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

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 260, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.76, 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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [B]  time = 0.85, size = 1765, normalized size = 6.79 \[ \text {result too large to display} \]

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)

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (h x +g \right )^{4}}\, dx \]

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 = 0.76, size = 456, normalized size = 1.75 \[ \frac {{\left ({\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{3} g^{3} - 3 \, a b^{2} g^{2} h + 3 \, a^{2} b g h^{2} - a^{3} h^{3}} - \frac {2 \, b^{2} \log \left (h x + g\right )}{b^{3} g^{3} - 3 \, a b^{2} g^{2} h + 3 \, a^{2} b g h^{2} - a^{3} h^{3}} + \frac {2 \, b h x + 3 \, b g - a h}{b^{2} g^{4} - 2 \, a b g^{3} h + a^{2} g^{2} h^{2} + {\left (b^{2} g^{2} h^{2} - 2 \, a b g h^{3} + a^{2} h^{4}\right )} x^{2} + 2 \, {\left (b^{2} g^{3} h - 2 \, a b g^{2} h^{2} + a^{2} g h^{3}\right )} x}\right )} b f p + {\left (\frac {2 \, d^{2} \log \left (d x + c\right )}{d^{3} g^{3} - 3 \, c d^{2} g^{2} h + 3 \, c^{2} d g h^{2} - c^{3} h^{3}} - \frac {2 \, d^{2} \log \left (h x + g\right )}{d^{3} g^{3} - 3 \, c d^{2} g^{2} h + 3 \, c^{2} d g h^{2} - c^{3} h^{3}} + \frac {2 \, d h x + 3 \, d g - c h}{d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2} + {\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} x^{2} + 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} x}\right )} d f q\right )} r}{6 \, f h} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{3 \, {\left (h x + g\right )}^{3} h} \]

