\(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{(g+h x) (i+j x)^3} \, dx\) [529]

Optimal result

Integrand size = 33, antiderivative size = 425 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=-\frac {b f p q}{2 (f i-e j) (h i-g j) (i+j x)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (h i-g j) (i+j x)^2}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2 (i+j x)}+\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^3}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^3}+\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2465, 2441, 2440, 2438, 2442, 46, 36, 31, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\frac {h^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}-\frac {h^2 \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}-\frac {b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2} \]

[In]

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

[Out]

-1/2*(b*f*p*q)/((f*i - e*j)*(h*i - g*j)*(i + j*x)) - (b*f*h*p*q*Log[e + f*x])/((f*i - e*j)*(h*i - g*j)^2) - (b
*f^2*p*q*Log[e + f*x])/(2*(f*i - e*j)^2*(h*i - g*j)) + (a + b*Log[c*(d*(e + f*x)^p)^q])/(2*(h*i - g*j)*(i + j*
x)^2) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q]))/((h*i - g*j)^2*(i + j*x)) + (h^2*(a + b*Log[c*(d*(e + f*x)^p)^q])
*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j)^3 + (b*f*h*p*q*Log[i + j*x])/((f*i - e*j)*(h*i - g*j)^2) + (b*f^2
*p*q*Log[i + j*x])/(2*(f*i - e*j)^2*(h*i - g*j)) - (h^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*
i - e*j)])/(h*i - g*j)^3 + (b*h^2*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^3 - (b*h^2*p*q*Pol
yLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^3

Rule 31

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

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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\frac {\frac {a (h i-g j)^2}{(i+j x)^2}+\frac {2 a h (h i-g j)}{i+j x}+\frac {b (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x)^2}+\frac {2 b h (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}+2 h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-\frac {2 b f h (h i-g j) p q (\log (e+f x)-\log (i+j x))}{f i-e j}-\frac {b f (h i-g j)^2 p q (f i-e j+f (i+j x) \log (e+f x)-f (i+j x) \log (i+j x))}{(f i-e j)^2 (i+j x)}-2 h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )+2 b h^2 p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 b h^2 p q \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )}{2 (h i-g j)^3} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x, algorithm="fricas")

[Out]

integral((b*log(((f*x + e)^p*d)^q*c) + a)/(h*j^3*x^4 + g*i^3 + (3*h*i*j^2 + g*j^3)*x^3 + 3*(h*i^2*j + g*i*j^2)
*x^2 + (h*i^3 + 3*g*i^2*j)*x), x)

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right ) \left (i + j x\right )^{3}}\, dx \]

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x+g)/(j*x+i)**3,x)

[Out]

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

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^3} \,d x \]

[In]

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

[Out]

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