3.26 \(\int (g+h x)^3 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\)

Optimal. Leaf size=276 \[ -\frac{p r x (b g-a h)^3}{4 b^3}-\frac{p r (g+h x)^2 (b g-a h)^2}{8 b^2 h}-\frac{p r (b g-a h)^4 \log (a+b x)}{4 b^4 h}+\frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{p r (g+h x)^3 (b g-a h)}{12 b h}-\frac{q r (g+h x)^2 (d g-c h)^2}{8 d^2 h}-\frac{q r x (d g-c h)^3}{4 d^3}-\frac{q r (d g-c h)^4 \log (c+d x)}{4 d^4 h}-\frac{q r (g+h x)^3 (d g-c h)}{12 d h}-\frac{p r (g+h x)^4}{16 h}-\frac{q r (g+h x)^4}{16 h} \]

[Out]

-((b*g - a*h)^3*p*r*x)/(4*b^3) - ((d*g - c*h)^3*q*r*x)/(4*d^3) - ((b*g - a*h)^2*p*r*(g + h*x)^2)/(8*b^2*h) - (
(d*g - c*h)^2*q*r*(g + h*x)^2)/(8*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^3)/(12*b*h) - ((d*g - c*h)*q*r*(g + h*x)
^3)/(12*d*h) - (p*r*(g + h*x)^4)/(16*h) - (q*r*(g + h*x)^4)/(16*h) - ((b*g - a*h)^4*p*r*Log[a + b*x])/(4*b^4*h
) - ((d*g - c*h)^4*q*r*Log[c + d*x])/(4*d^4*h) + ((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*h)

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Rubi [A]  time = 0.127419, antiderivative size = 276, 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} \[ -\frac{p r x (b g-a h)^3}{4 b^3}-\frac{p r (g+h x)^2 (b g-a h)^2}{8 b^2 h}-\frac{p r (b g-a h)^4 \log (a+b x)}{4 b^4 h}+\frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{p r (g+h x)^3 (b g-a h)}{12 b h}-\frac{q r (g+h x)^2 (d g-c h)^2}{8 d^2 h}-\frac{q r x (d g-c h)^3}{4 d^3}-\frac{q r (d g-c h)^4 \log (c+d x)}{4 d^4 h}-\frac{q r (g+h x)^3 (d g-c h)}{12 d h}-\frac{p r (g+h x)^4}{16 h}-\frac{q r (g+h x)^4}{16 h} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

-((b*g - a*h)^3*p*r*x)/(4*b^3) - ((d*g - c*h)^3*q*r*x)/(4*d^3) - ((b*g - a*h)^2*p*r*(g + h*x)^2)/(8*b^2*h) - (
(d*g - c*h)^2*q*r*(g + h*x)^2)/(8*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^3)/(12*b*h) - ((d*g - c*h)*q*r*(g + h*x)
^3)/(12*d*h) - (p*r*(g + h*x)^4)/(16*h) - (q*r*(g + h*x)^4)/(16*h) - ((b*g - a*h)^4*p*r*Log[a + b*x])/(4*b^4*h
) - ((d*g - c*h)^4*q*r*Log[c + d*x])/(4*d^4*h) + ((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*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)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{(b p r) \int \frac{(g+h x)^4}{a+b x} \, dx}{4 h}-\frac{(d q r) \int \frac{(g+h x)^4}{c+d x} \, dx}{4 h}\\ &=\frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{(b p r) \int \left (\frac{h (b g-a h)^3}{b^4}+\frac{(b g-a h)^4}{b^4 (a+b x)}+\frac{h (b g-a h)^2 (g+h x)}{b^3}+\frac{h (b g-a h) (g+h x)^2}{b^2}+\frac{h (g+h x)^3}{b}\right ) \, dx}{4 h}-\frac{(d q r) \int \left (\frac{h (d g-c h)^3}{d^4}+\frac{(d g-c h)^4}{d^4 (c+d x)}+\frac{h (d g-c h)^2 (g+h x)}{d^3}+\frac{h (d g-c h) (g+h x)^2}{d^2}+\frac{h (g+h x)^3}{d}\right ) \, dx}{4 h}\\ &=-\frac{(b g-a h)^3 p r x}{4 b^3}-\frac{(d g-c h)^3 q r x}{4 d^3}-\frac{(b g-a h)^2 p r (g+h x)^2}{8 b^2 h}-\frac{(d g-c h)^2 q r (g+h x)^2}{8 d^2 h}-\frac{(b g-a h) p r (g+h x)^3}{12 b h}-\frac{(d g-c h) q r (g+h x)^3}{12 d h}-\frac{p r (g+h x)^4}{16 h}-\frac{q r (g+h x)^4}{16 h}-\frac{(b g-a h)^4 p r \log (a+b x)}{4 b^4 h}-\frac{(d g-c h)^4 q r \log (c+d x)}{4 d^4 h}+\frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}\\ \end{align*}

