3.33 \(\int \frac{\log (e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^4} \, dx\)

Optimal. Leaf size=260 \[ \frac{b^2 p r}{3 h (g+h x) (b g-a h)^2}+\frac{b^3 p r \log (a+b x)}{3 h (b g-a h)^3}-\frac{b^3 p r \log (g+h x)}{3 h (b g-a h)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac{b p r}{6 h (g+h x)^2 (b g-a h)}+\frac{d^2 q r}{3 h (g+h x) (d g-c h)^2}+\frac{d^3 q r \log (c+d x)}{3 h (d g-c h)^3}-\frac{d^3 q r \log (g+h x)}{3 h (d g-c h)^3}+\frac{d q r}{6 h (g+h x)^2 (d g-c h)} \]

[Out]

(b*p*r)/(6*h*(b*g - a*h)*(g + h*x)^2) + (d*q*r)/(6*h*(d*g - c*h)*(g + h*x)^2) + (b^2*p*r)/(3*h*(b*g - a*h)^2*(
g + h*x)) + (d^2*q*r)/(3*h*(d*g - c*h)^2*(g + h*x)) + (b^3*p*r*Log[a + b*x])/(3*h*(b*g - a*h)^3) + (d^3*q*r*Lo
g[c + d*x])/(3*h*(d*g - c*h)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(3*h*(g + h*x)^3) - (b^3*p*r*Log[g + h*
x])/(3*h*(b*g - a*h)^3) - (d^3*q*r*Log[g + h*x])/(3*h*(d*g - c*h)^3)

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Rubi [A]  time = 0.151959, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2495, 44} \[ \frac{b^2 p r}{3 h (g+h x) (b g-a h)^2}+\frac{b^3 p r \log (a+b x)}{3 h (b g-a h)^3}-\frac{b^3 p r \log (g+h x)}{3 h (b g-a h)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac{b p r}{6 h (g+h x)^2 (b g-a h)}+\frac{d^2 q r}{3 h (g+h x) (d g-c h)^2}+\frac{d^3 q r \log (c+d x)}{3 h (d g-c h)^3}-\frac{d^3 q r \log (g+h x)}{3 h (d g-c h)^3}+\frac{d q r}{6 h (g+h x)^2 (d g-c h)} \]

Antiderivative was successfully verified.

[In]

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^4,x]

[Out]

(b*p*r)/(6*h*(b*g - a*h)*(g + h*x)^2) + (d*q*r)/(6*h*(d*g - c*h)*(g + h*x)^2) + (b^2*p*r)/(3*h*(b*g - a*h)^2*(
g + h*x)) + (d^2*q*r)/(3*h*(d*g - c*h)^2*(g + h*x)) + (b^3*p*r*Log[a + b*x])/(3*h*(b*g - a*h)^3) + (d^3*q*r*Lo
g[c + d*x])/(3*h*(d*g - c*h)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(3*h*(g + h*x)^3) - (b^3*p*r*Log[g + h*
x])/(3*h*(b*g - a*h)^3) - (d^3*q*r*Log[g + h*x])/(3*h*(d*g - c*h)^3)

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 44

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] && L
tQ[m + n + 2, 0])

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)^4} \, dx &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac{(b p r) \int \frac{1}{(a+b x) (g+h x)^3} \, dx}{3 h}+\frac{(d q r) \int \frac{1}{(c+d x) (g+h x)^3} \, dx}{3 h}\\ &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac{(b p r) \int \left (\frac{b^3}{(b g-a h)^3 (a+b x)}-\frac{h}{(b g-a h) (g+h x)^3}-\frac{b h}{(b g-a h)^2 (g+h x)^2}-\frac{b^2 h}{(b g-a h)^3 (g+h x)}\right ) \, dx}{3 h}+\frac{(d q r) \int \left (\frac{d^3}{(d g-c h)^3 (c+d x)}-\frac{h}{(d g-c h) (g+h x)^3}-\frac{d h}{(d g-c h)^2 (g+h x)^2}-\frac{d^2 h}{(d g-c h)^3 (g+h x)}\right ) \, dx}{3 h}\\ &=\frac{b p r}{6 h (b g-a h) (g+h x)^2}+\frac{d q r}{6 h (d g-c h) (g+h x)^2}+\frac{b^2 p r}{3 h (b g-a h)^2 (g+h x)}+\frac{d^2 q r}{3 h (d g-c h)^2 (g+h x)}+\frac{b^3 p r \log (a+b x)}{3 h (b g-a h)^3}+\frac{d^3 q r \log (c+d x)}{3 h (d g-c h)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}-\frac{b^3 p r \log (g+h x)}{3 h (b g-a h)^3}-\frac{d^3 q r \log (g+h x)}{3 h (d g-c h)^3}\\ \end{align*}

