3.198 \(\int (d+e x)^3 \log (c (a+\frac{b}{x})^p) \, dx\)

Optimal. Leaf size=139 \[ \frac{b e p x \left (6 a^2 d^2-4 a b d e+b^2 e^2\right )}{4 a^3}+\frac{b e^2 p x^2 (4 a d-b e)}{8 a^2}-\frac{p (a d-b e)^4 \log (a x+b)}{4 a^4 e}+\frac{(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 e}+\frac{b e^3 p x^3}{12 a}+\frac{d^4 p \log (x)}{4 e} \]

[Out]

(b*e*(6*a^2*d^2 - 4*a*b*d*e + b^2*e^2)*p*x)/(4*a^3) + (b*e^2*(4*a*d - b*e)*p*x^2)/(8*a^2) + (b*e^3*p*x^3)/(12*
a) + ((d + e*x)^4*Log[c*(a + b/x)^p])/(4*e) + (d^4*p*Log[x])/(4*e) - ((a*d - b*e)^4*p*Log[b + a*x])/(4*a^4*e)

________________________________________________________________________________________

Rubi [A]  time = 0.125612, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2463, 514, 72} \[ \frac{b e p x \left (6 a^2 d^2-4 a b d e+b^2 e^2\right )}{4 a^3}+\frac{b e^2 p x^2 (4 a d-b e)}{8 a^2}-\frac{p (a d-b e)^4 \log (a x+b)}{4 a^4 e}+\frac{(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 e}+\frac{b e^3 p x^3}{12 a}+\frac{d^4 p \log (x)}{4 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*Log[c*(a + b/x)^p],x]

[Out]

(b*e*(6*a^2*d^2 - 4*a*b*d*e + b^2*e^2)*p*x)/(4*a^3) + (b*e^2*(4*a*d - b*e)*p*x^2)/(8*a^2) + (b*e^3*p*x^3)/(12*
a) + ((d + e*x)^4*Log[c*(a + b/x)^p])/(4*e) + (d^4*p*Log[x])/(4*e) - ((a*d - b*e)^4*p*Log[b + a*x])/(4*a^4*e)

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[((
f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f
+ g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int (d+e x)^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 e}+\frac{(b p) \int \frac{(d+e x)^4}{\left (a+\frac{b}{x}\right ) x^2} \, dx}{4 e}\\ &=\frac{(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 e}+\frac{(b p) \int \frac{(d+e x)^4}{x (b+a x)} \, dx}{4 e}\\ &=\frac{(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 e}+\frac{(b p) \int \left (\frac{e^2 \left (6 a^2 d^2-4 a b d e+b^2 e^2\right )}{a^3}+\frac{d^4}{b x}+\frac{e^3 (4 a d-b e) x}{a^2}+\frac{e^4 x^2}{a}-\frac{(a d-b e)^4}{a^3 b (b+a x)}\right ) \, dx}{4 e}\\ &=\frac{b e \left (6 a^2 d^2-4 a b d e+b^2 e^2\right ) p x}{4 a^3}+\frac{b e^2 (4 a d-b e) p x^2}{8 a^2}+\frac{b e^3 p x^3}{12 a}+\frac{(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 e}+\frac{d^4 p \log (x)}{4 e}-\frac{(a d-b e)^4 p \log (b+a x)}{4 a^4 e}\\ \end{align*}

Mathematica [A]  time = 0.140229, size = 114, normalized size = 0.82 \[ \frac{\frac{b e^2 p x \left (2 a^2 \left (18 d^2+6 d e x+e^2 x^2\right )-3 a b e (8 d+e x)+6 b^2 e^2\right )}{6 a^3}-\frac{p (a d-b e)^4 \log (a x+b)}{a^4}+(d+e x)^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+d^4 p \log (x)}{4 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*Log[c*(a + b/x)^p],x]

[Out]

