\(\int \frac {(i+j x) (a+b \log (c (d (e+f x)^p)^q))}{g+h x} \, dx\) [525]

Optimal result

Integrand size = 31, antiderivative size = 129 \[ \int \frac {(i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\frac {a j x}{h}-\frac {b j p q x}{h}+\frac {b j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}+\frac {(h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^2}+\frac {b (h i-g j) p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2465, 2436, 2332, 2441, 2440, 2438, 2495} \[ \int \frac {(i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\frac {(h i-g j) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac {a j x}{h}+\frac {b j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}+\frac {b p q (h i-g j) \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^2}-\frac {b j p q x}{h} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.93 \[ \int \frac {(i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx=\frac {a h j x-b h j p q x+\frac {b h j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+(h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+b (h i-g j) p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )}{h^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((j*x+i)*(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)/(g + h*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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