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

Optimal. Leaf size=128 \[ -\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{b p r \log (a+b x)}{h (b g-a h)}-\frac{b p r \log (g+h x)}{h (b g-a h)}+\frac{d q r \log (c+d x)}{h (d g-c h)}-\frac{d q r \log (g+h x)}{h (d g-c h)} \]

[Out]

(b*p*r*Log[a + b*x])/(h*(b*g - a*h)) + (d*q*r*Log[c + d*x])/(h*(d*g - c*h)) - Log[e*(f*(a + b*x)^p*(c + d*x)^q
)^r]/(h*(g + h*x)) - (b*p*r*Log[g + h*x])/(h*(b*g - a*h)) - (d*q*r*Log[g + h*x])/(h*(d*g - c*h))

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Rubi [A]  time = 0.0524397, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2495, 36, 31} \[ -\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{b p r \log (a+b x)}{h (b g-a h)}-\frac{b p r \log (g+h x)}{h (b g-a h)}+\frac{d q r \log (c+d x)}{h (d g-c h)}-\frac{d q r \log (g+h x)}{h (d g-c h)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*p*r*Log[a + b*x])/(h*(b*g - a*h)) + (d*q*r*Log[c + d*x])/(h*(d*g - c*h)) - Log[e*(f*(a + b*x)^p*(c + d*x)^q
)^r]/(h*(g + h*x)) - (b*p*r*Log[g + h*x])/(h*(b*g - a*h)) - (d*q*r*Log[g + h*x])/(h*(d*g - c*h))

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -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]

Rubi steps

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

Mathematica [A]  time = 0.232556, size = 93, normalized size = 0.73 \[ \frac{-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x}+\frac{b p r (\log (a+b x)-\log (g+h x))}{b g-a h}+\frac{d q r (\log (c+d x)-\log (g+h x))}{d g-c h}}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)) + (b*p*r*(Log[a + b*x] - Log[g + h*x]))/(b*g - a*h) + (d*q*
r*(Log[c + d*x] - Log[g + h*x]))/(d*g - c*h))/h

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Maple [F]  time = 0.493, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{ \left ( hx+g \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.25269, size = 166, normalized size = 1.3 \begin{align*} \frac{{\left (b f p{\left (\frac{\log \left (b x + a\right )}{b g - a h} - \frac{\log \left (h x + g\right )}{b g - a h}\right )} + d f q{\left (\frac{\log \left (d x + c\right )}{d g - c h} - \frac{\log \left (h x + g\right )}{d g - c h}\right )}\right )} r}{f h} - \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h x + g\right )} h} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*f*p*(log(b*x + a)/(b*g - a*h) - log(h*x + g)/(b*g - a*h)) + d*f*q*(log(d*x + c)/(d*g - c*h) - log(h*x + g)/
(d*g - c*h)))*r/(f*h) - log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/((h*x + g)*h)

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [A]  time = 1.46868, size = 257, normalized size = 2.01 \begin{align*} \frac{b^{2} p r \log \left ({\left | -b x - a \right |}\right )}{b^{2} g h - a b h^{2}} + \frac{d^{2} q r \log \left ({\left | d x + c \right |}\right )}{d^{2} g h - c d h^{2}} - \frac{p r \log \left (b x + a\right )}{h^{2} x + g h} - \frac{q r \log \left (d x + c\right )}{h^{2} x + g h} - \frac{{\left (b d g p r - b c h p r + b d g q r - a d h q r\right )} \log \left (h x + g\right )}{b d g^{2} h - b c g h^{2} - a d g h^{2} + a c h^{3}} - \frac{r \log \left (f\right ) + 1}{h^{2} x + g h} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*p*r*log(abs(-b*x - a))/(b^2*g*h - a*b*h^2) + d^2*q*r*log(abs(d*x + c))/(d^2*g*h - c*d*h^2) - p*r*log(b*x +
a)/(h^2*x + g*h) - q*r*log(d*x + c)/(h^2*x + g*h) - (b*d*g*p*r - b*c*h*p*r + b*d*g*q*r - a*d*h*q*r)*log(h*x +
g)/(b*d*g^2*h - b*c*g*h^2 - a*d*g*h^2 + a*c*h^3) - (r*log(f) + 1)/(h^2*x + g*h)