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

Optimal. Leaf size=268 \[ \frac{b h p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac{b h p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x) (h i-g j)}+\frac{h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac{h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac{b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac{b f p q \log (i+j x)}{(f i-e j) (h i-g j)} \]

[Out]

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

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Rubi [A]  time = 0.595351, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2418, 2394, 2393, 2391, 2395, 36, 31, 2445} \[ \frac{b h p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac{b h p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x) (h i-g j)}+\frac{h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac{h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac{b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac{b f p q \log (i+j x)}{(f i-e j) (h i-g j)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2418

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 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 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 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 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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2445

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*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (528+j x)^2} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x) (528+j x)^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{h^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(528 h-g j)^2 (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(528 h-g j) (528+j x)^2}-\frac{h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(528 h-g j)^2 (528+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h^2 \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{(528 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(h j) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{528+j x} \, dx}{(528 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{j \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(528+j x)^2} \, dx}{528 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(528 h-g j) (528+j x)}+\frac{h \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 )}{(528 h-g j)^2}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (528+j x)}{528 f-e j}\right )}{(528 h-g j)^2}-\operatorname{Subst}\left (\frac{(b f h p q) \int \frac{\log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{(528 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b f h p q) \int \frac{\log \left (\frac{f (528+j x)}{528 f-e j}\right )}{e+f x} \, dx}{(528 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f p q) \int \frac{1}{(e+f x) (528+j x)} \, dx}{528 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(528 h-g j) (528+j x)}+\frac{h \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 )}{(528 h-g j)^2}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (528+j x)}{528 f-e j}\right )}{(528 h-g j)^2}-\operatorname{Subst}\left (\frac{(b h p q) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{(528 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b h p q) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{528 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(528 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b f^2 p q\right ) \int \frac{1}{e+f x} \, dx}{(528 f-e j) (528 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b f j p q) \int \frac{1}{528+j x} \, dx}{(528 f-e j) (528 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{b f p q \log (e+f x)}{(528 f-e j) (528 h-g j)}+\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(528 h-g j) (528+j x)}+\frac{h \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 )}{(528 h-g j)^2}+\frac{b f p q \log (528+j x)}{(528 f-e j) (528 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (528+j x)}{528 f-e j}\right )}{(528 h-g j)^2}+\frac{b h p q \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{(528 h-g j)^2}-\frac{b h p q \text{Li}_2\left (-\frac{j (e+f x)}{528 f-e j}\right )}{(528 h-g j)^2}\\ \end{align*}

Mathematica [A]  time = 0.306487, size = 225, normalized size = 0.84 \[ \frac{b h p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )-b h p q \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )+h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+\frac{a (h i-g j)}{i+j x}+\frac{b (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}-\frac{b f p q (h i-g j) (\log (e+f x)-\log (i+j x))}{f i-e j}}{(h i-g j)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.121, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }{ \left ( hx+g \right ) \left ( jx+i \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a{\left (\frac{h \log \left (h x + g\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} - \frac{h \log \left (j x + i\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} + \frac{1}{h i^{2} - g i j +{\left (h i j - g j^{2}\right )} x}\right )} + b \int \frac{\log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \left (c\right ) + \log \left (d^{q}\right )}{h j^{2} x^{3} + g i^{2} +{\left (2 \, h i j + g j^{2}\right )} x^{2} +{\left (h i^{2} + 2 \, g i j\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h j^{2} x^{3} + g i^{2} +{\left (2 \, h i j + g j^{2}\right )} x^{2} +{\left (h i^{2} + 2 \, g i j\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )}{\left (j x + i\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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