Mathematica [A]  time = 0.959729, size = 254, normalized size = 0.98 \[ \frac{\frac{r (g+h x) \left ((b g-a h)^2 (d g-c h)^2 (b d g (p+q)-h (a d q+b c p))-(g+h x) \left ((b g-a h) (d g-c h) \left (-2 a^2 d^2 h^2 q+4 a b d^2 g h q-2 b^2 \left (c^2 h^2 p-2 c d g h p+d^2 g^2 (p+q)\right )\right )-2 (g+h x) \left (b^3 p (d g-c h)^3 (\log (a+b x)-\log (g+h x))+d^3 q (b g-a h)^3 (\log (c+d x)-\log (g+h x))\right )\right )\right )}{(b g-a h)^3 (d g-c h)^3}-2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 h (g+h x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^4,x]

[Out]

(-2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (r*(g + h*x)*((b*g - a*h)^2*(d*g - c*h)^2*(b*d*g*(p + q) - h*(b*c*p
+ a*d*q)) - (g + h*x)*((b*g - a*h)*(d*g - c*h)*(4*a*b*d^2*g*h*q - 2*a^2*d^2*h^2*q - 2*b^2*(-2*c*d*g*h*p + c^2
*h^2*p + d^2*g^2*(p + q))) - 2*(g + h*x)*(b^3*(d*g - c*h)^3*p*(Log[a + b*x] - Log[g + h*x]) + d^3*(b*g - a*h)^
3*q*(Log[c + d*x] - Log[g + h*x])))))/((b*g - a*h)^3*(d*g - c*h)^3))/(6*h*(g + h*x)^3)

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Maple [F]  time = 0.519, 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 ) ^{4}}}\, 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)^4,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^4,x)

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Maxima [A]  time = 1.29783, size = 616, normalized size = 2.37 \begin{align*} \frac{{\left ({\left (\frac{2 \, b^{2} \log \left (b x + a\right )}{b^{3} g^{3} - 3 \, a b^{2} g^{2} h + 3 \, a^{2} b g h^{2} - a^{3} h^{3}} - \frac{2 \, b^{2} \log \left (h x + g\right )}{b^{3} g^{3} - 3 \, a b^{2} g^{2} h + 3 \, a^{2} b g h^{2} - a^{3} h^{3}} + \frac{2 \, b h x + 3 \, b g - a h}{b^{2} g^{4} - 2 \, a b g^{3} h + a^{2} g^{2} h^{2} +{\left (b^{2} g^{2} h^{2} - 2 \, a b g h^{3} + a^{2} h^{4}\right )} x^{2} + 2 \,{\left (b^{2} g^{3} h - 2 \, a b g^{2} h^{2} + a^{2} g h^{3}\right )} x}\right )} b f p +{\left (\frac{2 \, d^{2} \log \left (d x + c\right )}{d^{3} g^{3} - 3 \, c d^{2} g^{2} h + 3 \, c^{2} d g h^{2} - c^{3} h^{3}} - \frac{2 \, d^{2} \log \left (h x + g\right )}{d^{3} g^{3} - 3 \, c d^{2} g^{2} h + 3 \, c^{2} d g h^{2} - c^{3} h^{3}} + \frac{2 \, d h x + 3 \, d g - c h}{d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2} +{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} x^{2} + 2 \,{\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} x}\right )} d f q\right )} r}{6 \, f h} - \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{3 \,{\left (h x + g\right )}^{3} 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)^4,x, algorithm="maxima")

[Out]

