\(\int (g+h x) \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [28]

Optimal result

Integrand size = 27, antiderivative size = 160 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {(b g-a h) p r x}{2 b}-\frac {(d g-c h) q r x}{2 d}-\frac {p r (g+h x)^2}{4 h}-\frac {q r (g+h x)^2}{4 h}-\frac {(b g-a h)^2 p r \log (a+b x)}{2 b^2 h}-\frac {(d g-c h)^2 q r \log (c+d x)}{2 d^2 h}+\frac {(g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2581, 45} \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {p r (b g-a h)^2 \log (a+b x)}{2 b^2 h}+\frac {(g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h}-\frac {p r x (b g-a h)}{2 b}-\frac {q r (d g-c h)^2 \log (c+d x)}{2 d^2 h}-\frac {q r x (d g-c h)}{2 d}-\frac {p r (g+h x)^2}{4 h}-\frac {q r (g+h x)^2}{4 h} \]

[In]

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

[Out]

-1/2*((b*g - a*h)*p*r*x)/b - ((d*g - c*h)*q*r*x)/(2*d) - (p*r*(g + h*x)^2)/(4*h) - (q*r*(g + h*x)^2)/(4*h) - (
(b*g - a*h)^2*p*r*Log[a + b*x])/(2*b^2*h) - ((d*g - c*h)^2*q*r*Log[c + d*x])/(2*d^2*h) + ((g + h*x)^2*Log[e*(f
*(a + b*x)^p*(c + d*x)^q)^r])/(2*h)

Rule 45

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])

Rule 2581

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {2 a d^2 (-2 b g+a h) p r \log (a+b x)+b \left (2 b c (-2 d g+c h) q r \log (c+d x)+d x \left (r (-2 a d h p-2 b c h q+b d (p+q) (4 g+h x))-2 b d (2 g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )}{4 b^2 d^2} \]

[In]

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

[Out]

-1/4*(2*a*d^2*(-2*b*g + a*h)*p*r*Log[a + b*x] + b*(2*b*c*(-2*d*g + c*h)*q*r*Log[c + d*x] + d*x*(r*(-2*a*d*h*p
- 2*b*c*h*q + b*d*(p + q)*(4*g + h*x)) - 2*b*d*(2*g + h*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])))/(b^2*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(146)=292\).

Time = 11.89 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.74

method	result	size
parallelrisch	\(-\frac {-2 \ln \left (b x +a \right ) a b c d h p r -2 \ln \left (d x +c \right ) a b c d h q r +a b c d h p r +a b c d h q r -2 x a b \,d^{2} h p r -2 x \,b^{2} c d h q r -8 \ln \left (b x +a \right ) a b \,d^{2} g p r -4 \ln \left (b x +a \right ) b^{2} c d g p r -4 \ln \left (d x +c \right ) a b \,d^{2} g q r -8 \ln \left (d x +c \right ) b^{2} c d g q r -2 x^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{2} d^{2} h -4 x \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{2} d^{2} g +4 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a b \,d^{2} g +4 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{2} c d g +x^{2} b^{2} d^{2} h p r +x^{2} b^{2} d^{2} h q r +4 x \,b^{2} d^{2} g p r +4 x \,b^{2} d^{2} g q r +2 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a b c d h +2 \ln \left (b x +a \right ) a^{2} d^{2} h p r +2 \ln \left (d x +c \right ) b^{2} c^{2} h q r +2 a^{2} h p r \,d^{2}+2 b^{2} c^{2} h q r -4 a b \,d^{2} g q r -4 b^{2} c d g p r -4 a b \,d^{2} g p r -4 b^{2} c d g q r}{4 b^{2} d^{2}}\)	\(438\)

[In]

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

[Out]

