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]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, 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]
[Out]
Rule 65
Rule 2319
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 = 0.02, size = 61, normalized size = 0.90 \[ \frac {(a+b x)^{m+1} \left (n (a+b x) \, _2F_1\left (1,m+2;m+3;\frac {b x}{a}+1\right )+a (m+2) \log \left (c x^n\right )\right )}{a b (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} \log \left (c x^{n}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{m} \log \left (c x^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{m} \ln \left (c \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{m} \log \left (x^{n}\right )}{b {\left (m + 1\right )}} + \frac {-a n \int \frac {{\left (b x + a\right )}^{m}}{x}\,{d x} + \frac {{\left (b x + a\right )}^{m + 1} m \log \relax (c)}{m + 1} - \frac {{\left (b x + a\right )}^{m + 1} n}{m + 1} + \frac {{\left (b x + a\right )}^{m + 1} \log \relax (c)}{m + 1}}{b {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (c\,x^n\right )\,{\left (a+b\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 27.79, size = 233, normalized size = 3.43 \[ - n \left (\begin {cases} a^{m} x & \text {for}\: \left (b = 0 \wedge m \neq -1\right ) \vee b = 0 \\- \frac {b^{2} b^{m} m \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{m} \Phi \left (1 + \frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a b m \Gamma \left (m + 3\right ) + a b \Gamma \left (m + 3\right )} - \frac {2 b^{2} b^{m} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{m} \Phi \left (1 + \frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a b m \Gamma \left (m + 3\right ) + a b \Gamma \left (m + 3\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\\frac {\begin {cases} \log {\relax (a )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (a )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (a )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (a )} - \operatorname {Li}_{2}\left (\frac {b x e^{i \pi }}{a}\right ) & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} a^{m} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________