3.3.18 \(\int \frac {(h+i x)^2 (a+b \log (c (d+e x)^n))}{f+g x} \, dx\) [218]

Optimal. Leaf size=241 \[ \frac {a i (g h-f i) x}{g^2}-\frac {b i (e h-d i) n x}{2 e g}-\frac {b i (g h-f i) n x}{g^2}-\frac {b n (h+i x)^2}{4 g}-\frac {b (e h-d i)^2 n \log (d+e x)}{2 e^2 g}+\frac {b i (g h-f i) (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b (g h-f i)^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]

[Out]

a*i*(-f*i+g*h)*x/g^2-1/2*b*i*(-d*i+e*h)*n*x/e/g-b*i*(-f*i+g*h)*n*x/g^2-1/4*b*n*(i*x+h)^2/g-1/2*b*(-d*i+e*h)^2*
n*ln(e*x+d)/e^2/g+b*i*(-f*i+g*h)*(e*x+d)*ln(c*(e*x+d)^n)/e/g^2+1/2*(i*x+h)^2*(a+b*ln(c*(e*x+d)^n))/g+(-f*i+g*h
)^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/g^3+b*(-f*i+g*h)^2*n*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g^3

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 241, normalized size of antiderivative = 1.00, 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 = {2465, 2436, 2332, 2441, 2440, 2438, 2442, 45} \begin {gather*} \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} \end {gather*}

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 45

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])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, 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 2438

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 2440

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 2441

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 2442

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 2465

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]

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)) \text {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 ) \text {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 = 0.19, size = 224, normalized size = 0.93 \begin {gather*} \frac {-2 b d^2 g^2 i^2 n \log (d+e x)+e \left (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 \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 (4 g h-2 f i+g i x))+2 e (g h-f i)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )\right )+4 b e^2 (g h-f i)^2 n \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )}{4 e^2 g^3} \end {gather*}

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)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.50, size = 1605, normalized size = 6.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(1605\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________