1/6*((2*b^2*log(b*x + a)/(b^3*g^3 - 3*a*b^2*g^2*h + 3*a^2*b*g*h^2 - a^3*h^3) - 2*b^2*log(h*x + g)/(b^3*g^3 - 3
*a*b^2*g^2*h + 3*a^2*b*g*h^2 - a^3*h^3) + (2*b*h*x + 3*b*g - a*h)/(b^2*g^4 - 2*a*b*g^3*h + a^2*g^2*h^2 + (b^2*
g^2*h^2 - 2*a*b*g*h^3 + a^2*h^4)*x^2 + 2*(b^2*g^3*h - 2*a*b*g^2*h^2 + a^2*g*h^3)*x))*b*f*p + (2*d^2*log(d*x +
c)/(d^3*g^3 - 3*c*d^2*g^2*h + 3*c^2*d*g*h^2 - c^3*h^3) - 2*d^2*log(h*x + g)/(d^3*g^3 - 3*c*d^2*g^2*h + 3*c^2*d
*g*h^2 - c^3*h^3) + (2*d*h*x + 3*d*g - c*h)/(d^2*g^4 - 2*c*d*g^3*h + c^2*g^2*h^2 + (d^2*g^2*h^2 - 2*c*d*g*h^3
+ c^2*h^4)*x^2 + 2*(d^2*g^3*h - 2*c*d*g^2*h^2 + c^2*g*h^3)*x))*d*f*q)*r/(f*h) - 1/3*log(((b*x + a)^p*(d*x + c)
^q*f)^r*e)/((h*x + g)^3*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)^4,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)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.86354, size = 2383, normalized size = 9.17 \begin{align*} \text{result too large to display} \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)^4,x, algorithm="giac")

[Out]

