3.94 \(\int \frac{(f+g x)^3}{(a+b \log (c (d+e x)^n))^2} \, dx\)

Optimal. Leaf size=339 \[ \frac{9 g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{6 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{4 g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*e^4*E^(a/(b*n))*n^2*(c*(d + e*x
)^n)^n^(-1)) + (6*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^2*e^4*E^
((2*a)/(b*n))*n^2*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e
*x)^n]))/(b*n)])/(b^2*e^4*E^((3*a)/(b*n))*n^2*(c*(d + e*x)^n)^(3/n)) + (4*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a
+ b*Log[c*(d + e*x)^n]))/(b*n)])/(b^2*e^4*E^((4*a)/(b*n))*n^2*(c*(d + e*x)^n)^(4/n)) - ((d + e*x)*(f + g*x)^3)
/(b*e*n*(a + b*Log[c*(d + e*x)^n]))

________________________________________________________________________________________

Rubi [A]  time = 0.788404, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{9 g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{6 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{4 g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*e^4*E^(a/(b*n))*n^2*(c*(d + e*x
)^n)^n^(-1)) + (6*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^2*e^4*E^
((2*a)/(b*n))*n^2*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e
*x)^n]))/(b*n)])/(b^2*e^4*E^((3*a)/(b*n))*n^2*(c*(d + e*x)^n)^(3/n)) + (4*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a
+ b*Log[c*(d + e*x)^n]))/(b*n)])/(b^2*e^4*E^((4*a)/(b*n))*n^2*(c*(d + e*x)^n)^(4/n)) - ((d + e*x)*(f + g*x)^3)
/(b*e*n*(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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{4 \int \frac{(f+g x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b n}-\frac{(3 (e f-d g)) \int \frac{(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e n}\\ &=-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{4 \int \left (\frac{(e f-d g)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{3 g (e f-d g)^2 (d+e x)}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{3 g^2 (e f-d g) (d+e x)^2}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{g^3 (d+e x)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b n}-\frac{(3 (e f-d g)) \int \left (\frac{(e f-d g)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{2 g (e f-d g) (d+e x)}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{g^2 (d+e x)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b e n}\\ &=-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\left (4 g^3\right ) \int \frac{(d+e x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}-\frac{\left (3 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}+\frac{\left (12 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}-\frac{\left (6 g (e f-d g)^2\right ) \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}+\frac{\left (12 g (e f-d g)^2\right ) \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}-\frac{\left (3 (e f-d g)^3\right ) \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}+\frac{\left (4 (e f-d g)^3\right ) \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}\\ &=-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\left (4 g^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}-\frac{\left (3 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}+\frac{\left (12 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}-\frac{\left (6 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}+\frac{\left (12 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}-\frac{\left (3 (e f-d g)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}+\frac{\left (4 (e f-d g)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}\\ &=-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\left (4 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}-\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 \frac{e^{\frac{3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}+\frac{\left (12 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}-\frac{\left (6 g (e f-d g)^2 (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 e^4 n^2}+\frac{\left (12 g (e f-d g)^2 (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 e^4 n^2}-\frac{\left (3 (e f-d g)^3 (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 e^4 n^2}+\frac{\left (4 (e f-d g)^3 (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 e^4 n^2}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^3 (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 )}{b^2 e^4 n^2}+\frac{6 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} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{9 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} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac{4 e^{-\frac{4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac{(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end{align*}

Mathematica [B]  time = 1.01585, size = 1674, normalized size = 4.94 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

