\(\int (d+e x)^m \log (c (a+b x)^p) \, dx\) [208]

Optimal result

Integrand size = 18, antiderivative size = 89 \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {b p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {b (d+e x)}{b d-a e}\right )}{e (b d-a e) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2442, 70} \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac {b p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)} \]

[In]

Int[(d + e*x)^m*Log[c*(a + b*x)^p],x]

[Out]

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

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 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {b p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e) (2+m)}+\log \left (c (a+b x)^p\right )\right )}{e (1+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)