3.1.96 \(\int \frac {f+g x}{(a+b \log (c (d+e x)^n))^2} \, dx\) [96]

Optimal. Leaf size=177 \[ \frac {e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}+\frac {2 e^{-\frac {2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \begin {gather*} \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 )}{b^2 e^2 n^2}+\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^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{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)/(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*e^2*E^(a/(b*n))*n^2*(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^2*e^2*E^((2*a)/(b*n))*n^
2*(c*(d + e*x)^n)^(2/n)) - ((d + e*x)*(f + g*x))/(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}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=-\frac {(d+e x) (f+g x)}{b e n \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 n}-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e n}\\ &=-\frac {(d+e x) (f+g x)}{b e n \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 n}-\frac {(e f-d g) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^2 n}\\ &=-\frac {(d+e x) (f+g x)}{b e n \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 e n}+\frac {(2 (e f-d g)) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e n}-\frac {\left ((e f-d g) (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^2 n^2}\\ &=-\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 )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(2 g) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^2 n}+\frac {(2 (e f-d g)) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^2 n}\\ &=-\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 )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \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 ) \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^2 n^2}+\frac {\left (2 (e f-d g) (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^2 n^2}\\ &=\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 )}{b^2 e^2 n^2}+\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^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 208, normalized size = 1.18 \begin {gather*} -\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (b e e^{\frac {2 a}{b n}} n \left (c (d+e x)^n\right )^{2/n} (f+g x)-e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\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 )-2 g (d+e x) \text {Ei}\left (\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 )\right )}{b^2 e^2 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)/(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.70, size = 2300, normalized size = 12.99

method	result	size
risch	\(\text {Expression too large to display}\)	\(2300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(e*x+d)*(g*x+f)/b/e/n/(2*a+2*b*ln(c)+2*b*ln((e*x+d)^n)-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^
n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+
d)^n)^3)-1/b^2/n^2*f*((e*x+d)^n)^(-1/n)*c^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d
)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*
x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I
*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c
)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*x-1/b^2/e/n^2*f*((e*x+d)^n)^(-1/n)*c^(-1/n)*exp(-1/2*(-I*b*Pi*csgn
(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*cs
gn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^
2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d-2/b^2/n^2*g*((e*x+d)^n)^(
-2/n)*c^(-2/n)*exp(-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)
^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-
(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*
(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b
/n)*x^2-4/b^2/e/n^2*g*((e*x+d)^n)^(-2/n)*c^(-2/n)*exp(-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d
)^n)^3+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*c
sgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b
*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d*x-2/b^2/e^2/n^2*g*((e*x+d)^n)^(-2/n)*c^(-2/n)*exp(-(-I*b*Pi*csgn(I*c)
*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*
c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*c
sgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*P
i*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d^2+1/b^2/e/n^2*d*g*((e*x+d)^n)^(-
1/n)*c^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x
+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)
-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*cs
gn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2
*a)/b/n)*x+1/b^2/e^2/n^2*d^2*g*((e*x+d)^n)^(-1/n)*c^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(
I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*cs
gn(I*c*(e*x+d)^n)^3+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b
*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^
3+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)

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

[Out]

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

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Fricas [A]
time = 0.38, size = 244, normalized size = 1.38 \begin {gather*} -\frac {{\left ({\left (a d g - a f e + {\left (b d g n - b f n e\right )} \log \left (x e + d\right ) + {\left (b d g - b f 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 e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) + {\left ({\left (b g n x^{2} + b f n x\right )} e^{2} + {\left (b d g n x + b d f n\right )} e\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 2 \, {\left (b g n \log \left (x e + d\right ) + b g \log \left (c\right ) + a g\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 )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} n^{3} e^{2} \log \left (x e + d\right ) + b^{3} n^{2} e^{2} \log \left (c\right ) + a b^{2} n^{2} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 984 vs. \(2 (184) = 368\).
time = 3.30, size = 984, normalized size = 5.56 \begin {gather*} -\frac {{\left (x e + d\right )}^{2} b g n e}{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}} + \frac {{\left (x e + d\right )} b d g n e}{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}} - \frac {b d g n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} + 1\right )} \log \left (x e + d\right )}{{\left (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}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (x e + d\right )} b f n e^{2}}{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}} + \frac {b f n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} + 2\right )} \log \left (x e + d\right )}{{\left (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}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {2 \, b g n {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} + 1\right )} \log \left (x e + d\right )}{{\left (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}\right )} c^{\frac {2}{n}}} - \frac {b d g {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} + 1\right )} \log \left (c\right )}{{\left (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}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {a d g {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} + 1\right )}}{{\left (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}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {b f {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} + 2\right )} \log \left (c\right )}{{\left (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}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {2 \, b g {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} + 1\right )} \log \left (c\right )}{{\left (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}\right )} c^{\frac {2}{n}}} + \frac {a f {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} + 2\right )}}{{\left (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}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {2 \, a g {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} + 1\right )}}{{\left (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}\right )} c^{\frac {2}{n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f+g\,x}{{\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)/(a + b*log(c*(d + e*x)^n))^2,x)

[Out]

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

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