(-(b*d*e^3*E^((4*a)/(b*n))*f^3*n*(c*(d + e*x)^n)^(4/n)) - b*e^4*E^((4*a)/(b*n))*f^3*n*x*(c*(d + e*x)^n)^(4/n)
- 3*b*d*e^3*E^((4*a)/(b*n))*f^2*g*n*x*(c*(d + e*x)^n)^(4/n) - 3*b*e^4*E^((4*a)/(b*n))*f^2*g*n*x^2*(c*(d + e*x)
^n)^(4/n) - 3*b*d*e^3*E^((4*a)/(b*n))*f*g^2*n*x^2*(c*(d + e*x)^n)^(4/n) - 3*b*e^4*E^((4*a)/(b*n))*f*g^2*n*x^3*
(c*(d + e*x)^n)^(4/n) - b*d*e^3*E^((4*a)/(b*n))*g^3*n*x^3*(c*(d + e*x)^n)^(4/n) - b*e^4*E^((4*a)/(b*n))*g^3*n*
x^4*(c*(d + e*x)^n)^(4/n) + a*e^3*E^((3*a)/(b*n))*f^3*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log
[c*(d + e*x)^n])/(b*n)] - 3*a*d*e^2*E^((3*a)/(b*n))*f^2*g*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b
*Log[c*(d + e*x)^n])/(b*n)] + 3*a*d^2*e*E^((3*a)/(b*n))*f*g^2*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a
+ b*Log[c*(d + e*x)^n])/(b*n)] - a*d^3*E^((3*a)/(b*n))*g^3*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a +
b*Log[c*(d + e*x)^n])/(b*n)] + 6*a*e^2*E^((2*a)/(b*n))*f^2*g*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[
(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] - 12*a*d*e*E^((2*a)/(b*n))*f*g^2*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIn
tegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] + 6*a*d^2*E^((2*a)/(b*n))*g^3*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)
*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] + 9*a*e*E^(a/(b*n))*f*g^2*(d + e*x)^3*(c*(d + e*x)^n)^n^(
-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)] - 9*a*d*E^(a/(b*n))*g^3*(d + e*x)^3*(c*(d + e*x)^n)^n^
(-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)] + 4*a*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a + b*Log[c*(
d + e*x)^n]))/(b*n)] + b*e^3*E^((3*a)/(b*n))*f^3*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c*(d
+ e*x)^n])/(b*n)]*Log[c*(d + e*x)^n] - 3*b*d*e^2*E^((3*a)/(b*n))*f^2*g*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpInt
egralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*Log[c*(d + e*x)^n] + 3*b*d^2*e*E^((3*a)/(b*n))*f*g^2*(d + e*x)*(c*(d
+ e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*Log[c*(d + e*x)^n] - b*d^3*E^((3*a)/(b*n))*g^
3*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*Log[c*(d + e*x)^n] + 6*b*e^2
*E^((2*a)/(b*n))*f^2*g*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*L
og[c*(d + e*x)^n] - 12*b*d*e*E^((2*a)/(b*n))*f*g^2*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(2*(a + b*L
og[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] + 6*b*d^2*E^((2*a)/(b*n))*g^3*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*
ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] + 9*b*e*E^(a/(b*n))*f*g^2*(d + e*x)^3*(
c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] - 9*b*d*E^(a/(b*n
))*g^3*(d + e*x)^3*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^
n] + 4*b*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n])/(b^2*e^4*E^((
4*a)/(b*n))*n^2*(c*(d + e*x)^n)^(4/n)*(a + b*Log[c*(d + e*x)^n]))

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Maple [F]  time = 3.807, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{3}}{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}\, 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))^2,x)

[Out]

int((g*x+f)^3/(a+b*ln(c*(e*x+d)^n))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e g^{3} x^{4} + d f^{3} +{\left (3 \, e f g^{2} + d g^{3}\right )} x^{3} + 3 \,{\left (e f^{2} g + d f g^{2}\right )} x^{2} +{\left (e f^{3} + 3 \, d f^{2} g\right )} x}{b^{2} e n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} e n \log \left (c\right ) + a b e n} + \int \frac{4 \, e g^{3} x^{3} + e f^{3} + 3 \, d f^{2} g + 3 \,{\left (3 \, e f g^{2} + d g^{3}\right )} x^{2} + 6 \,{\left (e f^{2} g + d f g^{2}\right )} x}{b^{2} e n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} e n \log \left (c\right ) + a b e 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))^2,x, algorithm="maxima")

[Out]

-(e*g^3*x^4 + d*f^3 + (3*e*f*g^2 + d*g^3)*x^3 + 3*(e*f^2*g + d*f*g^2)*x^2 + (e*f^3 + 3*d*f^2*g)*x)/(b^2*e*n*lo
g((e*x + d)^n) + b^2*e*n*log(c) + a*b*e*n) + integrate((4*e*g^3*x^3 + e*f^3 + 3*d*f^2*g + 3*(3*e*f*g^2 + d*g^3
)*x^2 + 6*(e*f^2*g + d*f*g^2)*x)/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c) + a*b*e*n), x)

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Fricas [A]  time = 2.19106, size = 1562, normalized size = 4.61 \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)^3/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

(9*(a*e*f*g^2 - a*d*g^3 + (b*e*f*g^2 - b*d*g^3)*n*log(e*x + d) + (b*e*f*g^2 - b*d*g^3)*log(c))*e^((b*log(c) +
a)/(b*n))*log_integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*e^(3*(b*log(c) + a)/(b*n))) + 6*(a*e^2*f^2*g
- 2*a*d*e*f*g^2 + a*d^2*g^3 + (b*e^2*f^2*g - 2*b*d*e*f*g^2 + b*d^2*g^3)*n*log(e*x + d) + (b*e^2*f^2*g - 2*b*d*
e*f*g^2 + b*d^2*g^3)*log(c))*e^(2*(b*log(c) + a)/(b*n))*log_integral((e^2*x^2 + 2*d*e*x + d^2)*e^(2*(b*log(c)
+ a)/(b*n))) + (a*e^3*f^3 - 3*a*d*e^2*f^2*g + 3*a*d^2*e*f*g^2 - a*d^3*g^3 + (b*e^3*f^3 - 3*b*d*e^2*f^2*g + 3*b
*d^2*e*f*g^2 - b*d^3*g^3)*n*log(e*x + d) + (b*e^3*f^3 - 3*b*d*e^2*f^2*g + 3*b*d^2*e*f*g^2 - b*d^3*g^3)*log(c))
*e^(3*(b*log(c) + a)/(b*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - (b*e^4*g^3*n*x^4 + b*d*e^3*f^3*
n + (3*b*e^4*f*g^2 + b*d*e^3*g^3)*n*x^3 + 3*(b*e^4*f^2*g + b*d*e^3*f*g^2)*n*x^2 + (b*e^4*f^3 + 3*b*d*e^3*f^2*g
)*n*x)*e^(4*(b*log(c) + a)/(b*n)) + 4*(b*g^3*n*log(e*x + d) + b*g^3*log(c) + a*g^3)*log_integral((e^4*x^4 + 4*
d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)*e^(4*(b*log(c) + a)/(b*n))))*e^(-4*(b*log(c) + a)/(b*n))/(b^3*e^4
*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2)

