3.2.99 \(\int (d+e x)^2 \log (c (a+\frac {b}{x})^p) \, dx\) [199]

Optimal. Leaf size=102 \[ \frac {b e (3 a d-b e) p x}{3 a^2}+\frac {b e^2 p x^2}{6 a}+\frac {(d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {d^3 p \log (x)}{3 e}-\frac {(a d-b e)^3 p \log (b+a x)}{3 a^3 e} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 102, 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)^3 \log (a x+b)}{3 a^3 e}+\frac {b e p x (3 a d-b e)}{3 a^2}+\frac {(d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b e^2 p x^2}{6 a}+\frac {d^3 p \log (x)}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*Log[c*(a + b/x)^p],x]

[Out]

(b*e*(3*a*d - b*e)*p*x)/(3*a^2) + (b*e^2*p*x^2)/(6*a) + ((d + e*x)^3*Log[c*(a + b/x)^p])/(3*e) + (d^3*p*Log[x]
)/(3*e) - ((a*d - b*e)^3*p*Log[b + a*x])/(3*a^3*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)^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx &=\frac {(d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {(b p) \int \frac {(d+e x)^3}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {(b p) \int \frac {(d+e x)^3}{x (b+a x)} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {(b p) \int \left (\frac {e^2 (3 a d-b e)}{a^2}+\frac {d^3}{b x}+\frac {e^3 x}{a}-\frac {(a d-b e)^3}{a^2 b (b+a x)}\right ) \, dx}{3 e}\\ &=\frac {b e (3 a d-b e) p x}{3 a^2}+\frac {b e^2 p x^2}{6 a}+\frac {(d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {d^3 p \log (x)}{3 e}-\frac {(a d-b e)^3 p \log (b+a x)}{3 a^3 e}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 86, normalized size = 0.84 \begin {gather*} \frac {2 a^3 (d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )+p \left (a b e^2 x (6 a d-2 b e+a e x)+2 a^3 d^3 \log (x)-2 (a d-b e)^3 \log (b+a x)\right )}{6 a^3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*Log[c*(a + b/x)^p],x]

[Out]

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

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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{2} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*ln(c*(a+b/x)^p),x)

[Out]

int((e*x+d)^2*ln(c*(a+b/x)^p),x)

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Maxima [A]
time = 0.34, size = 101, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, b p {\left (\frac {a x^{2} e^{2} + 2 \, {\left (3 \, a d e - b e^{2}\right )} x}{a^{2}} + \frac {2 \, {\left (3 \, a^{2} d^{2} - 3 \, a b d e + b^{2} e^{2}\right )} \log \left (a x + b\right )}{a^{3}}\right )} + \frac {1}{3} \, {\left (x^{3} e^{2} + 3 \, d x^{2} e + 3 \, d^{2} 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)^2*log(c*(a+b/x)^p),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 153, normalized size = 1.50 \begin {gather*} \frac {6 \, a^{2} b d p x e + {\left (a^{2} b p x^{2} - 2 \, a b^{2} p x\right )} e^{2} + 2 \, {\left (3 \, a^{2} b d^{2} p - 3 \, a b^{2} d p e + b^{3} p e^{2}\right )} \log \left (a x + b\right ) + 2 \, {\left (a^{3} x^{3} e^{2} + 3 \, a^{3} d x^{2} e + 3 \, a^{3} d^{2} x\right )} \log \left (c\right ) + 2 \, {\left (a^{3} p x^{3} e^{2} + 3 \, a^{3} d p x^{2} e + 3 \, a^{3} d^{2} p x\right )} \log \left (\frac {a x + b}{x}\right )}{6 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*log(c*(a+b/x)^p),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (88) = 176\).
time = 0.99, size = 216, normalized size = 2.12 \begin {gather*} \begin {cases} d^{2} x \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + d e x^{2} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {e^{2} x^{3} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{3} + \frac {b d^{2} p \log {\left (x + \frac {b}{a} \right )}}{a} + \frac {b d e p x}{a} + \frac {b e^{2} p x^{2}}{6 a} - \frac {b^{2} d e p \log {\left (x + \frac {b}{a} \right )}}{a^{2}} - \frac {b^{2} e^{2} p x}{3 a^{2}} + \frac {b^{3} e^{2} p \log {\left (x + \frac {b}{a} \right )}}{3 a^{3}} & \text {for}\: a \neq 0 \\d^{2} p x + d^{2} x \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} + \frac {d e p x^{2}}{2} + d e x^{2} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} + \frac {e^{2} p x^{3}}{9} + \frac {e^{2} x^{3} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*ln(c*(a+b/x)**p),x)

[Out]