-1/4*(-2*ln(b*x+a)*a*b*c*d*h*p*r-2*ln(d*x+c)*a*b*c*d*h*q*r+a*b*c*d*h*p*r+a*b*c*d*h*q*r-2*x*a*b*d^2*h*p*r-2*x*b
^2*c*d*h*q*r-8*ln(b*x+a)*a*b*d^2*g*p*r-4*ln(b*x+a)*b^2*c*d*g*p*r-4*ln(d*x+c)*a*b*d^2*g*q*r-8*ln(d*x+c)*b^2*c*d
*g*q*r-2*x^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^2*d^2*h-4*x*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^2*d^2*g+4*ln(e*(f
*(b*x+a)^p*(d*x+c)^q)^r)*a*b*d^2*g+4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^2*c*d*g+x^2*b^2*d^2*h*p*r+x^2*b^2*d^2*h
*q*r+4*x*b^2*d^2*g*p*r+4*x*b^2*d^2*g*q*r+2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*a*b*c*d*h+2*ln(b*x+a)*a^2*d^2*h*p*r
+2*ln(d*x+c)*b^2*c^2*h*q*r+2*a^2*h*p*r*d^2+2*b^2*c^2*h*q*r-4*a*b*d^2*g*q*r-4*b^2*c*d*g*p*r-4*a*b*d^2*g*p*r-4*b
^2*c*d*g*q*r)/b^2/d^2

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.51 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {{\left (b^{2} d^{2} h p + b^{2} d^{2} h q\right )} r x^{2} + 2 \, {\left ({\left (2 \, b^{2} d^{2} g - a b d^{2} h\right )} p + {\left (2 \, b^{2} d^{2} g - b^{2} c d h\right )} q\right )} r x - 2 \, {\left (b^{2} d^{2} h p r x^{2} + 2 \, b^{2} d^{2} g p r x + {\left (2 \, a b d^{2} g - a^{2} d^{2} h\right )} p r\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} d^{2} h q r x^{2} + 2 \, b^{2} d^{2} g q r x + {\left (2 \, b^{2} c d g - b^{2} c^{2} h\right )} q r\right )} \log \left (d x + c\right ) - 2 \, {\left (b^{2} d^{2} h x^{2} + 2 \, b^{2} d^{2} g x\right )} \log \left (e\right ) - 2 \, {\left (b^{2} d^{2} h r x^{2} + 2 \, b^{2} d^{2} g r x\right )} \log \left (f\right )}{4 \, b^{2} d^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (141) = 282\).

