Integrand size = 31, antiderivative size = 469 \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=-\frac {2 a b i (e h-d i) n x}{e g}-\frac {2 a b i (g h-f i) n x}{g^2}+\frac {2 b^2 i (e h-d i) n^2 x}{e g}+\frac {2 b^2 i (g h-f i) n^2 x}{g^2}+\frac {b^2 i^2 n^2 (d+e x)^2}{4 e^2 g}-\frac {2 b^2 i (e h-d i) n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}-\frac {2 b^2 i (g h-f i) n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g}+\frac {i (e h-d i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}+\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {2 b (g h-f i)^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}-\frac {2 b^2 (g h-f i)^2 n^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]
-2*a*b*i*(-d*i+e*h)*n*x/e/g-2*a*b*i*(-f*i+g*h)*n*x/g^2+2*b^2*i*(-d*i+e*h)* n^2*x/e/g+2*b^2*i*(-f*i+g*h)*n^2*x/g^2+1/4*b^2*i^2*n^2*(e*x+d)^2/e^2/g-2*b ^2*i*(-d*i+e*h)*n*(e*x+d)*ln(c*(e*x+d)^n)/e^2/g-2*b^2*i*(-f*i+g*h)*n*(e*x+ d)*ln(c*(e*x+d)^n)/e/g^2-1/2*b*i^2*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^2/g +i*(-d*i+e*h)*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e^2/g+i*(-f*i+g*h)*(e*x+d)*( a+b*ln(c*(e*x+d)^n))^2/e/g^2+1/2*i^2*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^2 /g+(-f*i+g*h)^2*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(g*x+f)/(-d*g+e*f))/g^3+2*b*( -f*i+g*h)^2*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g^3-2 *b^2*(-f*i+g*h)^2*n^2*polylog(3,-g*(e*x+d)/(-d*g+e*f))/g^3
Time = 0.33 (sec) , antiderivative size = 876, normalized size of antiderivative = 1.87 \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {4 e^2 g i (2 g h-f i) x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 e^2 g^2 i^2 x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+4 e^2 (g h-f i)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+8 b e^2 g^2 h^2 n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+2 b i^2 n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e g (e x (4 f-g x)+2 d (2 f+g x))-2 \log (d+e x) \left (g (d+e x) (2 e f+d g-e g x)-2 e^2 f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+4 e^2 f^2 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )-16 b e g h i n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-g (d+e x) (-1+\log (d+e x))+e f \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )+8 b^2 e g h i n^2 \left (g \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )-e f \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )-b^2 i^2 n^2 \left (4 e f g \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )+g^2 \left (e x (6 d-e x)+\left (-6 d^2-4 d e x+2 e^2 x^2\right ) \log (d+e x)+2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)\right )-4 e^2 f^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )+4 b^2 e^2 g^2 h^2 n^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{4 e^2 g^3} \]
(4*e^2*g*i*(2*g*h - f*i)*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 2*e^2*g^2*i^2*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 4*e ^2*(g*h - f*i)^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g *x] + 8*b*e^2*g^2*h^2*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(Log [d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f ) + d*g)]) + 2*b*i^2*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e*g* (e*x*(4*f - g*x) + 2*d*(2*f + g*x)) - 2*Log[d + e*x]*(g*(d + e*x)*(2*e*f + d*g - e*g*x) - 2*e^2*f^2*Log[(e*(f + g*x))/(e*f - d*g)]) + 4*e^2*f^2*Poly Log[2, (g*(d + e*x))/(-(e*f) + d*g)]) - 16*b*e*g*h*i*n*(a - b*n*Log[d + e* x] + b*Log[c*(d + e*x)^n])*(-(g*(d + e*x)*(-1 + Log[d + e*x])) + e*f*(Log[ d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)])) + 8*b^2*e*g*h*i*n^2*(g*(2*e*x - 2*(d + e*x)*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2) - e*f*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)] + 2*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)])) - b^2*i^2*n^2*(4*e*f*g*(2*e*x - 2*(d + e*x )*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2) + g^2*(e*x*(6*d - e*x) + (-6*d^ 2 - 4*d*e*x + 2*e^2*x^2)*Log[d + e*x] + 2*(d^2 - e^2*x^2)*Log[d + e*x]^2) - 4*e^2*f^2*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)] + 2*Log[d + e*x ]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*x))/(- (e*f) + d*g)])) + 4*b^2*e^2*g^2*h^2*n^2*(Log[d + e*x]^2*Log[(e*(f + g*x...
