3.169 \(\int (a+b x)^m \log (c x^n) \, dx\)

Optimal. Leaf size=68 \[ \frac{(a+b x)^{m+1} \log \left (c x^n\right )}{b (m+1)}+\frac{n (a+b x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{b x}{a}+1\right )}{a b \left (m^2+3 m+2\right )} \]

[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))

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Rubi [A]  time = 0.0283179, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2319, 65} \[ \frac{(a+b x)^{m+1} \log \left (c x^n\right )}{b (m+1)}+\frac{n (a+b x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{b x}{a}+1\right )}{a b \left (m^2+3 m+2\right )} \]

Antiderivative was successfully verified.

[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 2319

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]))

Rule 65

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

Rubi steps

\begin{align*} \int (a+b x)^m \log \left (c x^n\right ) \, dx &=\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]  time = 0.0187284, size = 61, normalized size = 0.9 \[ \frac{(a+b x)^{m+1} \left (n (a+b x) \, _2F_1\left (1,m+2;m+3;\frac{b x}{a}+1\right )+a (m+2) \log \left (c x^n\right )\right )}{a b (m+1) (m+2)} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.535, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m}\ln \left ( c{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{m} \log \left (c x^{n}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^m*log(c*x^n), x)

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Sympy [A]  time = 36.7339, size = 233, normalized size = 3.43 \begin{align*} - n \left (\begin{cases} a^{m} x & \text{for}\: \left (b = 0 \wedge m \neq -1\right ) \vee b = 0 \\- \frac{b^{2} b^{m} m \left (\frac{a}{b} + x\right )^{2} \left (\frac{a}{b} + x\right )^{m} \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^{2} b^{m} \left (\frac{a}{b} + x\right )^{2} \left (\frac{a}{b} + x\right )^{m} \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} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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**2*b**m*m*(a/b + x)**2*(a/b + x)**m*lerchphi(1 +
b*x/a, 1, m + 2)*gamma(m + 2)/(a*b*m*gamma(m + 3) + a*b*gamma(m + 3)) - 2*b**2*b**m*(a/b + x)**2*(a/b + x)**m
*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)), (Piecewise((log(a)*log(x) - polylog(2, b*x*exp_polar(I*pi)/a), Abs(x) < 1), (-log(a)*log(1/x) - poly
log(2, b*x*exp_polar(I*pi)/a), 1/Abs(x) < 1), (-meijerg(((), (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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m} \log \left (c x^{n}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^m*log(c*x^n), x)