Mathematica [A]  time = 0.310036, size = 231, normalized size = 0.84 \[ \frac{\frac{1}{12} r \left (-\frac{p \left (6 b^2 (g+h x)^2 (b g-a h)^2+4 b^3 (g+h x)^3 (b g-a h)+12 b h x (b g-a h)^3+12 (b g-a h)^4 \log (a+b x)+3 b^4 (g+h x)^4\right )}{b^4}-\frac{q \left (6 d^2 (g+h x)^2 (d g-c h)^2+4 d^3 (g+h x)^3 (d g-c h)+12 d h x (d g-c h)^3+12 (d g-c h)^4 \log (c+d x)+3 d^4 (g+h x)^4\right )}{d^4}\right )+(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

((r*(-((p*(12*b*h*(b*g - a*h)^3*x + 6*b^2*(b*g - a*h)^2*(g + h*x)^2 + 4*b^3*(b*g - a*h)*(g + h*x)^3 + 3*b^4*(g
+ h*x)^4 + 12*(b*g - a*h)^4*Log[a + b*x]))/b^4) - (q*(12*d*h*(d*g - c*h)^3*x + 6*d^2*(d*g - c*h)^2*(g + h*x)^
2 + 4*d^3*(d*g - c*h)*(g + h*x)^3 + 3*d^4*(g + h*x)^4 + 12*(d*g - c*h)^4*Log[c + d*x]))/d^4))/12 + (g + h*x)^4
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*h)

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Maple [F]  time = 0.397, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{3}\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)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

[Out]

int((h*x+g)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

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Maxima [A]  time = 1.21627, size = 582, normalized size = 2.11 \begin{align*} \frac{1}{4} \,{\left (h^{3} x^{4} + 4 \, g h^{2} x^{3} + 6 \, g^{2} h x^{2} + 4 \, g^{3} 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{12 \,{\left (4 \, a b^{3} f g^{3} p - 6 \, a^{2} b^{2} f g^{2} h p + 4 \, a^{3} b f g h^{2} p - a^{4} f h^{3} p\right )} \log \left (b x + a\right )}{b^{4}} + \frac{12 \,{\left (4 \, c d^{3} f g^{3} q - 6 \, c^{2} d^{2} f g^{2} h q + 4 \, c^{3} d f g h^{2} q - c^{4} f h^{3} q\right )} \log \left (d x + c\right )}{d^{4}} - \frac{3 \, b^{3} d^{3} f h^{3}{\left (p + q\right )} x^{4} - 4 \,{\left (a b^{2} d^{3} f h^{3} p -{\left (4 \, d^{3} f g h^{2}{\left (p + q\right )} - c d^{2} f h^{3} q\right )} b^{3}\right )} x^{3} - 6 \,{\left (4 \, a b^{2} d^{3} f g h^{2} p - a^{2} b d^{3} f h^{3} p -{\left (6 \, d^{3} f g^{2} h{\left (p + q\right )} - 4 \, c d^{2} f g h^{2} q + c^{2} d f h^{3} q\right )} b^{3}\right )} x^{2} - 12 \,{\left (6 \, a b^{2} d^{3} f g^{2} h p - 4 \, a^{2} b d^{3} f g h^{2} p + a^{3} d^{3} f h^{3} p -{\left (4 \, d^{3} f g^{3}{\left (p + q\right )} - 6 \, c d^{2} f g^{2} h q + 4 \, c^{2} d f g h^{2} q - c^{3} f h^{3} q\right )} b^{3}\right )} x}{b^{3} d^{3}}\right )}}{48 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima")

[Out]

1/4*(h^3*x^4 + 4*g*h^2*x^3 + 6*g^2*h*x^2 + 4*g^3*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1/48*r*(12*(4*a*b^3
*f*g^3*p - 6*a^2*b^2*f*g^2*h*p + 4*a^3*b*f*g*h^2*p - a^4*f*h^3*p)*log(b*x + a)/b^4 + 12*(4*c*d^3*f*g^3*q - 6*c
^2*d^2*f*g^2*h*q + 4*c^3*d*f*g*h^2*q - c^4*f*h^3*q)*log(d*x + c)/d^4 - (3*b^3*d^3*f*h^3*(p + q)*x^4 - 4*(a*b^2
*d^3*f*h^3*p - (4*d^3*f*g*h^2*(p + q) - c*d^2*f*h^3*q)*b^3)*x^3 - 6*(4*a*b^2*d^3*f*g*h^2*p - a^2*b*d^3*f*h^3*p
- (6*d^3*f*g^2*h*(p + q) - 4*c*d^2*f*g*h^2*q + c^2*d*f*h^3*q)*b^3)*x^2 - 12*(6*a*b^2*d^3*f*g^2*h*p - 4*a^2*b*
d^3*f*g*h^2*p + a^3*d^3*f*h^3*p - (4*d^3*f*g^3*(p + q) - 6*c*d^2*f*g^2*h*q + 4*c^2*d*f*g*h^2*q - c^3*f*h^3*q)*
b^3)*x)/(b^3*d^3))/f

