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

Optimal. Leaf size=218 \[ -\frac{p r x (b g-a h)^2}{3 b^2}-\frac{p r (b g-a h)^3 \log (a+b x)}{3 b^3 h}+\frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{p r (g+h x)^2 (b g-a h)}{6 b h}-\frac{q r x (d g-c h)^2}{3 d^2}-\frac{q r (d g-c h)^3 \log (c+d x)}{3 d^3 h}-\frac{q r (g+h x)^2 (d g-c h)}{6 d h}-\frac{p r (g+h x)^3}{9 h}-\frac{q r (g+h x)^3}{9 h} \]

[Out]

-((b*g - a*h)^2*p*r*x)/(3*b^2) - ((d*g - c*h)^2*q*r*x)/(3*d^2) - ((b*g - a*h)*p*r*(g + h*x)^2)/(6*b*h) - ((d*g
- c*h)*q*r*(g + h*x)^2)/(6*d*h) - (p*r*(g + h*x)^3)/(9*h) - (q*r*(g + h*x)^3)/(9*h) - ((b*g - a*h)^3*p*r*Log[
a + b*x])/(3*b^3*h) - ((d*g - c*h)^3*q*r*Log[c + d*x])/(3*d^3*h) + ((g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)
^q)^r])/(3*h)

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Rubi [A]  time = 0.0987066, antiderivative size = 218, 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, 43} \[ -\frac{p r x (b g-a h)^2}{3 b^2}-\frac{p r (b g-a h)^3 \log (a+b x)}{3 b^3 h}+\frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{p r (g+h x)^2 (b g-a h)}{6 b h}-\frac{q r x (d g-c h)^2}{3 d^2}-\frac{q r (d g-c h)^3 \log (c+d x)}{3 d^3 h}-\frac{q r (g+h x)^2 (d g-c h)}{6 d h}-\frac{p r (g+h x)^3}{9 h}-\frac{q r (g+h x)^3}{9 h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*g - a*h)^2*p*r*x)/(3*b^2) - ((d*g - c*h)^2*q*r*x)/(3*d^2) - ((b*g - a*h)*p*r*(g + h*x)^2)/(6*b*h) - ((d*g
- c*h)*q*r*(g + h*x)^2)/(6*d*h) - (p*r*(g + h*x)^3)/(9*h) - (q*r*(g + h*x)^3)/(9*h) - ((b*g - a*h)^3*p*r*Log[
a + b*x])/(3*b^3*h) - ((d*g - c*h)^3*q*r*Log[c + d*x])/(3*d^3*h) + ((g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)
^q)^r])/(3*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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{(b p r) \int \frac{(g+h x)^3}{a+b x} \, dx}{3 h}-\frac{(d q r) \int \frac{(g+h x)^3}{c+d x} \, dx}{3 h}\\ &=\frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{(b p r) \int \left (\frac{h (b g-a h)^2}{b^3}+\frac{(b g-a h)^3}{b^3 (a+b x)}+\frac{h (b g-a h) (g+h x)}{b^2}+\frac{h (g+h x)^2}{b}\right ) \, dx}{3 h}-\frac{(d q r) \int \left (\frac{h (d g-c h)^2}{d^3}+\frac{(d g-c h)^3}{d^3 (c+d x)}+\frac{h (d g-c h) (g+h x)}{d^2}+\frac{h (g+h x)^2}{d}\right ) \, dx}{3 h}\\ &=-\frac{(b g-a h)^2 p r x}{3 b^2}-\frac{(d g-c h)^2 q r x}{3 d^2}-\frac{(b g-a h) p r (g+h x)^2}{6 b h}-\frac{(d g-c h) q r (g+h x)^2}{6 d h}-\frac{p r (g+h x)^3}{9 h}-\frac{q r (g+h x)^3}{9 h}-\frac{(b g-a h)^3 p r \log (a+b x)}{3 b^3 h}-\frac{(d g-c h)^3 q r \log (c+d x)}{3 d^3 h}+\frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\\ \end{align*}

