )\right )} b\right )} x +{\left (2 \, b e^{2} g x^{2} +{\left (e^{2} f + 3 \, d e g\right )} b x +{\left (d e f + d^{2} g\right )} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \,{\left (b^{4} e^{2} n^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{4} e^{2} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} e^{2} n^{2} \log \left (c\right ) + a^{2} b^{2} e^{2} n^{2} + 2 \,{\left (b^{4} e^{2} n^{2} \log \left (c\right ) + a b^{3} e^{2} n^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}} + \int \frac{4 \, e g x + e f + 3 \, d g}{2 \,{\left (b^{3} e n^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e n^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((2*a*e^2*g + (e^2*g*n + 2*e^2*g*log(c))*b)*x^2 + (d*e*f + d^2*g)*a + (d*e*f*n + (d*e*f + d^2*g)*log(c))*
b + ((e^2*f + 3*d*e*g)*a + (e^2*f*n + d*e*g*n + (e^2*f + 3*d*e*g)*log(c))*b)*x + (2*b*e^2*g*x^2 + (e^2*f + 3*d
*e*g)*b*x + (d*e*f + d^2*g)*b)*log((e*x + d)^n))/(b^4*e^2*n^2*log((e*x + d)^n)^2 + b^4*e^2*n^2*log(c)^2 + 2*a*
b^3*e^2*n^2*log(c) + a^2*b^2*e^2*n^2 + 2*(b^4*e^2*n^2*log(c) + a*b^3*e^2*n^2)*log((e*x + d)^n)) + integrate(1/
2*(4*e*g*x + e*f + 3*d*g)/(b^3*e*n^2*log((e*x + d)^n) + b^3*e*n^2*log(c) + a*b^2*e*n^2), x)

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Fricas [B]  time = 2.23634, size = 1353, normalized size = 5.18 \begin{align*} \frac{{\left ({\left ({\left (b^{2} e f - b^{2} d g\right )} n^{2} \log \left (e x + d\right )^{2} + a^{2} e f - a^{2} d g +{\left (b^{2} e f - b^{2} d g\right )} \log \left (c\right )^{2} + 2 \,{\left ({\left (b^{2} e f - b^{2} d g\right )} n \log \left (c\right ) +{\left (a b e f - a b d g\right )} n\right )} \log \left (e x + d\right ) + 2 \,{\left (a b e f - a b d g\right )} \log \left (c\right )\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} \logintegral \left ({\left (e x + d\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right ) -{\left (b^{2} d e f n^{2} +{\left (b^{2} e^{2} g n^{2} + 2 \, a b e^{2} g n\right )} x^{2} +{\left (a b d e f + a b d^{2} g\right )} n +{\left ({\left (b^{2} e^{2} f + b^{2} d e g\right )} n^{2} +{\left (a b e^{2} f + 3 \, a b d e g\right )} n\right )} x +{\left (2 \, b^{2} e^{2} g n^{2} x^{2} +{\left (b^{2} e^{2} f + 3 \, b^{2} d e g\right )} n^{2} x +{\left (b^{2} d e f + b^{2} d^{2} g\right )} n^{2}\right )} \log \left (e x + d\right ) +{\left (2 \, b^{2} e^{2} g n x^{2} +{\left (b^{2} e^{2} f + 3 \, b^{2} d e g\right )} n x +{\left (b^{2} d e f + b^{2} d^{2} g\right )} n\right )} \log \left (c\right )\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 4 \,{\left (b^{2} g n^{2} \log \left (e x + d\right )^{2} + b^{2} g \log \left (c\right )^{2} + 2 \, a b g \log \left (c\right ) + a^{2} g + 2 \,{\left (b^{2} g n \log \left (c\right ) + a b g n\right )} \log \left (e x + d\right )\right )} \logintegral \left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \,{\left (b^{5} e^{2} n^{5} \log \left (e x + d\right )^{2} + b^{5} e^{2} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e^{2} n^{3} \log \left (c\right ) + a^{2} b^{3} e^{2} n^{3} + 2 \,{\left (b^{5} e^{2} n^{4} \log \left (c\right ) + a b^{4} e^{2} n^{4}\right )} \log \left (e x + d\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(((b^2*e*f - b^2*d*g)*n^2*log(e*x + d)^2 + a^2*e*f - a^2*d*g + (b^2*e*f - b^2*d*g)*log(c)^2 + 2*((b^2*e*f
- b^2*d*g)*n*log(c) + (a*b*e*f - a*b*d*g)*n)*log(e*x + d) + 2*(a*b*e*f - a*b*d*g)*log(c))*e^((b*log(c) + a)/(b
*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - (b^2*d*e*f*n^2 + (b^2*e^2*g*n^2 + 2*a*b*e^2*g*n)*x^2 +
(a*b*d*e*f + a*b*d^2*g)*n + ((b^2*e^2*f + b^2*d*e*g)*n^2 + (a*b*e^2*f + 3*a*b*d*e*g)*n)*x + (2*b^2*e^2*g*n^2*
x^2 + (b^2*e^2*f + 3*b^2*d*e*g)*n^2*x + (b^2*d*e*f + b^2*d^2*g)*n^2)*log(e*x + d) + (2*b^2*e^2*g*n*x^2 + (b^2*
e^2*f + 3*b^2*d*e*g)*n*x + (b^2*d*e*f + b^2*d^2*g)*n)*log(c))*e^(2*(b*log(c) + a)/(b*n)) + 4*(b^2*g*n^2*log(e*
x + d)^2 + b^2*g*log(c)^2 + 2*a*b*g*log(c) + a^2*g + 2*(b^2*g*n*log(c) + a*b*g*n)*log(e*x + d))*log_integral((
e^2*x^2 + 2*d*e*x + d^2)*e^(2*(b*log(c) + a)/(b*n))))*e^(-2*(b*log(c) + a)/(b*n))/(b^5*e^2*n^5*log(e*x + d)^2
+ b^5*e^2*n^3*log(c)^2 + 2*a*b^4*e^2*n^3*log(c) + a^2*b^3*e^2*n^3 + 2*(b^5*e^2*n^4*log(c) + a*b^4*e^2*n^4)*log
(e*x + d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.5777, size = 5554, normalized size = 21.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e + d)^2*b^2*g*n^2*e*log(x*e + d)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^
4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e + d)*b^2*
d*g*n^2*e*log(x*e + d)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x
*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - 1/2*b^2*d*g*n^2*Ei(log(c)/n + a/(
b*n) + log(x*e + d))*e^(-a/(b*n) + 1)*log(x*e + d)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)
*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1