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Fricas [B]  time = 1.24885, size = 1377, normalized size = 4.99 \begin{align*} -\frac{3 \,{\left (b^{4} d^{4} h^{3} p + b^{4} d^{4} h^{3} q\right )} r x^{4} + 4 \,{\left ({\left (4 \, b^{4} d^{4} g h^{2} - a b^{3} d^{4} h^{3}\right )} p +{\left (4 \, b^{4} d^{4} g h^{2} - b^{4} c d^{3} h^{3}\right )} q\right )} r x^{3} + 6 \,{\left ({\left (6 \, b^{4} d^{4} g^{2} h - 4 \, a b^{3} d^{4} g h^{2} + a^{2} b^{2} d^{4} h^{3}\right )} p +{\left (6 \, b^{4} d^{4} g^{2} h - 4 \, b^{4} c d^{3} g h^{2} + b^{4} c^{2} d^{2} h^{3}\right )} q\right )} r x^{2} + 12 \,{\left ({\left (4 \, b^{4} d^{4} g^{3} - 6 \, a b^{3} d^{4} g^{2} h + 4 \, a^{2} b^{2} d^{4} g h^{2} - a^{3} b d^{4} h^{3}\right )} p +{\left (4 \, b^{4} d^{4} g^{3} - 6 \, b^{4} c d^{3} g^{2} h + 4 \, b^{4} c^{2} d^{2} g h^{2} - b^{4} c^{3} d h^{3}\right )} q\right )} r x - 12 \,{\left (b^{4} d^{4} h^{3} p r x^{4} + 4 \, b^{4} d^{4} g h^{2} p r x^{3} + 6 \, b^{4} d^{4} g^{2} h p r x^{2} + 4 \, b^{4} d^{4} g^{3} p r x +{\left (4 \, a b^{3} d^{4} g^{3} - 6 \, a^{2} b^{2} d^{4} g^{2} h + 4 \, a^{3} b d^{4} g h^{2} - a^{4} d^{4} h^{3}\right )} p r\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} d^{4} h^{3} q r x^{4} + 4 \, b^{4} d^{4} g h^{2} q r x^{3} + 6 \, b^{4} d^{4} g^{2} h q r x^{2} + 4 \, b^{4} d^{4} g^{3} q r x +{\left (4 \, b^{4} c d^{3} g^{3} - 6 \, b^{4} c^{2} d^{2} g^{2} h + 4 \, b^{4} c^{3} d g h^{2} - b^{4} c^{4} h^{3}\right )} q r\right )} \log \left (d x + c\right ) - 12 \,{\left (b^{4} d^{4} h^{3} x^{4} + 4 \, b^{4} d^{4} g h^{2} x^{3} + 6 \, b^{4} d^{4} g^{2} h x^{2} + 4 \, b^{4} d^{4} g^{3} x\right )} \log \left (e\right ) - 12 \,{\left (b^{4} d^{4} h^{3} r x^{4} + 4 \, b^{4} d^{4} g h^{2} r x^{3} + 6 \, b^{4} d^{4} g^{2} h r x^{2} + 4 \, b^{4} d^{4} g^{3} r x\right )} \log \left (f\right )}{48 \, b^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas")

[Out]

