\(\int \frac {(i+j x)^2 (a+b \log (c (d (e+f x)^p)^q))}{g+h x} \, dx\) [524]

Optimal result

Integrand size = 33, antiderivative size = 258 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\frac {a j (h i-g j) x}{h^2}-\frac {b j (f i-e j) p q x}{2 f h}-\frac {b j (h i-g j) p q x}{h^2}-\frac {b p q (i+j x)^2}{4 h}-\frac {b (f i-e j)^2 p q \log (e+f x)}{2 f^2 h}+\frac {b j (h i-g j) (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {b (h i-g j)^2 p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2465, 2436, 2332, 2441, 2440, 2438, 2442, 45, 2495} \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\frac {(h i-g j)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^3}+\frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}+\frac {a j x (h i-g j)}{h^2}+\frac {b j (e+f x) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac {b p q (f i-e j)^2 \log (e+f x)}{2 f^2 h}+\frac {b p q (h i-g j)^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {b j p q x (f i-e j)}{2 f h}-\frac {b j p q x (h i-g j)}{h^2}-\frac {b p q (i+j x)^2}{4 h} \]

[In]

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

[Out]

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

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {(i+j x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\int \left (\frac {j (h i-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^2}+\frac {(h i-g j)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^2 (g+h x)}+\frac {j (i+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {j \int (i+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(j (h i-g j)) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(h i-g j)^2 \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {a j (h i-g j) x}{h^2}+\frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\text {Subst}\left (\frac {(b j (h i-g j)) \int \log \left (c d^q (e+f x)^{p q}\right ) \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(b f p q) \int \frac {(i+j x)^2}{e+f x} \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (b f (h i-g j)^2 p q\right ) \int \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {a j (h i-g j) x}{h^2}+\frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\text {Subst}\left (\frac {(b j (h i-g j)) \text {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(b f p q) \int \left (\frac {j (f i-e j)}{f^2}+\frac {(f i-e j)^2}{f^2 (e+f x)}+\frac {j (i+j x)}{f}\right ) \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (b (h i-g j)^2 p q\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {a j (h i-g j) x}{h^2}-\frac {b j (f i-e j) p q x}{2 f h}-\frac {b j (h i-g j) p q x}{h^2}-\frac {b p q (i+j x)^2}{4 h}-\frac {b (f i-e j)^2 p q \log (e+f x)}{2 f^2 h}+\frac {b j (h i-g j) (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {b (h i-g j)^2 p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.90 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\frac {-2 b e^2 h^2 j^2 p q \log (e+f x)+f \left (h j x (2 a f (4 h i-2 g j+h j x)+b p q (2 e h j-f (8 h i-4 g j+h j x)))+4 a f (h i-g j)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 b \log \left (c \left (d (e+f x)^p\right )^q\right ) \left (h j (e (4 h i-2 g j)+f x (4 h i-2 g j+h j x))+2 f (h i-g j)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )\right )+4 b f^2 (h i-g j)^2 p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )}{4 f^2 h^3} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (j x +i \right )^{2} \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}{h x +g}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}}{h x + g} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right ) \left (i + j x\right )^{2}}{g + h x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}}{h x + g} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}}{h x + g} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\int \frac {{\left (i+j\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}{g+h\,x} \,d x \]

[In]

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

[Out]

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