\(\int (d+e x)^m \log (c (a+\frac {b}{x})^p) \, dx\) [209]

Optimal result

Integrand size = 20, antiderivative size = 135 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {a p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {a (d+e x)}{a d-b e}\right )}{e (a d-b e) (1+m) (2+m)}-\frac {p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)} \]

[Out]

a*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],a*(e*x+d)/(a*d-b*e))/e/(a*d-b*e)/(1+m)/(2+m)-p*(e*x+d)^(2+m)*hyperg
eom([1, 2+m],[3+m],1+e*x/d)/d/e/(m^2+3*m+2)+(e*x+d)^(1+m)*ln(c*(a+b/x)^p)/e/(1+m)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2513, 528, 88, 67, 70} \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (m+1)}+\frac {a p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {a (d+e x)}{a d-b e}\right )}{e (m+1) (m+2) (a d-b e)}-\frac {p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {e x}{d}+1\right )}{d e \left (m^2+3 m+2\right )} \]

[In]

Int[(d + e*x)^m*Log[c*(a + b/x)^p],x]

[Out]

(a*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (a*(d + e*x))/(a*d - b*e)])/(e*(a*d - b*e)*(1 + m)*(
2 + m)) - (p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (e*x)/d])/(d*e*(2 + 3*m + m^2)) + ((d +
e*x)^(1 + m)*Log[c*(a + b/x)^p])/(e*(1 + m))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 88

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, 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*} \text {integral}& = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}+\frac {(b p) \int \frac {(d+e x)^{1+m}}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}+\frac {(b p) \int \frac {(d+e x)^{1+m}}{x (b+a x)} \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}+\frac {p \int \frac {(d+e x)^{1+m}}{x} \, dx}{e (1+m)}-\frac {(a p) \int \frac {(d+e x)^{1+m}}{b+a x} \, dx}{e (1+m)} \\ & = \frac {a p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {a (d+e x)}{a d-b e}\right )}{e (a d-b e) (1+m) (2+m)}-\frac {p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.91 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (-a d p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {a (d+e x)}{a d-b e}\right )+(a d-b e) \left (p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )-d (2+m) \log \left (c \left (a+\frac {b}{x}\right )^p\right )\right )\right )}{d e (-a d+b e) (1+m) (2+m)} \]

[In]

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

[Out]

((d + e*x)^(1 + m)*(-(a*d*p*(d + e*x)*Hypergeometric2F1[1, 2 + m, 3 + m, (a*(d + e*x))/(a*d - b*e)]) + (a*d -
b*e)*(p*(d + e*x)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (e*x)/d] - d*(2 + m)*Log[c*(a + b/x)^p])))/(d*e*(-(a*
d) + b*e)*(1 + m)*(2 + m))

Maple [F]

\[\int \left (e x +d \right )^{m} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

integral((e*x + d)^m*log(c*((a*x + b)/x)^p), x)

Sympy [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\int \left (d + e x\right )^{m} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}\, dx \]

[In]

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

[Out]

Integral((d + e*x)**m*log(c*(a + b/x)**p), x)

Maxima [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

(e*x + d)*(e*x + d)^m*log((a*x + b)^p)/(e*(m + 1)) - integrate(-(b*e*(m + 1)*log(c) - a*d*p + (e*(m + 1)*log(c
) - e*p)*a*x - (a*e*(m + 1)*x + b*e*(m + 1))*log(x^p))*(e*x + d)^m/(a*e*(m + 1)*x + b*e*(m + 1)), x)

Giac [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

integrate((e*x + d)^m*log((a + b/x)^p*c), x)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]

[In]

int(log(c*(a + b/x)^p)*(d + e*x)^m,x)

[Out]

int(log(c*(a + b/x)^p)*(d + e*x)^m, x)