3.6.28 \(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{(g+h x) (i+j x)^2} \, dx\) [528]

Optimal. Leaf size=268 \[ -\frac {b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(h i-g j) (i+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {b f p q \log (i+j x)}{(f i-e j) (h i-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {b h p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {b h p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \]

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 268, normalized size of antiderivative = 1.00, 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 = {2465, 2441, 2440, 2438, 2442, 36, 31, 2495} \begin {gather*} \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)} \end {gather*}

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

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Mathematica [A]
time = 0.21, size = 527, normalized size = 1.97 \begin {gather*} \frac {a (f i-e j) (h i-g j)-b (f i-e j) (h i-g j) p q \log (e+f x)+b e j (-h i+g j) p q \log (e+f x)+b f j (-h i+g j) p q x \log (e+f x)+b (f i-e j) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )+a h (f i-e j) (i+j x) \log (g+h x)-b h (f i-e j) p q (i+j x) \log (e+f x) \log (g+h x)+b h (f i-e j) (i+j x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b h (f i-e j) p q (i+j x) \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+b f i (h i-g j) p q \log (i+j x)+b f j (h i-g j) p q x \log (i+j x)-a h (f i-e j) (i+j x) \log (i+j x)+b h (f i-e j) p q (i+j x) \log (e+f x) \log (i+j x)-b h (f i-e j) (i+j x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)-b h (f i-e j) p q (i+j x) \log (e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+b h (f i-e j) p q (i+j x) \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-b h (f i-e j) p q (i+j x) \text {Li}_2\left (\frac {j (e+f x)}{-f i+e j}\right )}{(f i-e j) (h i-g j)^2 (i+j x)} \end {gather*}

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

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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )^{2}}\, dx\]

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{2}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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