Mathematica [A]  time = 0.253959, size = 209, normalized size = 0.96 \[ \frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac{r \left (b \left (6 a^2 d^3 h^3 p x-3 a b d^3 h p \left (g^2+6 g h x+h^2 x^2\right )+b^2 d \left (6 c^2 h^3 q x-3 c d h q \left (g^2+6 g h x+h^2 x^2\right )+d^2 (p+q) \left (18 g^2 h x+5 g^3+9 g h^2 x^2+2 h^3 x^3\right )\right )+6 b^2 q (d g-c h)^3 \log (c+d x)\right )+6 d^3 p (b g-a h)^3 \log (a+b x)\right )}{6 b^3 d^3}}{3 h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(r*(6*d^3*(b*g - a*h)^3*p*Log[a + b*x] + b*(6*a^2*d^3*h^3*p*x - 3*a*b*d^3*h*p*(g^2 + 6*g*h*x + h^2*x^2) + b^
2*d*(6*c^2*h^3*q*x - 3*c*d*h*q*(g^2 + 6*g*h*x + h^2*x^2) + d^2*(p + q)*(5*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h
^3*x^3)) + 6*b^2*(d*g - c*h)^3*q*Log[c + d*x])))/(6*b^3*d^3) + (g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r
])/(3*h)

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Maple [F]  time = 0.431, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{2}\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.25796, size = 363, normalized size = 1.67 \begin{align*} \frac{1}{3} \,{\left (h^{2} x^{3} + 3 \, g h x^{2} + 3 \, g^{2} x\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac{r{\left (\frac{6 \,{\left (3 \, a b^{2} f g^{2} p - 3 \, a^{2} b f g h p + a^{3} f h^{2} p\right )} \log \left (b x + a\right )}{b^{3}} + \frac{6 \,{\left (3 \, c d^{2} f g^{2} q - 3 \, c^{2} d f g h q + c^{3} f h^{2} q\right )} \log \left (d x + c\right )}{d^{3}} - \frac{2 \, b^{2} d^{2} f h^{2}{\left (p + q\right )} x^{3} - 3 \,{\left (a b d^{2} f h^{2} p -{\left (3 \, d^{2} f g h{\left (p + q\right )} - c d f h^{2} q\right )} b^{2}\right )} x^{2} - 6 \,{\left (3 \, a b d^{2} f g h p - a^{2} d^{2} f h^{2} p -{\left (3 \, d^{2} f g^{2}{\left (p + q\right )} - 3 \, c d f g h q + c^{2} f h^{2} q\right )} b^{2}\right )} x}{b^{2} d^{2}}\right )}}{18 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(h^2*x^3 + 3*g*h*x^2 + 3*g^2*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1/18*r*(6*(3*a*b^2*f*g^2*p - 3*a^2*
b*f*g*h*p + a^3*f*h^2*p)*log(b*x + a)/b^3 + 6*(3*c*d^2*f*g^2*q - 3*c^2*d*f*g*h*q + c^3*f*h^2*q)*log(d*x + c)/d
^3 - (2*b^2*d^2*f*h^2*(p + q)*x^3 - 3*(a*b*d^2*f*h^2*p - (3*d^2*f*g*h*(p + q) - c*d*f*h^2*q)*b^2)*x^2 - 6*(3*a
*b*d^2*f*g*h*p - a^2*d^2*f*h^2*p - (3*d^2*f*g^2*(p + q) - 3*c*d*f*g*h*q + c^2*f*h^2*q)*b^2)*x)/(b^2*d^2))/f

