3.2.98 \(\int (d+e x)^3 \log (c (a+\frac {b}{x})^p) \, dx\) [198]

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

[Out]

1/4*b*e*(6*a^2*d^2-4*a*b*d*e+b^2*e^2)*p*x/a^3+1/8*b*e^2*(4*a*d-b*e)*p*x^2/a^2+1/12*b*e^3*p*x^3/a+1/4*(e*x+d)^4
*ln(c*(a+b/x)^p)/e+1/4*d^4*p*ln(x)/e-1/4*(a*d-b*e)^4*p*ln(a*x+b)/a^4/e

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Rubi [A]
time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2513, 528, 84} \begin {gather*} -\frac {p (a d-b e)^4 \log (a x+b)}{4 a^4 e}+\frac {b e^2 p x^2 (4 a d-b e)}{8 a^2}+\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 {(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} \end {gather*}

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 84

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]

Rule 528

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 2513

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]

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*}

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Mathematica [A]
time = 0.10, size = 114, normalized size = 0.82 \begin {gather*} \frac {\frac {b e^2 p x \left (6 b^2 e^2-3 a b e (8 d+e x)+2 a^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )}{6 a^3}+(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )+d^4 p \log (x)-\frac {(a d-b e)^4 p \log (b+a x)}{a^4}}{4 e} \end {gather*}

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)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{3} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )\, dx\]

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)

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Maxima [A]
time = 0.36, size = 160, normalized size = 1.15 \begin {gather*} \frac {1}{24} \, b p {\left (\frac {2 \, a^{2} x^{3} e^{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 (x^{4} e^{3} + 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e + 4 \, d^{3} x\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \end {gather*}

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

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Fricas [A]
time = 0.39, size = 236, normalized size = 1.70 \begin {gather*} \frac {36 \, a^{3} b d^{2} p x e + {\left (2 \, a^{3} b p x^{3} - 3 \, a^{2} b^{2} p x^{2} + 6 \, a b^{3} p x\right )} e^{3} + 12 \, {\left (a^{3} b d p x^{2} - 2 \, a^{2} b^{2} d p x\right )} e^{2} + 6 \, {\left (4 \, a^{3} b d^{3} p - 6 \, a^{2} b^{2} d^{2} p e + 4 \, a b^{3} d p e^{2} - b^{4} p e^{3}\right )} \log \left (a x + b\right ) + 6 \, {\left (a^{4} x^{4} e^{3} + 4 \, a^{4} d x^{3} e^{2} + 6 \, a^{4} d^{2} x^{2} e + 4 \, a^{4} d^{3} x\right )} \log \left (c\right ) + 6 \, {\left (a^{4} p x^{4} e^{3} + 4 \, a^{4} d p x^{3} e^{2} + 6 \, a^{4} d^{2} p x^{2} e + 4 \, a^{4} d^{3} p x\right )} \log \left (\frac {a x + b}{x}\right )}{24 \, a^{4}} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (128) = 256\).
time = 1.71, size = 355, normalized size = 2.55 \begin {gather*} \begin {cases} d^{3} x \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {3 d^{2} e x^{2} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{2} + d e^{2} x^{3} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {e^{3} x^{4} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \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 + d^{3} x \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} + \frac {3 d^{2} e p x^{2}}{4} + \frac {3 d^{2} e x^{2} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )}}{2} + \frac {d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} + \frac {e^{3} p x^{4}}{16} + \frac {e^{3} x^{4} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}

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*x*log(c*(a + b/x)**p) + 3*d**2*e*x**2*log(c*(a + b/x)**p)/2 + d*e**2*x**3*log(c*(a + b/x)**p)
+ e**3*x**4*log(c*(a + b/x)**p)/4 + b*d**3*p*log(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 + d**3*x*log(c*(b/x)**p) + 3*d**2*e*p*x**2/4 + 3*d**2*e*x**2*log(c*(b/x)**p)/2 + d*e**2*p*x**3/3 + d*e
**2*x**3*log(c*(b/x)**p) + e**3*p*x**4/16 + e**3*x**4*log(c*(b/x)**p)/4, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1659 vs. \(2 (126) = 252\).
time = 5.02, size = 1659, normalized size = 11.94 \begin {gather*} \text {Too large to display} \end {gather*}

