\(\int (a+b x)^m \log (c x^n) \, dx\) [169]

Optimal result

Integrand size = 14, antiderivative size = 68 \[ \int (a+b x)^m \log \left (c x^n\right ) \, dx=\frac {n (a+b x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {b x}{a}\right )}{a b \left (2+3 m+m^2\right )}+\frac {(a+b x)^{1+m} \log \left (c x^n\right )}{b (1+m)} \]

[Out]

n*(b*x+a)^(2+m)*hypergeom([1, 2+m],[3+m],1+b*x/a)/a/b/(m^2+3*m+2)+(b*x+a)^(1+m)*ln(c*x^n)/b/(1+m)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2356, 67} \[ \int (a+b x)^m \log \left (c x^n\right ) \, dx=\frac {(a+b x)^{m+1} \log \left (c x^n\right )}{b (m+1)}+\frac {n (a+b x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {b x}{a}+1\right )}{a b \left (m^2+3 m+2\right )} \]

[In]

Int[(a + b*x)^m*Log[c*x^n],x]

[Out]

(n*(a + b*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (b*x)/a])/(a*b*(2 + 3*m + m^2)) + ((a + b*x)^(1 +
m)*Log[c*x^n])/(b*(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 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+m} \log \left (c x^n\right )}{b (1+m)}-\frac {n \int \frac {(a+b x)^{1+m}}{x} \, dx}{b (1+m)} \\ & = \frac {n (a+b x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {b x}{a}\right )}{a b \left (2+3 m+m^2\right )}+\frac {(a+b x)^{1+m} \log \left (c x^n\right )}{b (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int (a+b x)^m \log \left (c x^n\right ) \, dx=\frac {(a+b x)^{1+m} \left (n (a+b x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {b x}{a}\right )+a (2+m) \log \left (c x^n\right )\right )}{a b (1+m) (2+m)} \]

[In]

Integrate[(a + b*x)^m*Log[c*x^n],x]

[Out]

((a + b*x)^(1 + m)*(n*(a + b*x)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (b*x)/a] + a*(2 + m)*Log[c*x^n]))/(a*b*
(1 + m)*(2 + m))

Maple [F]

\[\int \left (b x +a \right )^{m} \ln \left (c \,x^{n}\right )d x\]

[In]

int((b*x+a)^m*ln(c*x^n),x)

[Out]

int((b*x+a)^m*ln(c*x^n),x)

Fricas [F]

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

[In]

integrate((b*x+a)^m*log(c*x^n),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*log(c*x^n), x)

Sympy [A] (verification not implemented)

Time = 6.24 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.50 \[ \int (a+b x)^m \log \left (c x^n\right ) \, dx=- n \left (\begin {cases} a^{m} x & \text {for}\: \left (b = 0 \wedge m \neq -1\right ) \vee b = 0 \\- \frac {b^{m + 2} m \left (\frac {a}{b} + x\right )^{m + 2} \Phi \left (1 + \frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a b m \Gamma \left (m + 3\right ) + a b \Gamma \left (m + 3\right )} - \frac {2 b^{m + 2} \left (\frac {a}{b} + x\right )^{m + 2} \Phi \left (1 + \frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a b m \Gamma \left (m + 3\right ) + a b \Gamma \left (m + 3\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (a \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (a \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (a \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (a \right )} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} a^{m} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]

[In]

integrate((b*x+a)**m*ln(c*x**n),x)

[Out]

-n*Piecewise((a**m*x, Eq(b, 0) | (Eq(b, 0) & Ne(m, -1))), (-b**(m + 2)*m*(a/b + x)**(m + 2)*lerchphi(1 + b*x/a
, 1, m + 2)*gamma(m + 2)/(a*b*m*gamma(m + 3) + a*b*gamma(m + 3)) - 2*b**(m + 2)*(a/b + x)**(m + 2)*lerchphi(1
+ b*x/a, 1, m + 2)*gamma(m + 2)/(a*b*m*gamma(m + 3) + a*b*gamma(m + 3)), (m > -oo) & (m < oo) & Ne(m, -1)), (P
iecewise((-polylog(2, b*x*exp_polar(I*pi)/a), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(a)*log(x) - polylog(2, b*x*
exp_polar(I*pi)/a), Abs(x) < 1), (-log(a)*log(1/x) - polylog(2, b*x*exp_polar(I*pi)/a), 1/Abs(x) < 1), (-meije
rg(((), (1, 1)), ((0, 0), ()), x)*log(a) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(a) - polylog(2, b*x*exp_
polar(I*pi)/a), True))/b, True)) + Piecewise((a**m*x, Eq(b, 0)), (Piecewise(((a + b*x)**(m + 1)/(m + 1), Ne(m,
-1)), (log(a + b*x), True))/b, True))*log(c*x**n)

Maxima [F]

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

[In]

integrate((b*x+a)^m*log(c*x^n),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((b*x+a)^m*log(c*x^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*log(c*x^n), x)

Mupad [F(-1)]

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

[In]

int(log(c*x^n)*(a + b*x)^m,x)

[Out]

int(log(c*x^n)*(a + b*x)^m, x)