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Fricas [B]  time = 1.16108, size = 917, normalized size = 4.21 \begin{align*} -\frac{2 \,{\left (b^{3} d^{3} h^{2} p + b^{3} d^{3} h^{2} q\right )} r x^{3} + 3 \,{\left ({\left (3 \, b^{3} d^{3} g h - a b^{2} d^{3} h^{2}\right )} p +{\left (3 \, b^{3} d^{3} g h - b^{3} c d^{2} h^{2}\right )} q\right )} r x^{2} + 6 \,{\left ({\left (3 \, b^{3} d^{3} g^{2} - 3 \, a b^{2} d^{3} g h + a^{2} b d^{3} h^{2}\right )} p +{\left (3 \, b^{3} d^{3} g^{2} - 3 \, b^{3} c d^{2} g h + b^{3} c^{2} d h^{2}\right )} q\right )} r x - 6 \,{\left (b^{3} d^{3} h^{2} p r x^{3} + 3 \, b^{3} d^{3} g h p r x^{2} + 3 \, b^{3} d^{3} g^{2} p r x +{\left (3 \, a b^{2} d^{3} g^{2} - 3 \, a^{2} b d^{3} g h + a^{3} d^{3} h^{2}\right )} p r\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} d^{3} h^{2} q r x^{3} + 3 \, b^{3} d^{3} g h q r x^{2} + 3 \, b^{3} d^{3} g^{2} q r x +{\left (3 \, b^{3} c d^{2} g^{2} - 3 \, b^{3} c^{2} d g h + b^{3} c^{3} h^{2}\right )} q r\right )} \log \left (d x + c\right ) - 6 \,{\left (b^{3} d^{3} h^{2} x^{3} + 3 \, b^{3} d^{3} g h x^{2} + 3 \, b^{3} d^{3} g^{2} x\right )} \log \left (e\right ) - 6 \,{\left (b^{3} d^{3} h^{2} r x^{3} + 3 \, b^{3} d^{3} g h r x^{2} + 3 \, b^{3} d^{3} g^{2} r x\right )} \log \left (f\right )}{18 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*(b^3*d^3*h^2*p + b^3*d^3*h^2*q)*r*x^3 + 3*((3*b^3*d^3*g*h - a*b^2*d^3*h^2)*p + (3*b^3*d^3*g*h - b^3*c
*d^2*h^2)*q)*r*x^2 + 6*((3*b^3*d^3*g^2 - 3*a*b^2*d^3*g*h + a^2*b*d^3*h^2)*p + (3*b^3*d^3*g^2 - 3*b^3*c*d^2*g*h
+ b^3*c^2*d*h^2)*q)*r*x - 6*(b^3*d^3*h^2*p*r*x^3 + 3*b^3*d^3*g*h*p*r*x^2 + 3*b^3*d^3*g^2*p*r*x + (3*a*b^2*d^3
*g^2 - 3*a^2*b*d^3*g*h + a^3*d^3*h^2)*p*r)*log(b*x + a) - 6*(b^3*d^3*h^2*q*r*x^3 + 3*b^3*d^3*g*h*q*r*x^2 + 3*b
^3*d^3*g^2*q*r*x + (3*b^3*c*d^2*g^2 - 3*b^3*c^2*d*g*h + b^3*c^3*h^2)*q*r)*log(d*x + c) - 6*(b^3*d^3*h^2*x^3 +
3*b^3*d^3*g*h*x^2 + 3*b^3*d^3*g^2*x)*log(e) - 6*(b^3*d^3*h^2*r*x^3 + 3*b^3*d^3*g*h*r*x^2 + 3*b^3*d^3*g^2*r*x)*
log(f))/(b^3*d^3)

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

[Out]