Time = 0.83 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2865, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx\) |
\(\Big \downarrow \) 2865 |
\(\displaystyle \int \left (\frac {i (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 (f+g x)}+\frac {i (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac {b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}+\frac {2 b n (g h-f i)^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {(g h-f i)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^3}+\frac {i (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {2 a b i n x (e h-d i)}{e g}-\frac {2 a b i n x (g h-f i)}{g^2}-\frac {2 b^2 i n (d+e x) (e h-d i) \log \left (c (d+e x)^n\right )}{e^2 g}-\frac {2 b^2 i n (d+e x) (g h-f i) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {b^2 i^2 n^2 (d+e x)^2}{4 e^2 g}-\frac {2 b^2 n^2 (g h-f i)^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {2 b^2 i n^2 x (e h-d i)}{e g}+\frac {2 b^2 i n^2 x (g h-f i)}{g^2}\) |
(-2*a*b*i*(e*h - d*i)*n*x)/(e*g) - (2*a*b*i*(g*h - f*i)*n*x)/g^2 + (2*b^2* i*(e*h - d*i)*n^2*x)/(e*g) + (2*b^2*i*(g*h - f*i)*n^2*x)/g^2 + (b^2*i^2*n^ 2*(d + e*x)^2)/(4*e^2*g) - (2*b^2*i*(e*h - d*i)*n*(d + e*x)*Log[c*(d + e*x )^n])/(e^2*g) - (2*b^2*i*(g*h - f*i)*n*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^ 2) - (b*i^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g) + (i*(e*h - d*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g) + (i*(g*h - f*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) + (i^2*(d + e*x)^2*(a + b*Lo g[c*(d + e*x)^n])^2)/(2*e^2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n]) ^2*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (2*b*(g*h - f*i)^2*n*(a + b*Log[c *(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3 - (2*b^2*(g*h - f*i)^2*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g^3
3.3.24.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> 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] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
\[\int \frac {\left (i x +h \right )^{2} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{g x +f}d x\]
\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{g x + f} \,d x } \]
integral((a^2*i^2*x^2 + 2*a^2*h*i*x + a^2*h^2 + (b^2*i^2*x^2 + 2*b^2*h*i*x + b^2*h^2)*log((e*x + d)^n*c)^2 + 2*(a*b*i^2*x^2 + 2*a*b*h*i*x + a*b*h^2) *log((e*x + d)^n*c))/(g*x + f), x)
\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (h + i x\right )^{2}}{f + g x}\, dx \]
\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{g x + f} \,d x } \]
2*a^2*h*i*(x/g - f*log(g*x + f)/g^2) + 1/2*a^2*i^2*(2*f^2*log(g*x + f)/g^3 + (g*x^2 - 2*f*x)/g^2) + a^2*h^2*log(g*x + f)/g + integrate((b^2*h^2*log( c)^2 + 2*a*b*h^2*log(c) + (b^2*i^2*log(c)^2 + 2*a*b*i^2*log(c))*x^2 + (b^2 *i^2*x^2 + 2*b^2*h*i*x + b^2*h^2)*log((e*x + d)^n)^2 + 2*(b^2*h*i*log(c)^2 + 2*a*b*h*i*log(c))*x + 2*(b^2*h^2*log(c) + a*b*h^2 + (b^2*i^2*log(c) + a *b*i^2)*x^2 + 2*(b^2*h*i*log(c) + a*b*h*i)*x)*log((e*x + d)^n))/(g*x + f), x)
\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\int \frac {{\left (h+i\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{f+g\,x} \,d x \]