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

Optimal. Leaf size=219 \[ \frac{2 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^3 n}+\frac{g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n} \]

[Out]

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

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Rubi [A]  time = 0.293258, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{2 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^3 n}+\frac{g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.401409, size = 197, normalized size = 0.9 \[ \frac{e^{-\frac{3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (e^{\frac{2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )-g (d+e x) \left (-2 e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-g (d+e x) \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )\right )}{b e^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.612, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{2}}{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08685, size = 473, normalized size = 2.16 \begin{align*} \frac{{\left (g^{2} \logintegral \left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} e^{\left (\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + 2 \,{\left (e f g - d g^{2}\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\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 ) +{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left ({\left (e x + d\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b e^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.39778, size = 455, normalized size = 2.08 \begin{align*} \frac{d^{2} g^{2}{\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} - 3\right )}}{b c^{\left (\frac{1}{n}\right )} n} - \frac{2 \, d f 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} - 2\right )}}{b c^{\left (\frac{1}{n}\right )} n} - \frac{2 \, d g^{2}{\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} - 3\right )}}{b c^{\frac{2}{n}} n} + \frac{f^{2}{\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 )}}{b c^{\left (\frac{1}{n}\right )} n} + \frac{2 \, f 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} - 2\right )}}{b c^{\frac{2}{n}} n} + \frac{g^{2}{\rm Ei}\left (\frac{3 \, \log \left (c\right )}{n} + \frac{3 \, a}{b n} + 3 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{3 \, a}{b n} - 3\right )}}{b c^{\frac{3}{n}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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