Timed out

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Giac [B]  time = 1.30521, size = 837, normalized size = 3.84 \begin{align*} -\frac{1}{9} \,{\left (h^{2} p r + h^{2} q r - 3 \, h^{2} r \log \left (f\right ) - 3 \, h^{2}\right )} x^{3} + \frac{1}{3} \,{\left (h^{2} p r x^{3} + 3 \, g h p r x^{2} + 3 \, g^{2} p r x\right )} \log \left (b x + a\right ) + \frac{1}{3} \,{\left (h^{2} q r x^{3} + 3 \, g h q r x^{2} + 3 \, g^{2} q r x\right )} \log \left (d x + c\right ) - \frac{{\left (3 \, b d g h p r - a d h^{2} p r + 3 \, b d g h q r - b c h^{2} q r - 6 \, b d g h r \log \left (f\right ) - 6 \, b d g h\right )} x^{2}}{6 \, b d} - \frac{{\left (3 \, b^{2} d^{2} g^{2} p r - 3 \, a b d^{2} g h p r + a^{2} d^{2} h^{2} p r + 3 \, b^{2} d^{2} g^{2} q r - 3 \, b^{2} c d g h q r + b^{2} c^{2} h^{2} q r - 3 \, b^{2} d^{2} g^{2} r \log \left (f\right ) - 3 \, b^{2} d^{2} g^{2}\right )} x}{3 \, b^{2} d^{2}} + \frac{{\left (3 \, a b^{2} d^{3} g^{2} p r - 3 \, a^{2} b d^{3} g h p r + a^{3} d^{3} h^{2} p r + 3 \, b^{3} c d^{2} g^{2} q r - 3 \, b^{3} c^{2} d g h q r + b^{3} c^{3} h^{2} q r\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{6 \, b^{3} d^{3}} + \frac{{\left (3 \, a b^{3} c d^{3} g^{2} p r - 3 \, a^{2} b^{2} d^{4} g^{2} p r - 3 \, a^{2} b^{2} c d^{3} g h p r + 3 \, a^{3} b d^{4} g h p r + a^{3} b c d^{3} h^{2} p r - a^{4} d^{4} h^{2} p r - 3 \, b^{4} c^{2} d^{2} g^{2} q r + 3 \, a b^{3} c d^{3} g^{2} q r + 3 \, b^{4} c^{3} d g h q r - 3 \, a b^{3} c^{2} d^{2} g h q r - b^{4} c^{4} h^{2} q r + a b^{3} c^{3} d h^{2} q r\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | -b c + a d \right |}}{2 \, b d x + b c + a d +{\left | -b c + a d \right |}} \right |}\right )}{6 \, b^{3} d^{3}{\left | -b c + a d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(h^2*p*r + h^2*q*r - 3*h^2*r*log(f) - 3*h^2)*x^3 + 1/3*(h^2*p*r*x^3 + 3*g*h*p*r*x^2 + 3*g^2*p*r*x)*log(b*
x + a) + 1/3*(h^2*q*r*x^3 + 3*g*h*q*r*x^2 + 3*g^2*q*r*x)*log(d*x + c) - 1/6*(3*b*d*g*h*p*r - a*d*h^2*p*r + 3*b
*d*g*h*q*r - b*c*h^2*q*r - 6*b*d*g*h*r*log(f) - 6*b*d*g*h)*x^2/(b*d) - 1/3*(3*b^2*d^2*g^2*p*r - 3*a*b*d^2*g*h*
p*r + a^2*d^2*h^2*p*r + 3*b^2*d^2*g^2*q*r - 3*b^2*c*d*g*h*q*r + b^2*c^2*h^2*q*r - 3*b^2*d^2*g^2*r*log(f) - 3*b
^2*d^2*g^2)*x/(b^2*d^2) + 1/6*(3*a*b^2*d^3*g^2*p*r - 3*a^2*b*d^3*g*h*p*r + a^3*d^3*h^2*p*r + 3*b^3*c*d^2*g^2*q
*r - 3*b^3*c^2*d*g*h*q*r + b^3*c^3*h^2*q*r)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^3*d^3) + 1/6*(3*a*b^3*c
*d^3*g^2*p*r - 3*a^2*b^2*d^4*g^2*p*r - 3*a^2*b^2*c*d^3*g*h*p*r + 3*a^3*b*d^4*g*h*p*r + a^3*b*c*d^3*h^2*p*r - a
^4*d^4*h^2*p*r - 3*b^4*c^2*d^2*g^2*q*r + 3*a*b^3*c*d^3*g^2*q*r + 3*b^4*c^3*d*g*h*q*r - 3*a*b^3*c^2*d^2*g*h*q*r
- b^4*c^4*h^2*q*r + a*b^3*c^3*d*h^2*q*r)*log(abs((2*b*d*x + b*c + a*d - abs(-b*c + a*d))/(2*b*d*x + b*c + a*d
+ abs(-b*c + a*d))))/(b^3*d^3*abs(-b*c + a*d))