((b*e^2*p*x*(6*b^2*e^2 - 3*a*b*e*(8*d + e*x) + 2*a^2*(18*d^2 + 6*d*e*x + e^2*x^2)))/(6*a^3) + (d + e*x)^4*Log[
c*(a + b/x)^p] + d^4*p*Log[x] - ((a*d - b*e)^4*p*Log[b + a*x])/a^4)/(4*e)

________________________________________________________________________________________

Maple [F]  time = 0.582, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*ln(c*(a+b/x)^p),x)

[Out]

int((e*x+d)^3*ln(c*(a+b/x)^p),x)

________________________________________________________________________________________

Maxima [A]  time = 1.05346, size = 224, normalized size = 1.61 \begin{align*} \frac{1}{24} \, b p{\left (\frac{2 \, a^{2} e^{3} x^{3} + 3 \,{\left (4 \, a^{2} d e^{2} - a b e^{3}\right )} x^{2} + 6 \,{\left (6 \, a^{2} d^{2} e - 4 \, a b d e^{2} + b^{2} e^{3}\right )} x}{a^{3}} + \frac{6 \,{\left (4 \, a^{3} d^{3} - 6 \, a^{2} b d^{2} e + 4 \, a b^{2} d e^{2} - b^{3} e^{3}\right )} \log \left (a x + b\right )}{a^{4}}\right )} + \frac{1}{4} \,{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*log(c*(a+b/x)^p),x, algorithm="maxima")

[Out]

1/24*b*p*((2*a^2*e^3*x^3 + 3*(4*a^2*d*e^2 - a*b*e^3)*x^2 + 6*(6*a^2*d^2*e - 4*a*b*d*e^2 + b^2*e^3)*x)/a^3 + 6*
(4*a^3*d^3 - 6*a^2*b*d^2*e + 4*a*b^2*d*e^2 - b^3*e^3)*log(a*x + b)/a^4) + 1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e
*x^2 + 4*d^3*x)*log((a + b/x)^p*c)

________________________________________________________________________________________

Fricas [A]  time = 1.68249, size = 498, normalized size = 3.58 \begin{align*} \frac{2 \, a^{3} b e^{3} p x^{3} + 3 \,{\left (4 \, a^{3} b d e^{2} - a^{2} b^{2} e^{3}\right )} p x^{2} + 6 \,{\left (6 \, a^{3} b d^{2} e - 4 \, a^{2} b^{2} d e^{2} + a b^{3} e^{3}\right )} p x + 6 \,{\left (4 \, a^{3} b d^{3} - 6 \, a^{2} b^{2} d^{2} e + 4 \, a b^{3} d e^{2} - b^{4} e^{3}\right )} p \log \left (a x + b\right ) + 6 \,{\left (a^{4} e^{3} x^{4} + 4 \, a^{4} d e^{2} x^{3} + 6 \, a^{4} d^{2} e x^{2} + 4 \, a^{4} d^{3} x\right )} \log \left (c\right ) + 6 \,{\left (a^{4} e^{3} p x^{4} + 4 \, a^{4} d e^{2} p x^{3} + 6 \, a^{4} d^{2} e p x^{2} + 4 \, a^{4} d^{3} p x\right )} \log \left (\frac{a x + b}{x}\right )}{24 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*log(c*(a+b/x)^p),x, algorithm="fricas")

[Out]

1/24*(2*a^3*b*e^3*p*x^3 + 3*(4*a^3*b*d*e^2 - a^2*b^2*e^3)*p*x^2 + 6*(6*a^3*b*d^2*e - 4*a^2*b^2*d*e^2 + a*b^3*e
^3)*p*x + 6*(4*a^3*b*d^3 - 6*a^2*b^2*d^2*e + 4*a*b^3*d*e^2 - b^4*e^3)*p*log(a*x + b) + 6*(a^4*e^3*x^4 + 4*a^4*
d*e^2*x^3 + 6*a^4*d^2*e*x^2 + 4*a^4*d^3*x)*log(c) + 6*(a^4*e^3*p*x^4 + 4*a^4*d*e^2*p*x^3 + 6*a^4*d^2*e*p*x^2 +
4*a^4*d^3*p*x)*log((a*x + b)/x))/a^4