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

[Out]

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

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Giac [B]  time = 1.84836, size = 4691, normalized size = 13.84 \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)^3/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")

[Out]

-(x*e + d)^4*b*g^3*n*e^6/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*(x*e + d)^3*b*
d*g^3*n*e^6/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - 3*(x*e + d)^2*b*d^2*g^3*n*e^6
/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + (x*e + d)*b*d^3*g^3*n*e^6/(b^3*n^3*e^10*
log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - b*d^3*g^3*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a
/(b*n) + 6)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 3*(x*e
+ d)^3*b*f*g^2*n*e^7/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 6*(x*e + d)^2*b*d*f
*g^2*n*e^7/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - 3*(x*e + d)*b*d^2*f*g^2*n*e^7/
(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*b*d^2*f*g^2*n*Ei(log(c)/n + a/(b*n) + l
og(x*e + d))*e^(-a/(b*n) + 7)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)
*c^(1/n)) + 6*b*d^2*g^3*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 6)*log(x*e + d)/((b^3*n^
3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) - b*d^3*g^3*Ei(log(c)/n + a/(b*n) + log(x
*e + d))*e^(-a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n))
- 3*(x*e + d)^2*b*f^2*g*n*e^8/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*(x*e + d)
*b*d*f^2*g*n*e^8/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - a*d^3*g^3*Ei(log(c)/n +
a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c
^(1/n)) - 3*b*d*f^2*g*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 8)*log(x*e + d)/((b^3*n^3*e^10*log
(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 12*b*d*f*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(
x*e + d))*e^(-2*a/(b*n) + 7)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*
c^(2/n)) - 9*b*d*g^3*n*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 6)*log(x*e + d)/((b^3*n^3*e
^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + 3*b*d^2*f*g^2*Ei(log(c)/n + a/(b*n) + log(
x*e + d))*e^(-a/(b*n) + 7)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n))
+ 6*b*d^2*g^3*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e +
d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) - (x*e + d)*b*f^3*n*e^9/(b^3*n^3*e^10*log(x*e + d) + b^3*n
^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*a*d^2*f*g^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 7)/((b^3*
n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*a*d^2*g^3*Ei(2*log(c)/n + 2*a/(b*n)
+ 2*log(x*e + d))*e^(-2*a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2
/n)) + b*f^3*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 9)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d)
+ b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*b*f^2*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^
(-2*a/(b*n) + 8)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) + 9
*b*f*g^2*n*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 7)*log(x*e + d)/((b^3*n^3*e^10*log(x*e
+ d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + 4*b*g^3*n*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x*e + d))*
e^(-4*a/(b*n) + 6)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(4/n)) -
3*b*d*f^2*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 8)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n
^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 12*b*d*f*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(
b*n) + 7)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) - 9*b*d*g^3*Ei(3
*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*l
og(c) + a*b^2*n^2*e^10)*c^(3/n)) - 3*a*d*f^2*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 8)/((b^3*n^
3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 12*a*d*f*g^2*Ei(2*log(c)/n + 2*a/(b*n)
+ 2*log(x*e + d))*e^(-2*a/(b*n) + 7)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/
n)) - 9*a*d*g^3*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e + d) + b
^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + b*f^3*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 9)*l
og(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*b*f^2*g*Ei(2*log(c)/n +
2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 8)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b
^2*n^2*e^10)*c^(2/n)) + 9*b*f*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 7)*log(c)/((b^3*
n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + 4*b*g^3*Ei(4*log(c)/n + 4*a/(b*n) + 4
*log(x*e + d))*e^(-4*a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c
^(4/n)) + a*f^3*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 9)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e
^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*a*f^2*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) +
8)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) + 9*a*f*g^2*Ei(3*log(c)/n + 3*
a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 7)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^1
0)*c^(3/n)) + 4*a*g^3*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x*e + d))*e^(-4*a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e +
d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(4/n))