/n)) - 1/2*(x*e + d)^2*b^2*g*n^2*e/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n
^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e + d)*b^2*d*g
*n^2*e/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^
3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - 1/2*(x*e + d)*b^2*f*n^2*e^2*log(x*e + d)/(b^5*n^5
*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2
+ 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*b^2*f*n^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n)
+ 2)*log(x*e + d)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e
+ d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 2*b^2*g*n^2*Ei(2*log(c)/n
+ 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)*log(x*e + d)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*l
og(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^
3*e^3)*c^(2/n)) - (x*e + d)^2*b^2*g*n*e*log(c)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c)
+ 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e
+ d)*b^2*d*g*n*e*log(c)/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(
x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - b^2*d*g*n*Ei(log(c)/n + a/(b*n)
+ log(x*e + d))*e^(-a/(b*n) + 1)*log(x*e + d)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)
*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1
/n)) - 1/2*(x*e + d)*b^2*f*n^2*e^2/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n
^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) - (x*e + d)^2*a*b*g*n*e
/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*
log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3) + 1/2*(x*e + d)*a*b*d*g*n*e/(b^5*n^5*e^3*log(x*e + d)^2 +
2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log
(c) + a^2*b^3*n^3*e^3) - a*b*d*g*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 1)*log(x*e + d)/((b^5*n
^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^
2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) - 1/2*(x*e + d)*b^2*f*n*e^2*log(c)/(b^5*n^5*e^3*log(x*e
+ d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^
3*e^3*log(c) + a^2*b^3*n^3*e^3) + b^2*f*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 2)*log(x*e + d)*
log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n
^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 4*b^2*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2
*log(x*e + d))*e^(-2*a/(b*n) + 1)*log(x*e + d)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d
)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(
2/n)) - 1/2*b^2*d*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 1)*log(c)^2/((b^5*n^5*e^3*log(x*e + d)
^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3
*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) - 1/2*(x*e + d)*a*b*f*n*e^2/(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*lo
g(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3
*e^3) + a*b*f*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 2)*log(x*e + d)/((b^5*n^5*e^3*log(x*e + d)
^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3
*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 4*a*b*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)
*log(x*e + d)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d)
+ b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(2/n)) - a*b*d*g*Ei(log(c)/n + a/(b*n) +
log(x*e + d))*e^(-a/(b*n) + 1)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b
^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 1/2*b^2*
f*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 2)*log(c)^2/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3
*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*
n^3*e^3)*c^(1/n)) + 2*b^2*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)*log(c)^2/((b^5*n^5*
e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 +
2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(2/n)) - 1/2*a^2*d*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/
(b*n) + 1)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b
^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + a*b*f*Ei(log(c)/n + a/(b*n) + log(x
*e + d))*e^(-a/(b*n) + 2)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^
4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 4*a*b*g*Ei(2*
log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)*log(c)/((b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*
log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n
^3*e^3)*c^(2/n)) + 1/2*a^2*f*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 2)/((b^5*n^5*e^3*log(x*e + d)
^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*log(c)^2 + 2*a*b^4*n^3*e^3
*log(c) + a^2*b^3*n^3*e^3)*c^(1/n)) + 2*a^2*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 1)/(
(b^5*n^5*e^3*log(x*e + d)^2 + 2*b^5*n^4*e^3*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^3*log(x*e + d) + b^5*n^3*e^3*l
og(c)^2 + 2*a*b^4*n^3*e^3*log(c) + a^2*b^3*n^3*e^3)*c^(2/n))