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

Optimal. Leaf size=241 \[ \frac{b n (g h-f i)^2 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g^3}+\frac{(g h-f i)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{a i x (g h-f i)}{g^2}+\frac{b i (d+e x) (g h-f i) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{b n (e h-d i)^2 \log (d+e x)}{2 e^2 g}-\frac{b i n x (e h-d i)}{2 e g}-\frac{b i n x (g h-f i)}{g^2}-\frac{b n (h+i x)^2}{4 g} \]

[Out]

(a*i*(g*h - f*i)*x)/g^2 - (b*i*(e*h - d*i)*n*x)/(2*e*g) - (b*i*(g*h - f*i)*n*x)/g^2 - (b*n*(h + i*x)^2)/(4*g)
- (b*(e*h - d*i)^2*n*Log[d + e*x])/(2*e^2*g) + (b*i*(g*h - f*i)*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + ((h +
i*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f -
d*g)])/g^3 + (b*(g*h - f*i)^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3

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Rubi [A]  time = 0.222943, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2418, 2389, 2295, 2394, 2393, 2391, 2395, 43} \[ \frac{b n (g h-f i)^2 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g^3}+\frac{(g h-f i)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{a i x (g h-f i)}{g^2}+\frac{b i (d+e x) (g h-f i) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{b n (e h-d i)^2 \log (d+e x)}{2 e^2 g}-\frac{b i n x (e h-d i)}{2 e g}-\frac{b i n x (g h-f i)}{g^2}-\frac{b n (h+i x)^2}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*i*(g*h - f*i)*x)/g^2 - (b*i*(e*h - d*i)*n*x)/(2*e*g) - (b*i*(g*h - f*i)*n*x)/g^2 - (b*n*(h + i*x)^2)/(4*g)
- (b*(e*h - d*i)^2*n*Log[d + e*x])/(2*e^2*g) + (b*i*(g*h - f*i)*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + ((h +
i*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f -
d*g)])/g^3 + (b*(g*h - f*i)^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2395

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(h+218 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx &=\int \left (\frac{218 (-218 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{218 (h+218 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{(-218 f+g h)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}\right ) \, dx\\ &=\frac{218 \int (h+218 x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}-\frac{(218 (218 f-g h)) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}+\frac{(218 f-g h)^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g^2}\\ &=-\frac{218 a (218 f-g h) x}{g^2}+\frac{(h+218 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{(218 f-g h)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g^3}-\frac{(218 b (218 f-g h)) \int \log \left (c (d+e x)^n\right ) \, dx}{g^2}-\frac{(b e n) \int \frac{(h+218 x)^2}{d+e x} \, dx}{2 g}-\frac{\left (b e (218 f-g h)^2 n\right ) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^3}\\ &=-\frac{218 a (218 f-g h) x}{g^2}+\frac{(h+218 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{(218 f-g h)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g^3}-\frac{(218 b (218 f-g h)) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}-\frac{(b e n) \int \left (\frac{218 (-218 d+e h)}{e^2}+\frac{218 (h+218 x)}{e}+\frac{(-218 d+e h)^2}{e^2 (d+e x)}\right ) \, dx}{2 g}-\frac{\left (b (218 f-g h)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^3}\\ &=-\frac{218 a (218 f-g h) x}{g^2}+\frac{109 b (218 d-e h) n x}{e g}+\frac{218 b (218 f-g h) n x}{g^2}-\frac{b n (h+218 x)^2}{4 g}-\frac{b (218 d-e h)^2 n \log (d+e x)}{2 e^2 g}-\frac{218 b (218 f-g h) (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{(h+218 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{(218 f-g h)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g^3}+\frac{b (218 f-g h)^2 n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^3}\\ \end{align*}

Mathematica [A]  time = 0.261613, size = 224, normalized size = 0.93 \[ \frac{4 b e^2 n (g h-f i)^2 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+e \left (g i x (2 a e (-2 f i+4 g h+g i x)+b n (2 d g i-e (-4 f i+8 g h+g i x)))+4 a e (g h-f i)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )+2 b \log \left (c (d+e x)^n\right ) \left (g i (d (4 g h-2 f i)+e x (-2 f i+4 g h+g i x))+2 e (g h-f i)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )\right )\right )-2 b d^2 g^2 i^2 n \log (d+e x)}{4 e^2 g^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*d^2*g^2*i^2*n*Log[d + e*x] + e*(g*i*x*(2*a*e*(4*g*h - 2*f*i + g*i*x) + b*n*(2*d*g*i - e*(8*g*h - 4*f*i +
g*i*x))) + 4*a*e*(g*h - f*i)^2*Log[(e*(f + g*x))/(e*f - d*g)] + 2*b*Log[c*(d + e*x)^n]*(g*i*(d*(4*g*h - 2*f*i
) + e*x*(4*g*h - 2*f*i + g*i*x)) + 2*e*(g*h - f*i)^2*Log[(e*(f + g*x))/(e*f - d*g)])) + 4*b*e^2*(g*h - f*i)^2*
n*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)])/(4*e^2*g^3)