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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mupad [B]  time = 5.48, size = 977, normalized size = 3.76 \[ \frac {\frac {3\,b^2\,d^2\,g^3\,p\,r+3\,b^2\,d^2\,g^3\,q\,r-a\,b\,c^2\,h^3\,p\,r-a^2\,c\,d\,h^3\,q\,r+3\,b^2\,c^2\,g\,h^2\,p\,r+3\,a^2\,d^2\,g\,h^2\,q\,r-a\,b\,d^2\,g^2\,h\,p\,r-6\,a\,b\,d^2\,g^2\,h\,q\,r-6\,b^2\,c\,d\,g^2\,h\,p\,r-b^2\,c\,d\,g^2\,h\,q\,r+2\,a\,b\,c\,d\,g\,h^2\,p\,r+2\,a\,b\,c\,d\,g\,h^2\,q\,r}{2\,\left (a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4\right )}+\frac {x\,\left (b^2\,c^2\,h^3\,p\,r+a^2\,d^2\,h^3\,q\,r+b^2\,d^2\,g^2\,h\,p\,r+b^2\,d^2\,g^2\,h\,q\,r-2\,a\,b\,d^2\,g\,h^2\,q\,r-2\,b^2\,c\,d\,g\,h^2\,p\,r\right )}{a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4}}{3\,g^2\,h+6\,g\,h^2\,x+3\,h^3\,x^2}+\frac {\ln \left (g+h\,x\right )\,\left (g^2\,\left (3\,c\,h\,p\,r\,b^3\,d^2+3\,a\,h\,q\,r\,b^2\,d^3\right )-g^3\,\left (b^3\,d^3\,p\,r+b^3\,d^3\,q\,r\right )-g\,\left (3\,q\,r\,a^2\,b\,d^3\,h^2+3\,p\,r\,b^3\,c^2\,d\,h^2\right )+b^3\,c^3\,h^3\,p\,r+a^3\,d^3\,h^3\,q\,r\right )}{3\,a^3\,c^3\,h^7-9\,a^3\,c^2\,d\,g\,h^6+9\,a^3\,c\,d^2\,g^2\,h^5-3\,a^3\,d^3\,g^3\,h^4-9\,a^2\,b\,c^3\,g\,h^6+27\,a^2\,b\,c^2\,d\,g^2\,h^5-27\,a^2\,b\,c\,d^2\,g^3\,h^4+9\,a^2\,b\,d^3\,g^4\,h^3+9\,a\,b^2\,c^3\,g^2\,h^5-27\,a\,b^2\,c^2\,d\,g^3\,h^4+27\,a\,b^2\,c\,d^2\,g^4\,h^3-9\,a\,b^2\,d^3\,g^5\,h^2-3\,b^3\,c^3\,g^3\,h^4+9\,b^3\,c^2\,d\,g^4\,h^3-9\,b^3\,c\,d^2\,g^5\,h^2+3\,b^3\,d^3\,g^6\,h}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {x}{3}+\frac {g}{3\,h}\right )}{{\left (g+h\,x\right )}^4}-\frac {b^3\,p\,r\,\ln \left (a+b\,x\right )}{3\,a^3\,h^4-9\,a^2\,b\,g\,h^3+9\,a\,b^2\,g^2\,h^2-3\,b^3\,g^3\,h}-\frac {d^3\,q\,r\,\ln \left (c+d\,x\right )}{3\,c^3\,h^4-9\,c^2\,d\,g\,h^3+9\,c\,d^2\,g^2\,h^2-3\,d^3\,g^3\,h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*b^2*d^2*g^3*p*r + 3*b^2*d^2*g^3*q*r - a*b*c^2*h^3*p*r - a^2*c*d*h^3*q*r + 3*b^2*c^2*g*h^2*p*r + 3*a^2*d^2*
g*h^2*q*r - a*b*d^2*g^2*h*p*r - 6*a*b*d^2*g^2*h*q*r - 6*b^2*c*d*g^2*h*p*r - b^2*c*d*g^2*h*q*r + 2*a*b*c*d*g*h^
2*p*r + 2*a*b*c*d*g*h^2*q*r)/(2*(a^2*c^2*h^4 + b^2*d^2*g^4 + a^2*d^2*g^2*h^2 + b^2*c^2*g^2*h^2 - 2*a*b*c^2*g*h
^3 - 2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*c*d*g^3*h + 4*a*b*c*d*g^2*h^2)) + (x*(b^2*c^2*h^3*p*r + a^2*d^2
*h^3*q*r + b^2*d^2*g^2*h*p*r + b^2*d^2*g^2*h*q*r - 2*a*b*d^2*g*h^2*q*r - 2*b^2*c*d*g*h^2*p*r))/(a^2*c^2*h^4 +
b^2*d^2*g^4 + a^2*d^2*g^2*h^2 + b^2*c^2*g^2*h^2 - 2*a*b*c^2*g*h^3 - 2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*
c*d*g^3*h + 4*a*b*c*d*g^2*h^2))/(3*g^2*h + 3*h^3*x^2 + 6*g*h^2*x) + (log(g + h*x)*(g^2*(3*a*b^2*d^3*h*q*r + 3*
b^3*c*d^2*h*p*r) - g^3*(b^3*d^3*p*r + b^3*d^3*q*r) - g*(3*a^2*b*d^3*h^2*q*r + 3*b^3*c^2*d*h^2*p*r) + b^3*c^3*h
^3*p*r + a^3*d^3*h^3*q*r))/(3*a^3*c^3*h^7 + 3*b^3*d^3*g^6*h - 3*a^3*d^3*g^3*h^4 - 3*b^3*c^3*g^3*h^4 - 9*a^2*b*
c^3*g*h^6 - 9*a^3*c^2*d*g*h^6 + 9*a*b^2*c^3*g^2*h^5 - 9*a*b^2*d^3*g^5*h^2 + 9*a^2*b*d^3*g^4*h^3 + 9*a^3*c*d^2*
g^2*h^5 - 9*b^3*c*d^2*g^5*h^2 + 9*b^3*c^2*d*g^4*h^3 + 27*a*b^2*c*d^2*g^4*h^3 - 27*a*b^2*c^2*d*g^3*h^4 - 27*a^2
*b*c*d^2*g^3*h^4 + 27*a^2*b*c^2*d*g^2*h^5) - (log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(x/3 + g/(3*h)))/(g + h*x)^
4 - (b^3*p*r*log(a + b*x))/(3*a^3*h^4 - 3*b^3*g^3*h + 9*a*b^2*g^2*h^2 - 9*a^2*b*g*h^3) - (d^3*q*r*log(c + d*x)
)/(3*c^3*h^4 - 3*d^3*g^3*h + 9*c*d^2*g^2*h^2 - 9*c^2*d*g*h^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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