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*(24*a^7*b^2*d^3*p*log(-a + (a*x + b)/x) - 36*a^6*b^3*d^2*p*e*log(-a + (a*x + b)/x) + 36*a^6*b^3*d^2*p*e
- 96*(a*x + b)*a^6*b^2*d^3*p*log(-a + (a*x + b)/x)/x + 24*a^5*b^4*d*p*e^2*log(-a + (a*x + b)/x) + 144*(a*x + b
)*a^5*b^3*d^2*p*e*log(-a + (a*x + b)/x)/x + 24*a^7*b^2*d^3*log(c) - 36*a^6*b^3*d^2*e*log(c) + 24*(a*x + b)*a^6
*b^2*d^3*p*log((a*x + b)/x)/x - 72*(a*x + b)*a^5*b^3*d^2*p*e*log((a*x + b)/x)/x - 36*a^5*b^4*d*p*e^2 - 108*(a*
x + b)*a^5*b^3*d^2*p*e/x + 144*(a*x + b)^2*a^5*b^2*d^3*p*log(-a + (a*x + b)/x)/x^2 - 6*a^4*b^5*p*e^3*log(-a +
(a*x + b)/x) - 96*(a*x + b)*a^4*b^4*d*p*e^2*log(-a + (a*x + b)/x)/x - 216*(a*x + b)^2*a^4*b^3*d^2*p*e*log(-a +
(a*x + b)/x)/x^2 - 72*(a*x + b)*a^6*b^2*d^3*log(c)/x + 24*a^5*b^4*d*e^2*log(c) + 72*(a*x + b)*a^5*b^3*d^2*e*l
og(c)/x - 72*(a*x + b)^2*a^5*b^2*d^3*p*log((a*x + b)/x)/x^2 + 72*(a*x + b)*a^4*b^4*d*p*e^2*log((a*x + b)/x)/x
+ 180*(a*x + b)^2*a^4*b^3*d^2*p*e*log((a*x + b)/x)/x^2 + 11*a^4*b^5*p*e^3 + 96*(a*x + b)*a^4*b^4*d*p*e^2/x + 1
08*(a*x + b)^2*a^4*b^3*d^2*p*e/x^2 - 96*(a*x + b)^3*a^4*b^2*d^3*p*log(-a + (a*x + b)/x)/x^3 + 24*(a*x + b)*a^3
*b^5*p*e^3*log(-a + (a*x + b)/x)/x + 144*(a*x + b)^2*a^3*b^4*d*p*e^2*log(-a + (a*x + b)/x)/x^2 + 144*(a*x + b)
^3*a^3*b^3*d^2*p*e*log(-a + (a*x + b)/x)/x^3 + 72*(a*x + b)^2*a^5*b^2*d^3*log(c)/x^2 - 6*a^4*b^5*e^3*log(c) -
24*(a*x + b)*a^4*b^4*d*e^2*log(c)/x - 36*(a*x + b)^2*a^4*b^3*d^2*e*log(c)/x^2 + 72*(a*x + b)^3*a^4*b^2*d^3*p*l
og((a*x + b)/x)/x^3 - 24*(a*x + b)*a^3*b^5*p*e^3*log((a*x + b)/x)/x - 144*(a*x + b)^2*a^3*b^4*d*p*e^2*log((a*x
+ b)/x)/x^2 - 144*(a*x + b)^3*a^3*b^3*d^2*p*e*log((a*x + b)/x)/x^3 - 26*(a*x + b)*a^3*b^5*p*e^3/x - 84*(a*x +
b)^2*a^3*b^4*d*p*e^2/x^2 - 36*(a*x + b)^3*a^3*b^3*d^2*p*e/x^3 + 24*(a*x + b)^4*a^3*b^2*d^3*p*log(-a + (a*x +
b)/x)/x^4 - 36*(a*x + b)^2*a^2*b^5*p*e^3*log(-a + (a*x + b)/x)/x^2 - 96*(a*x + b)^3*a^2*b^4*d*p*e^2*log(-a + (
a*x + b)/x)/x^3 - 36*(a*x + b)^4*a^2*b^3*d^2*p*e*log(-a + (a*x + b)/x)/x^4 - 24*(a*x + b)^3*a^4*b^2*d^3*log(c)
/x^3 - 24*(a*x + b)^4*a^3*b^2*d^3*p*log((a*x + b)/x)/x^4 + 36*(a*x + b)^2*a^2*b^5*p*e^3*log((a*x + b)/x)/x^2 +
96*(a*x + b)^3*a^2*b^4*d*p*e^2*log((a*x + b)/x)/x^3 + 36*(a*x + b)^4*a^2*b^3*d^2*p*e*log((a*x + b)/x)/x^4 + 2
1*(a*x + b)^2*a^2*b^5*p*e^3/x^2 + 24*(a*x + b)^3*a^2*b^4*d*p*e^2/x^3 + 24*(a*x + b)^3*a*b^5*p*e^3*log(-a + (a*
x + b)/x)/x^3 + 24*(a*x + b)^4*a*b^4*d*p*e^2*log(-a + (a*x + b)/x)/x^4 - 24*(a*x + b)^3*a*b^5*p*e^3*log((a*x +
b)/x)/x^3 - 24*(a*x + b)^4*a*b^4*d*p*e^2*log((a*x + b)/x)/x^4 - 6*(a*x + b)^3*a*b^5*p*e^3/x^3 - 6*(a*x + b)^4
*b^5*p*e^3*log(-a + (a*x + b)/x)/x^4 + 6*(a*x + b)^4*b^5*p*e^3*log((a*x + b)/x)/x^4)/((a^8 - 4*(a*x + b)*a^7/x
+ 6*(a*x + b)^2*a^6/x^2 - 4*(a*x + b)^3*a^5/x^3 + (a*x + b)^4*a^4/x^4)*b)

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Mupad [B]
time = 0.34, size = 184, normalized size = 1.32 \begin {gather*} x\,\left (\frac {b\,\left (\frac {b^2\,e^3\,p}{4\,a^2}-\frac {b\,d\,e^2\,p}{a}\right )}{a}+\frac {3\,b\,d^2\,e\,p}{2\,a}\right )+\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )\,\left (d^3\,x+\frac {3\,d^2\,e\,x^2}{2}+d\,e^2\,x^3+\frac {e^3\,x^4}{4}\right )-x^2\,\left (\frac {b^2\,e^3\,p}{8\,a^2}-\frac {b\,d\,e^2\,p}{2\,a}\right )-\frac {\ln \left (b+a\,x\right )\,\left (-4\,p\,a^3\,b\,d^3+6\,p\,a^2\,b^2\,d^2\,e-4\,p\,a\,b^3\,d\,e^2+p\,b^4\,e^3\right )}{4\,a^4}+\frac {b\,e^3\,p\,x^3}{12\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x)^p)*(d + e*x)^3,x)

[Out]

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

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