Piecewise((d**2*x*log(c*(a + b/x)**p) + d*e*x**2*log(c*(a + b/x)**p) + e**2*x**3*log(c*(a + b/x)**p)/3 + b*d**
2*p*log(x + b/a)/a + b*d*e*p*x/a + b*e**2*p*x**2/(6*a) - b**2*d*e*p*log(x + b/a)/a**2 - b**2*e**2*p*x/(3*a**2)
+ b**3*e**2*p*log(x + b/a)/(3*a**3), Ne(a, 0)), (d**2*p*x + d**2*x*log(c*(b/x)**p) + d*e*p*x**2/2 + d*e*x**2*
log(c*(b/x)**p) + e**2*p*x**3/9 + e**2*x**3*log(c*(b/x)**p)/3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (92) = 184\).
time = 3.87, size = 918, normalized size = 9.00 \begin {gather*} -\frac {6 \, a^{5} b^{2} d^{2} p \log \left (-a + \frac {a x + b}{x}\right ) - 6 \, a^{4} b^{3} d p e \log \left (-a + \frac {a x + b}{x}\right ) + 6 \, a^{4} b^{3} d p e - \frac {18 \, {\left (a x + b\right )} a^{4} b^{2} d^{2} p \log \left (-a + \frac {a x + b}{x}\right )}{x} + 2 \, a^{3} b^{4} p e^{2} \log \left (-a + \frac {a x + b}{x}\right ) + \frac {18 \, {\left (a x + b\right )} a^{3} b^{3} d p e \log \left (-a + \frac {a x + b}{x}\right )}{x} + 6 \, a^{5} b^{2} d^{2} \log \left (c\right ) - 6 \, a^{4} b^{3} d e \log \left (c\right ) + \frac {6 \, {\left (a x + b\right )} a^{4} b^{2} d^{2} p \log \left (\frac {a x + b}{x}\right )}{x} - \frac {12 \, {\left (a x + b\right )} a^{3} b^{3} d p e \log \left (\frac {a x + b}{x}\right )}{x} - 3 \, a^{3} b^{4} p e^{2} - \frac {12 \, {\left (a x + b\right )} a^{3} b^{3} d p e}{x} + \frac {18 \, {\left (a x + b\right )}^{2} a^{3} b^{2} d^{2} p \log \left (-a + \frac {a x + b}{x}\right )}{x^{2}} - \frac {6 \, {\left (a x + b\right )} a^{2} b^{4} p e^{2} \log \left (-a + \frac {a x + b}{x}\right )}{x} - \frac {18 \, {\left (a x + b\right )}^{2} a^{2} b^{3} d p e \log \left (-a + \frac {a x + b}{x}\right )}{x^{2}} - \frac {12 \, {\left (a x + b\right )} a^{4} b^{2} d^{2} \log \left (c\right )}{x} + 2 \, a^{3} b^{4} e^{2} \log \left (c\right ) + \frac {6 \, {\left (a x + b\right )} a^{3} b^{3} d e \log \left (c\right )}{x} - \frac {12 \, {\left (a x + b\right )}^{2} a^{3} b^{2} d^{2} p \log \left (\frac {a x + b}{x}\right )}{x^{2}} + \frac {6 \, {\left (a x + b\right )} a^{2} b^{4} p e^{2} \log \left (\frac {a x + b}{x}\right )}{x} + \frac {18 \, {\left (a x + b\right )}^{2} a^{2} b^{3} d p e \log \left (\frac {a x + b}{x}\right )}{x^{2}} + \frac {5 \, {\left (a x + b\right )} a^{2} b^{4} p e^{2}}{x} + \frac {6 \, {\left (a x + b\right )}^{2} a^{2} b^{3} d p e}{x^{2}} - \frac {6 \, {\left (a x + b\right )}^{3} a^{2} b^{2} d^{2} p \log \left (-a + \frac {a x + b}{x}\right )}{x^{3}} + \frac {6 \, {\left (a x + b\right )}^{2} a b^{4} p e^{2} \log \left (-a + \frac {a x + b}{x}\right )}{x^{2}} + \frac {6 \, {\left (a x + b\right )}^{3} a b^{3} d p e \log \left (-a + \frac {a x + b}{x}\right )}{x^{3}} + \frac {6 \, {\left (a x + b\right )}^{2} a^{3} b^{2} d^{2} \log \left (c\right )}{x^{2}} + \frac {6 \, {\left (a x + b\right )}^{3} a^{2} b^{2} d^{2} p \log \left (\frac {a x + b}{x}\right )}{x^{3}} - \frac {6 \, {\left (a x + b\right )}^{2} a b^{4} p e^{2} \log \left (\frac {a x + b}{x}\right )}{x^{2}} - \frac {6 \, {\left (a x + b\right )}^{3} a b^{3} d p e \log \left (\frac {a x + b}{x}\right )}{x^{3}} - \frac {2 \, {\left (a x + b\right )}^{2} a b^{4} p e^{2}}{x^{2}} - \frac {2 \, {\left (a x + b\right )}^{3} b^{4} p e^{2} \log \left (-a + \frac {a x + b}{x}\right )}{x^{3}} + \frac {2 \, {\left (a x + b\right )}^{3} b^{4} p e^{2} \log \left (\frac {a x + b}{x}\right )}{x^{3}}}{6 \, {\left (a^{6} - \frac {3 \, {\left (a x + b\right )} a^{5}}{x} + \frac {3 \, {\left (a x + b\right )}^{2} a^{4}}{x^{2}} - \frac {{\left (a x + b\right )}^{3} a^{3}}{x^{3}}\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*log(c*(a+b/x)^p),x, algorithm="giac")