Time = 25.80 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.14 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\begin {cases} \left (g x + \frac {h x^{2}}{2}\right ) \log {\left (e \left (a^{p} c^{q} f\right )^{r} \right )} & \text {for}\: b = 0 \wedge d = 0 \\- \frac {c^{2} h \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}}{2 d^{2}} + \frac {c g \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}}{d} + \frac {c h q r x}{2 d} - g q r x + g x \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )} - \frac {h q r x^{2}}{4} + \frac {h x^{2} \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}}{2} & \text {for}\: b = 0 \\- \frac {a^{2} h \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{2 b^{2}} + \frac {a g \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{b} + \frac {a h p r x}{2 b} - g p r x + g x \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )} - \frac {h p r x^{2}}{4} + \frac {h x^{2} \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{2} & \text {for}\: d = 0 \\\frac {a^{2} h q r \log {\left (\frac {c}{d} + x \right )}}{2 b^{2}} - \frac {a^{2} h \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{2 b^{2}} - \frac {a g q r \log {\left (\frac {c}{d} + x \right )}}{b} + \frac {a g \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{b} + \frac {a h p r x}{2 b} - \frac {c^{2} h q r \log {\left (\frac {c}{d} + x \right )}}{2 d^{2}} + \frac {c g q r \log {\left (\frac {c}{d} + x \right )}}{d} + \frac {c h q r x}{2 d} - g p r x - g q r x + g x \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )} - \frac {h p r x^{2}}{4} - \frac {h q r x^{2}}{4} + \frac {h x^{2} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((g*x + h*x**2/2)*log(e*(a**p*c**q*f)**r), Eq(b, 0) & Eq(d, 0)), (-c**2*h*log(e*(a**p*f*(c + d*x)**q
)**r)/(2*d**2) + c*g*log(e*(a**p*f*(c + d*x)**q)**r)/d + c*h*q*r*x/(2*d) - g*q*r*x + g*x*log(e*(a**p*f*(c + d*
x)**q)**r) - h*q*r*x**2/4 + h*x**2*log(e*(a**p*f*(c + d*x)**q)**r)/2, Eq(b, 0)), (-a**2*h*log(e*(c**q*f*(a + b
*x)**p)**r)/(2*b**2) + a*g*log(e*(c**q*f*(a + b*x)**p)**r)/b + a*h*p*r*x/(2*b) - g*p*r*x + g*x*log(e*(c**q*f*(
a + b*x)**p)**r) - h*p*r*x**2/4 + h*x**2*log(e*(c**q*f*(a + b*x)**p)**r)/2, Eq(d, 0)), (a**2*h*q*r*log(c/d + x
)/(2*b**2) - a**2*h*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)/(2*b**2) - a*g*q*r*log(c/d + x)/b + a*g*log(e*(f*(
a + b*x)**p*(c + d*x)**q)**r)/b + a*h*p*r*x/(2*b) - c**2*h*q*r*log(c/d + x)/(2*d**2) + c*g*q*r*log(c/d + x)/d
+ c*h*q*r*x/(2*d) - g*p*r*x - g*q*r*x + g*x*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r) - h*p*r*x**2/4 - h*q*r*x**
2/4 + h*x**2*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)/2, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{2} \, {\left (h x^{2} + 2 \, g x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {r {\left (\frac {2 \, {\left (2 \, a b f g p - a^{2} f h p\right )} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left (2 \, c d f g q - c^{2} f h q\right )} \log \left (d x + c\right )}{d^{2}} - \frac {b d f h {\left (p + q\right )} x^{2} - 2 \, {\left (a d f h p - {\left (2 \, d f g {\left (p + q\right )} - c f h q\right )} b\right )} x}{b d}\right )}}{4 \, f} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 3.02 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.12 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {1}{4} \, {\left (h p r + h q r - 2 \, h r \log \left (f\right ) - 2 \, h \log \left (e\right )\right )} x^{2} + \frac {1}{2} \, {\left (h p r x^{2} + 2 \, g p r x\right )} \log \left (b x + a\right ) + \frac {1}{2} \, {\left (h q r x^{2} + 2 \, g q r x\right )} \log \left (d x + c\right ) - \frac {{\left (2 \, b d g p r - a d h p r + 2 \, b d g q r - b c h q r - 2 \, b d g r \log \left (f\right ) - 2 \, b d g \log \left (e\right )\right )} x}{2 \, b d} + \frac {{\left (2 \, a b g p r - a^{2} h p r\right )} \log \left (-b x - a\right )}{2 \, b^{2}} + \frac {{\left (2 \, c d g q r - c^{2} h q r\right )} \log \left (d x + c\right )}{2 \, d^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96 \[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {h\,x^2}{2}+g\,x\right )-x\,\left (\frac {r\,\left (b\,c\,h\,p+2\,b\,d\,g\,p+a\,d\,h\,q+2\,b\,d\,g\,q\right )}{2\,b\,d}-\frac {h\,r\,\left (p+q\right )\,\left (2\,a\,d+2\,b\,c\right )}{4\,b\,d}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^2\,h\,p\,r-2\,a\,b\,g\,p\,r\right )}{2\,b^2}-\frac {\ln \left (c+d\,x\right )\,\left (c^2\,h\,q\,r-2\,c\,d\,g\,q\,r\right )}{2\,d^2}-\frac {h\,r\,x^2\,\left (p+q\right )}{4} \]

[In]

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

[Out]

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