3.1.95 \(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^2} \, dx\) [95]

Optimal. Leaf size=259 \[ \frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^3 n^2}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^3 n^2}+\frac {3 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^3 n^2}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

(-d*g+e*f)^2*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^3/exp(a/b/n)/n^2/((c*(e*x+d)^n)^(1/n))+4*g*(-d*g+e*f)
*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^3/exp(2*a/b/n)/n^2/((c*(e*x+d)^n)^(2/n))+3*g^2*(e*x+d)^3*Ei(3
*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^3/exp(3*a/b/n)/n^2/((c*(e*x+d)^n)^(3/n))-(e*x+d)*(g*x+f)^2/b/e/n/(a+b*ln(c*(
e*x+d)^n))

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \begin {gather*} \frac {4 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \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^3 n^2}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \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^3 n^2}+\frac {3 g^2 e^{-\frac {3 a}{b n}} (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^3 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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2209

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

Rule 2337

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2436

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 2437

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 2446

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 2447

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1015\) vs. \(2(259)=518\).
time = 0.35, size = 1015, normalized size = 3.92 \begin {gather*} \frac {e^{-\frac {3 a}{b n}} \left (c (d+e x)^n\right )^{-3/n} \left (-b d e^2 e^{\frac {3 a}{b n}} f^2 n \left (c (d+e x)^n\right )^{3/n}-b e^3 e^{\frac {3 a}{b n}} f^2 n x \left (c (d+e x)^n\right )^{3/n}-2 b d e^2 e^{\frac {3 a}{b n}} f g n x \left (c (d+e x)^n\right )^{3/n}-2 b e^3 e^{\frac {3 a}{b n}} f g n x^2 \left (c (d+e x)^n\right )^{3/n}-b d e^2 e^{\frac {3 a}{b n}} g^2 n x^2 \left (c (d+e x)^n\right )^{3/n}-b e^3 e^{\frac {3 a}{b n}} g^2 n x^3 \left (c (d+e x)^n\right )^{3/n}+a e^2 e^{\frac {2 a}{b n}} f^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )-2 a d e e^{\frac {2 a}{b n}} f g (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+a d^2 e^{\frac {2 a}{b n}} g^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+4 a e e^{\frac {a}{b n}} f g (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-4 a d e^{\frac {a}{b n}} g^2 (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+3 a g^2 (d+e x)^3 \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+b e^2 e^{\frac {2 a}{b n}} f^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )-2 b d e e^{\frac {2 a}{b n}} f g (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+b d^2 e^{\frac {2 a}{b n}} g^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+4 b e e^{\frac {a}{b n}} f g (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )-4 b d e^{\frac {a}{b n}} g^2 (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+3 b g^2 (d+e x)^3 \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.01, size = 5123, normalized size = 19.78

method	result	size
risch	\(\text {Expression too large to display}\)	\(5123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 433, normalized size = 1.67 \begin {gather*} -\frac {{\left (4 \, {\left (a d g^{2} - a f g e + {\left (b d g^{2} n - b f g n e\right )} \log \left (x e + d\right ) + {\left (b d g^{2} - b f g e\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) - {\left (a d^{2} g^{2} - 2 \, a d f g e + a f^{2} e^{2} + {\left (b d^{2} g^{2} n - 2 \, b d f g n e + b f^{2} n e^{2}\right )} \log \left (x e + d\right ) + {\left (b d^{2} g^{2} - 2 \, b d f g e + b f^{2} e^{2}\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (x e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) + {\left ({\left (b g^{2} n x^{3} + 2 \, b f g n x^{2} + b f^{2} n x\right )} e^{3} + {\left (b d g^{2} n x^{2} + 2 \, b d f g n x + b d f^{2} n\right )} e^{2}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 3 \, {\left (b g^{2} n \log \left (x e + d\right ) + b g^{2} \log \left (c\right ) + a g^{2}\right )} \operatorname {log\_integral}\left ({\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} n^{3} e^{3} \log \left (x e + d\right ) + b^{3} n^{2} e^{3} \log \left (c\right ) + a b^{2} n^{2} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2041 vs. \(2 (269) = 538\).
time = 6.25, size = 2041, normalized size = 7.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")

[Out]

-(x*e + d)^3*b*g^2*n*e^3/(b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6) + 2*(x*e + d)^2*b*d*g
^2*n*e^3/(b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6) - (x*e + d)*b*d^2*g^2*n*e^3/(b^3*n^3*
e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6) + b*d^2*g^2*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(
-a/(b*n) + 3)*log(x*e + d)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n)) - 2*(x*e
+ d)^2*b*f*g*n*e^4/(b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6) + 2*(x*e + d)*b*d*f*g*n*e^4
/(b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6) - 2*b*d*f*g*n*Ei(log(c)/n + a/(b*n) + log(x*e
+ d))*e^(-a/(b*n) + 4)*log(x*e + d)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n))
- 4*b*d*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 3)*log(x*e + d)/((b^3*n^3*e^6*log(x
*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(2/n)) + b*d^2*g^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a
/(b*n) + 3)*log(c)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n)) - (x*e + d)*b*f^2
*n*e^5/(b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6) + a*d^2*g^2*Ei(log(c)/n + a/(b*n) + log
(x*e + d))*e^(-a/(b*n) + 3)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n)) + b*f^2*
n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 5)*log(x*e + d)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6
*log(c) + a*b^2*n^2*e^6)*c^(1/n)) + 4*b*f*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 4)*l
og(x*e + d)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(2/n)) + 3*b*g^2*n*Ei(3*log(c)/
n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 3)*log(x*e + d)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c
) + a*b^2*n^2*e^6)*c^(3/n)) - 2*b*d*f*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 4)*log(c)/((b^3*n^
3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n)) - 4*b*d*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*lo
g(x*e + d))*e^(-2*a/(b*n) + 3)*log(c)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(2/n)
) - 2*a*d*f*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 4)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*
log(c) + a*b^2*n^2*e^6)*c^(1/n)) - 4*a*d*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 3)/((
b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(2/n)) + b*f^2*Ei(log(c)/n + a/(b*n) + log(x*
e + d))*e^(-a/(b*n) + 5)*log(c)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n)) + 4*
b*f*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 4)*log(c)/((b^3*n^3*e^6*log(x*e + d) + b^3*n
^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(2/n)) + 3*b*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) +
3)*log(c)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(3/n)) + a*f^2*Ei(log(c)/n + a/(b
*n) + log(x*e + d))*e^(-a/(b*n) + 5)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(1/n))
+ 4*a*f*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 4)/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2
*e^6*log(c) + a*b^2*n^2*e^6)*c^(2/n)) + 3*a*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 3)
/((b^3*n^3*e^6*log(x*e + d) + b^3*n^2*e^6*log(c) + a*b^2*n^2*e^6)*c^(3/n))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________