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

Optimal. Leaf size=89 \[ \frac{(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac{b p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)} \]

[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))

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Rubi [A]  time = 0.045602, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2395, 68} \[ \frac{(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac{b p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)} \]

Antiderivative was successfully verified.

[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 2395

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]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx &=\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]  time = 0.0600622, size = 77, normalized size = 0.87 \[ \frac{(d+e x)^{m+1} \left (\log \left (c (a+b x)^p\right )+\frac{b p (d+e x) \, _2F_1\left (1,m+2;m+3;\frac{b (d+e x)}{b d-a e}\right )}{(m+2) (b d-a e)}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 1.257, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m}\ln \left ( c \left ( bx+a \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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((e*x+d)^m*log(c*(b*x+a)^p),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 (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)