________________________________________________________________________________________

Sympy [A]  time = 13.9199, size = 484, normalized size = 3.48 \begin{align*} \begin{cases} d^{3} p x \log{\left (a + \frac{b}{x} \right )} + d^{3} x \log{\left (c \right )} + \frac{3 d^{2} e p x^{2} \log{\left (a + \frac{b}{x} \right )}}{2} + \frac{3 d^{2} e x^{2} \log{\left (c \right )}}{2} + d e^{2} p x^{3} \log{\left (a + \frac{b}{x} \right )} + d e^{2} x^{3} \log{\left (c \right )} + \frac{e^{3} p x^{4} \log{\left (a + \frac{b}{x} \right )}}{4} + \frac{e^{3} x^{4} \log{\left (c \right )}}{4} + \frac{b d^{3} p \log{\left (x + \frac{b}{a} \right )}}{a} + \frac{3 b d^{2} e p x}{2 a} + \frac{b d e^{2} p x^{2}}{2 a} + \frac{b e^{3} p x^{3}}{12 a} - \frac{3 b^{2} d^{2} e p \log{\left (x + \frac{b}{a} \right )}}{2 a^{2}} - \frac{b^{2} d e^{2} p x}{a^{2}} - \frac{b^{2} e^{3} p x^{2}}{8 a^{2}} + \frac{b^{3} d e^{2} p \log{\left (x + \frac{b}{a} \right )}}{a^{3}} + \frac{b^{3} e^{3} p x}{4 a^{3}} - \frac{b^{4} e^{3} p \log{\left (x + \frac{b}{a} \right )}}{4 a^{4}} & \text{for}\: a \neq 0 \\d^{3} p x \log{\left (b \right )} - d^{3} p x \log{\left (x \right )} + d^{3} p x + d^{3} x \log{\left (c \right )} + \frac{3 d^{2} e p x^{2} \log{\left (b \right )}}{2} - \frac{3 d^{2} e p x^{2} \log{\left (x \right )}}{2} + \frac{3 d^{2} e p x^{2}}{4} + \frac{3 d^{2} e x^{2} \log{\left (c \right )}}{2} + d e^{2} p x^{3} \log{\left (b \right )} - d e^{2} p x^{3} \log{\left (x \right )} + \frac{d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log{\left (c \right )} + \frac{e^{3} p x^{4} \log{\left (b \right )}}{4} - \frac{e^{3} p x^{4} \log{\left (x \right )}}{4} + \frac{e^{3} p x^{4}}{16} + \frac{e^{3} x^{4} \log{\left (c \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*ln(c*(a+b/x)**p),x)

[Out]

Piecewise((d**3*p*x*log(a + b/x) + d**3*x*log(c) + 3*d**2*e*p*x**2*log(a + b/x)/2 + 3*d**2*e*x**2*log(c)/2 + d
*e**2*p*x**3*log(a + b/x) + d*e**2*x**3*log(c) + e**3*p*x**4*log(a + b/x)/4 + e**3*x**4*log(c)/4 + b*d**3*p*lo
g(x + b/a)/a + 3*b*d**2*e*p*x/(2*a) + b*d*e**2*p*x**2/(2*a) + b*e**3*p*x**3/(12*a) - 3*b**2*d**2*e*p*log(x + b
/a)/(2*a**2) - b**2*d*e**2*p*x/a**2 - b**2*e**3*p*x**2/(8*a**2) + b**3*d*e**2*p*log(x + b/a)/a**3 + b**3*e**3*
p*x/(4*a**3) - b**4*e**3*p*log(x + b/a)/(4*a**4), Ne(a, 0)), (d**3*p*x*log(b) - d**3*p*x*log(x) + d**3*p*x + d
**3*x*log(c) + 3*d**2*e*p*x**2*log(b)/2 - 3*d**2*e*p*x**2*log(x)/2 + 3*d**2*e*p*x**2/4 + 3*d**2*e*x**2*log(c)/
2 + d*e**2*p*x**3*log(b) - d*e**2*p*x**3*log(x) + d*e**2*p*x**3/3 + d*e**2*x**3*log(c) + e**3*p*x**4*log(b)/4
- e**3*p*x**4*log(x)/4 + e**3*p*x**4/16 + e**3*x**4*log(c)/4, True))

