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

Optimal. Leaf size=181 \[ \frac {f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac {a f x}{g^2}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}+\frac {b f^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {b d n x}{2 e g}+\frac {b f n x}{g^2}-\frac {b n x^2}{4 g} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ \frac {b f^2 n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac {a f x}{g^2}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}+\frac {b d n x}{2 e g}+\frac {b f n x}{g^2}-\frac {b n x^2}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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 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 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 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 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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 170, normalized size = 0.94 \[ \frac {f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac {a f x}{g^2}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {b n \left (-\frac {2 d^2 \log (d+e x)}{e^2}+\frac {2 d x}{e}-x^2\right )}{4 g}+\frac {b f^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {b f n x}{g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a x^{2}}{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.27, size = 724, normalized size = 4.00 \[ -\frac {b f x \ln \left (\left (e x +d \right )^{n}\right )}{g^{2}}+\frac {b \,f^{2} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g^{3}}-\frac {b d f n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{e \,g^{2}}-\frac {b \,f^{2} n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{g^{3}}-\frac {b f x \ln \relax (c )}{g^{2}}+\frac {b \,f^{2} \ln \relax (c ) \ln \left (g x +f \right )}{g^{3}}-\frac {i \pi b f x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g^{2}}-\frac {i \pi b f x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g^{2}}+\frac {i \pi b \,f^{2} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{3}}+\frac {i \pi b \,f^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{3}}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{4 g}+\frac {a \,f^{2} \ln \left (g x +f \right )}{g^{3}}+\frac {b \,x^{2} \ln \relax (c )}{2 g}-\frac {i \pi b \,f^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{2 g^{3}}+\frac {i \pi b f x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2 g^{2}}+\frac {b d f n}{2 e \,g^{2}}-\frac {b \,f^{2} n \ln \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right ) \ln \left (g x +f \right )}{g^{3}}+\frac {b \,x^{2} \ln \left (\left (e x +d \right )^{n}\right )}{2 g}-\frac {b \,d^{2} n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{2 e^{2} g}+\frac {a \,x^{2}}{2 g}+\frac {5 b \,f^{2} n}{4 g^{3}}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{4 g}+\frac {b f n x}{g^{2}}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4 g}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4 g}-\frac {i \pi b \,f^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 g^{3}}+\frac {i \pi b f x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 g^{2}}-\frac {b n \,x^{2}}{4 g}+\frac {b d n x}{2 e g}-\frac {a f x}{g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {2 \, f^{2} \log \left (g x + f\right )}{g^{3}} + \frac {g x^{2} - 2 \, f x}{g^{2}}\right )} + b \int \frac {x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + x^{2} \log \relax (c)}{g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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