-1/48*(3*(b^4*d^4*h^3*p + b^4*d^4*h^3*q)*r*x^4 + 4*((4*b^4*d^4*g*h^2 - a*b^3*d^4*h^3)*p + (4*b^4*d^4*g*h^2 - b
^4*c*d^3*h^3)*q)*r*x^3 + 6*((6*b^4*d^4*g^2*h - 4*a*b^3*d^4*g*h^2 + a^2*b^2*d^4*h^3)*p + (6*b^4*d^4*g^2*h - 4*b
^4*c*d^3*g*h^2 + b^4*c^2*d^2*h^3)*q)*r*x^2 + 12*((4*b^4*d^4*g^3 - 6*a*b^3*d^4*g^2*h + 4*a^2*b^2*d^4*g*h^2 - a^
3*b*d^4*h^3)*p + (4*b^4*d^4*g^3 - 6*b^4*c*d^3*g^2*h + 4*b^4*c^2*d^2*g*h^2 - b^4*c^3*d*h^3)*q)*r*x - 12*(b^4*d^
4*h^3*p*r*x^4 + 4*b^4*d^4*g*h^2*p*r*x^3 + 6*b^4*d^4*g^2*h*p*r*x^2 + 4*b^4*d^4*g^3*p*r*x + (4*a*b^3*d^4*g^3 - 6
*a^2*b^2*d^4*g^2*h + 4*a^3*b*d^4*g*h^2 - a^4*d^4*h^3)*p*r)*log(b*x + a) - 12*(b^4*d^4*h^3*q*r*x^4 + 4*b^4*d^4*
g*h^2*q*r*x^3 + 6*b^4*d^4*g^2*h*q*r*x^2 + 4*b^4*d^4*g^3*q*r*x + (4*b^4*c*d^3*g^3 - 6*b^4*c^2*d^2*g^2*h + 4*b^4
*c^3*d*g*h^2 - b^4*c^4*h^3)*q*r)*log(d*x + c) - 12*(b^4*d^4*h^3*x^4 + 4*b^4*d^4*g*h^2*x^3 + 6*b^4*d^4*g^2*h*x^
2 + 4*b^4*d^4*g^3*x)*log(e) - 12*(b^4*d^4*h^3*r*x^4 + 4*b^4*d^4*g*h^2*r*x^3 + 6*b^4*d^4*g^2*h*r*x^2 + 4*b^4*d^
4*g^3*r*x)*log(f))/(b^4*d^4)

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

[Out]

Timed out

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Giac [B]  time = 1.32799, size = 1254, normalized size = 4.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")

[Out]

-1/16*(h^3*p*r + h^3*q*r - 4*h^3*r*log(f) - 4*h^3)*x^4 - 1/12*(4*b*d*g*h^2*p*r - a*d*h^3*p*r + 4*b*d*g*h^2*q*r
- b*c*h^3*q*r - 12*b*d*g*h^2*r*log(f) - 12*b*d*g*h^2)*x^3/(b*d) + 1/4*(h^3*p*r*x^4 + 4*g*h^2*p*r*x^3 + 6*g^2*
h*p*r*x^2 + 4*g^3*p*r*x)*log(b*x + a) + 1/4*(h^3*q*r*x^4 + 4*g*h^2*q*r*x^3 + 6*g^2*h*q*r*x^2 + 4*g^3*q*r*x)*lo
g(d*x + c) - 1/8*(6*b^2*d^2*g^2*h*p*r - 4*a*b*d^2*g*h^2*p*r + a^2*d^2*h^3*p*r + 6*b^2*d^2*g^2*h*q*r - 4*b^2*c*
d*g*h^2*q*r + b^2*c^2*h^3*q*r - 12*b^2*d^2*g^2*h*r*log(f) - 12*b^2*d^2*g^2*h)*x^2/(b^2*d^2) - 1/4*(4*b^3*d^3*g
^3*p*r - 6*a*b^2*d^3*g^2*h*p*r + 4*a^2*b*d^3*g*h^2*p*r - a^3*d^3*h^3*p*r + 4*b^3*d^3*g^3*q*r - 6*b^3*c*d^2*g^2
*h*q*r + 4*b^3*c^2*d*g*h^2*q*r - b^3*c^3*h^3*q*r - 4*b^3*d^3*g^3*r*log(f) - 4*b^3*d^3*g^3)*x/(b^3*d^3) + 1/8*(
4*a*b^3*d^4*g^3*p*r - 6*a^2*b^2*d^4*g^2*h*p*r + 4*a^3*b*d^4*g*h^2*p*r - a^4*d^4*h^3*p*r + 4*b^4*c*d^3*g^3*q*r
- 6*b^4*c^2*d^2*g^2*h*q*r + 4*b^4*c^3*d*g*h^2*q*r - b^4*c^4*h^3*q*r)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(
b^4*d^4) + 1/8*(4*a*b^4*c*d^4*g^3*p*r - 4*a^2*b^3*d^5*g^3*p*r - 6*a^2*b^3*c*d^4*g^2*h*p*r + 6*a^3*b^2*d^5*g^2*
h*p*r + 4*a^3*b^2*c*d^4*g*h^2*p*r - 4*a^4*b*d^5*g*h^2*p*r - a^4*b*c*d^4*h^3*p*r + a^5*d^5*h^3*p*r - 4*b^5*c^2*
d^3*g^3*q*r + 4*a*b^4*c*d^4*g^3*q*r + 6*b^5*c^3*d^2*g^2*h*q*r - 6*a*b^4*c^2*d^3*g^2*h*q*r - 4*b^5*c^4*d*g*h^2*
q*r + 4*a*b^4*c^3*d^2*g*h^2*q*r + b^5*c^5*h^3*q*r - a*b^4*c^4*d*h^3*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^4*d^4*abs(-b*c + a*d))