________________________________________________________________________________________

Giac [B]  time = 1.23204, size = 437, normalized size = 3.14 \begin{align*} \frac{6 \, a^{4} p x^{4} e^{3} \log \left (a x + b\right ) + 24 \, a^{4} d p x^{3} e^{2} \log \left (a x + b\right ) + 36 \, a^{4} d^{2} p x^{2} e \log \left (a x + b\right ) - 6 \, a^{4} p x^{4} e^{3} \log \left (x\right ) - 24 \, a^{4} d p x^{3} e^{2} \log \left (x\right ) - 36 \, a^{4} d^{2} p x^{2} e \log \left (x\right ) + 24 \, a^{4} d^{3} p x \log \left (a x + b\right ) + 6 \, a^{4} x^{4} e^{3} \log \left (c\right ) + 24 \, a^{4} d x^{3} e^{2} \log \left (c\right ) + 36 \, a^{4} d^{2} x^{2} e \log \left (c\right ) - 24 \, a^{4} d^{3} p x \log \left (x\right ) + 2 \, a^{3} b p x^{3} e^{3} + 12 \, a^{3} b d p x^{2} e^{2} + 36 \, a^{3} b d^{2} p x e + 24 \, a^{3} b d^{3} p \log \left (a x + b\right ) - 36 \, a^{2} b^{2} d^{2} p e \log \left (a x + b\right ) + 24 \, a^{4} d^{3} x \log \left (c\right ) - 3 \, a^{2} b^{2} p x^{2} e^{3} - 24 \, a^{2} b^{2} d p x e^{2} + 24 \, a b^{3} d p e^{2} \log \left (a x + b\right ) + 6 \, a b^{3} p x e^{3} - 6 \, b^{4} p e^{3} \log \left (a x + b\right )}{24 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*log(c*(a+b/x)^p),x, algorithm="giac")

[Out]

1/24*(6*a^4*p*x^4*e^3*log(a*x + b) + 24*a^4*d*p*x^3*e^2*log(a*x + b) + 36*a^4*d^2*p*x^2*e*log(a*x + b) - 6*a^4
*p*x^4*e^3*log(x) - 24*a^4*d*p*x^3*e^2*log(x) - 36*a^4*d^2*p*x^2*e*log(x) + 24*a^4*d^3*p*x*log(a*x + b) + 6*a^
4*x^4*e^3*log(c) + 24*a^4*d*x^3*e^2*log(c) + 36*a^4*d^2*x^2*e*log(c) - 24*a^4*d^3*p*x*log(x) + 2*a^3*b*p*x^3*e
^3 + 12*a^3*b*d*p*x^2*e^2 + 36*a^3*b*d^2*p*x*e + 24*a^3*b*d^3*p*log(a*x + b) - 36*a^2*b^2*d^2*p*e*log(a*x + b)
+ 24*a^4*d^3*x*log(c) - 3*a^2*b^2*p*x^2*e^3 - 24*a^2*b^2*d*p*x*e^2 + 24*a*b^3*d*p*e^2*log(a*x + b) + 6*a*b^3*
p*x*e^3 - 6*b^4*p*e^3*log(a*x + b))/a^4