1/3*b^4*p*r*log(abs(b*x + a))/(b^4*g^3*h - 3*a*b^3*g^2*h^2 + 3*a^2*b^2*g*h^3 - a^3*b*h^4) + 1/3*d^4*q*r*log(ab
s(d*x + c))/(d^4*g^3*h - 3*c*d^3*g^2*h^2 + 3*c^2*d^2*g*h^3 - c^3*d*h^4) - 1/3*p*r*log(b*x + a)/(h^4*x^3 + 3*g*
h^3*x^2 + 3*g^2*h^2*x + g^3*h) - 1/3*q*r*log(d*x + c)/(h^4*x^3 + 3*g*h^3*x^2 + 3*g^2*h^2*x + g^3*h) - 1/3*(b^3
*d^3*g^3*p*r - 3*b^3*c*d^2*g^2*h*p*r + 3*b^3*c^2*d*g*h^2*p*r - b^3*c^3*h^3*p*r + b^3*d^3*g^3*q*r - 3*a*b^2*d^3
*g^2*h*q*r + 3*a^2*b*d^3*g*h^2*q*r - a^3*d^3*h^3*q*r)*log(h*x + g)/(b^3*d^3*g^6*h - 3*b^3*c*d^2*g^5*h^2 - 3*a*
b^2*d^3*g^5*h^2 + 3*b^3*c^2*d*g^4*h^3 + 9*a*b^2*c*d^2*g^4*h^3 + 3*a^2*b*d^3*g^4*h^3 - b^3*c^3*g^3*h^4 - 9*a*b^
2*c^2*d*g^3*h^4 - 9*a^2*b*c*d^2*g^3*h^4 - a^3*d^3*g^3*h^4 + 3*a*b^2*c^3*g^2*h^5 + 9*a^2*b*c^2*d*g^2*h^5 + 3*a^
3*c*d^2*g^2*h^5 - 3*a^2*b*c^3*g*h^6 - 3*a^3*c^2*d*g*h^6 + a^3*c^3*h^7) + 1/6*(2*b^2*d^2*g^2*h^2*p*r*x^2 - 4*b^
2*c*d*g*h^3*p*r*x^2 + 2*b^2*c^2*h^4*p*r*x^2 + 2*b^2*d^2*g^2*h^2*q*r*x^2 - 4*a*b*d^2*g*h^3*q*r*x^2 + 2*a^2*d^2*
h^4*q*r*x^2 + 5*b^2*d^2*g^3*h*p*r*x - 10*b^2*c*d*g^2*h^2*p*r*x - a*b*d^2*g^2*h^2*p*r*x + 5*b^2*c^2*g*h^3*p*r*x
+ 2*a*b*c*d*g*h^3*p*r*x - a*b*c^2*h^4*p*r*x + 5*b^2*d^2*g^3*h*q*r*x - b^2*c*d*g^2*h^2*q*r*x - 10*a*b*d^2*g^2*
h^2*q*r*x + 2*a*b*c*d*g*h^3*q*r*x + 5*a^2*d^2*g*h^3*q*r*x - a^2*c*d*h^4*q*r*x + 3*b^2*d^2*g^4*p*r - 6*b^2*c*d*
g^3*h*p*r - a*b*d^2*g^3*h*p*r + 3*b^2*c^2*g^2*h^2*p*r + 2*a*b*c*d*g^2*h^2*p*r - a*b*c^2*g*h^3*p*r + 3*b^2*d^2*
g^4*q*r - b^2*c*d*g^3*h*q*r - 6*a*b*d^2*g^3*h*q*r + 2*a*b*c*d*g^2*h^2*q*r + 3*a^2*d^2*g^2*h^2*q*r - a^2*c*d*g*
h^3*q*r - 2*b^2*d^2*g^4*r*log(f) + 4*b^2*c*d*g^3*h*r*log(f) + 4*a*b*d^2*g^3*h*r*log(f) - 2*b^2*c^2*g^2*h^2*r*l
og(f) - 8*a*b*c*d*g^2*h^2*r*log(f) - 2*a^2*d^2*g^2*h^2*r*log(f) + 4*a*b*c^2*g*h^3*r*log(f) + 4*a^2*c*d*g*h^3*r
*log(f) - 2*a^2*c^2*h^4*r*log(f) - 2*b^2*d^2*g^4 + 4*b^2*c*d*g^3*h + 4*a*b*d^2*g^3*h - 2*b^2*c^2*g^2*h^2 - 8*a
*b*c*d*g^2*h^2 - 2*a^2*d^2*g^2*h^2 + 4*a*b*c^2*g*h^3 + 4*a^2*c*d*g*h^3 - 2*a^2*c^2*h^4)/(b^2*d^2*g^4*h^4*x^3 -
2*b^2*c*d*g^3*h^5*x^3 - 2*a*b*d^2*g^3*h^5*x^3 + b^2*c^2*g^2*h^6*x^3 + 4*a*b*c*d*g^2*h^6*x^3 + a^2*d^2*g^2*h^6
*x^3 - 2*a*b*c^2*g*h^7*x^3 - 2*a^2*c*d*g*h^7*x^3 + a^2*c^2*h^8*x^3 + 3*b^2*d^2*g^5*h^3*x^2 - 6*b^2*c*d*g^4*h^4
*x^2 - 6*a*b*d^2*g^4*h^4*x^2 + 3*b^2*c^2*g^3*h^5*x^2 + 12*a*b*c*d*g^3*h^5*x^2 + 3*a^2*d^2*g^3*h^5*x^2 - 6*a*b*
c^2*g^2*h^6*x^2 - 6*a^2*c*d*g^2*h^6*x^2 + 3*a^2*c^2*g*h^7*x^2 + 3*b^2*d^2*g^6*h^2*x - 6*b^2*c*d*g^5*h^3*x - 6*
a*b*d^2*g^5*h^3*x + 3*b^2*c^2*g^4*h^4*x + 12*a*b*c*d*g^4*h^4*x + 3*a^2*d^2*g^4*h^4*x - 6*a*b*c^2*g^3*h^5*x - 6
*a^2*c*d*g^3*h^5*x + 3*a^2*c^2*g^2*h^6*x + b^2*d^2*g^7*h - 2*b^2*c*d*g^6*h^2 - 2*a*b*d^2*g^6*h^2 + b^2*c^2*g^5
*h^3 + 4*a*b*c*d*g^5*h^3 + a^2*d^2*g^5*h^3 - 2*a*b*c^2*g^4*h^4 - 2*a^2*c*d*g^4*h^4 + a^2*c^2*g^3*h^5)