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Maple [C]  time = 0.536, size = 1605, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*ln(c)/g^2*ln(g*x+f)*f*h*i-b/e*n/g^2*i^2*d*ln((g*x+f)*e+d*g-f*e)*f+2*b/e*n/g*i*d*ln((g*x+f)*e+d*g-f*e)*h+2
*b*n/g^2*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f*h*i+1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*
x+d)^n)*i^2/g^2*f*x-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i/g*h*x-1/2*I*b*Pi*csgn(I*c)*csgn(I
*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g^3*ln(g*x+f)*f^2*i^2-2*a/g^2*ln(g*x+f)*f*h*i+b*ln(c)/g^3*ln(g*x+f)*f^2*i^2-b*
ln(c)*i^2/g^2*f*x+2*b*ln(c)*i/g*h*x+I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*ln(g*x+f)*f*h*i-1/2*I*b*Pi*csgn(I*c)*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g*ln(g*x+f)*h^2-1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i^2/g^2*f
*x+2*a*i/g*h*x+a/g^3*ln(g*x+f)*f^2*i^2+b*ln(c)/g*ln(g*x+f)*h^2+1/2*b*ln(c)*i^2/g*x^2-b*n/g^3*ln(g*x+f)*ln(((g*
x+f)*e+d*g-f*e)/(d*g-e*f))*f^2*i^2+1/2*b/e*n/g*i^2*d*x-1/2*b/e^2*n/g*i^2*d^2*ln((g*x+f)*e+d*g-f*e)-I*b*Pi*csgn
(I*c)*csgn(I*c*(e*x+d)^n)^2/g^2*ln(g*x+f)*f*h*i+1/2*b*ln((e*x+d)^n)*i^2/g*x^2+1/2*b/e*n/g^2*i^2*d*f+2*b*n/g^2*
dilog(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f*h*i+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i/g*h*x+b*ln((e*x+d)
^n)/g*ln(g*x+f)*h^2-1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i^2/g^2*f*x-b*ln((e*x+d)^n)*i^2/g^2*f*x-I*b*Pi*
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g^2*ln(g*x+f)*f*h*i+5/4*b*n/g^3*f^2*i^2-b*n/g*dilog(((g*x+f)*e+d*g-f*e
)/(d*g-e*f))*h^2-1/4*b*n/g*i^2*x^2-a*i^2/g^2*f*x+1/2*a*i^2/g*x^2+a/g*ln(g*x+f)*h^2+1/2*I*b*Pi*csgn(I*c)*csgn(I
*c*(e*x+d)^n)^2/g*ln(g*x+f)*h^2+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i^2/g^2*f*x+1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*
x+d)^n)^2*i^2/g*x^2+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g*ln(g*x+f)*h^2-I*b*Pi*csgn(I*c*(e*x+d)
^n)^3*i/g*h*x-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^3*ln(g*x+f)*f^2*i^2-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g*ln(g*x
+f)*h^2+1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i^2/g*x^2-2*b*n/g^2*f*h*i-b*n/g^3*dilog(((g*x+f)*e+
d*g-f*e)/(d*g-e*f))*f^2*i^2-b*n/g*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*h^2+b*n/g^2*i^2*x*f-2*b*n/g*i*h*
x+2*b*ln((e*x+d)^n)*i/g*h*x+b*ln((e*x+d)^n)/g^3*ln(g*x+f)*f^2*i^2+I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*
(e*x+d)^n)/g^2*ln(g*x+f)*f*h*i-2*b*ln((e*x+d)^n)/g^2*ln(g*x+f)*f*h*i-1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i^2/g*x^
2+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i/g*h*x+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g^3*ln(g*x
+f)*f^2*i^2-1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i^2/g*x^2+1/2*I*b*Pi*csgn(I*c)*csgn(I*c
*(e*x+d)^n)^2/g^3*ln(g*x+f)*f^2*i^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, a h i{\left (\frac{x}{g} - \frac{f \log \left (g x + f\right )}{g^{2}}\right )} + \frac{1}{2} \, a i^{2}{\left (\frac{2 \, f^{2} \log \left (g x + f\right )}{g^{3}} + \frac{g x^{2} - 2 \, f x}{g^{2}}\right )} + \frac{a h^{2} \log \left (g x + f\right )}{g} + \int \frac{b i^{2} x^{2} \log \left (c\right ) + 2 \, b h i x \log \left (c\right ) + b h^{2} \log \left (c\right ) +{\left (b i^{2} x^{2} + 2 \, b h i x + b h^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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