[Out]

-1/6*(6*a^5*b^2*d^2*p*log(-a + (a*x + b)/x) - 6*a^4*b^3*d*p*e*log(-a + (a*x + b)/x) + 6*a^4*b^3*d*p*e - 18*(a*
x + b)*a^4*b^2*d^2*p*log(-a + (a*x + b)/x)/x + 2*a^3*b^4*p*e^2*log(-a + (a*x + b)/x) + 18*(a*x + b)*a^3*b^3*d*
p*e*log(-a + (a*x + b)/x)/x + 6*a^5*b^2*d^2*log(c) - 6*a^4*b^3*d*e*log(c) + 6*(a*x + b)*a^4*b^2*d^2*p*log((a*x
+ b)/x)/x - 12*(a*x + b)*a^3*b^3*d*p*e*log((a*x + b)/x)/x - 3*a^3*b^4*p*e^2 - 12*(a*x + b)*a^3*b^3*d*p*e/x +
18*(a*x + b)^2*a^3*b^2*d^2*p*log(-a + (a*x + b)/x)/x^2 - 6*(a*x + b)*a^2*b^4*p*e^2*log(-a + (a*x + b)/x)/x - 1
8*(a*x + b)^2*a^2*b^3*d*p*e*log(-a + (a*x + b)/x)/x^2 - 12*(a*x + b)*a^4*b^2*d^2*log(c)/x + 2*a^3*b^4*e^2*log(
c) + 6*(a*x + b)*a^3*b^3*d*e*log(c)/x - 12*(a*x + b)^2*a^3*b^2*d^2*p*log((a*x + b)/x)/x^2 + 6*(a*x + b)*a^2*b^
4*p*e^2*log((a*x + b)/x)/x + 18*(a*x + b)^2*a^2*b^3*d*p*e*log((a*x + b)/x)/x^2 + 5*(a*x + b)*a^2*b^4*p*e^2/x +
6*(a*x + b)^2*a^2*b^3*d*p*e/x^2 - 6*(a*x + b)^3*a^2*b^2*d^2*p*log(-a + (a*x + b)/x)/x^3 + 6*(a*x + b)^2*a*b^4
*p*e^2*log(-a + (a*x + b)/x)/x^2 + 6*(a*x + b)^3*a*b^3*d*p*e*log(-a + (a*x + b)/x)/x^3 + 6*(a*x + b)^2*a^3*b^2
*d^2*log(c)/x^2 + 6*(a*x + b)^3*a^2*b^2*d^2*p*log((a*x + b)/x)/x^3 - 6*(a*x + b)^2*a*b^4*p*e^2*log((a*x + b)/x
)/x^2 - 6*(a*x + b)^3*a*b^3*d*p*e*log((a*x + b)/x)/x^3 - 2*(a*x + b)^2*a*b^4*p*e^2/x^2 - 2*(a*x + b)^3*b^4*p*e
^2*log(-a + (a*x + b)/x)/x^3 + 2*(a*x + b)^3*b^4*p*e^2*log((a*x + b)/x)/x^3)/((a^6 - 3*(a*x + b)*a^5/x + 3*(a*
x + b)^2*a^4/x^2 - (a*x + b)^3*a^3/x^3)*b)

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Mupad [B]
time = 0.32, size = 111, normalized size = 1.09 \begin {gather*} \ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )\,\left (d^2\,x+d\,e\,x^2+\frac {e^2\,x^3}{3}\right )-x\,\left (\frac {b^2\,e^2\,p}{3\,a^2}-\frac {b\,d\,e\,p}{a}\right )+\frac {\ln \left (b+a\,x\right )\,\left (3\,p\,a^2\,b\,d^2-3\,p\,a\,b^2\,d\,e+p\,b^3\,e^2\right )}{3\,a^3}+\frac {b\,e^2\,p\,x^2}{6\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x)^p)*(d + e*x)^2,x)

[Out]

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

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