3.57 \(\int (f x)^m \log (c (d+e x)^p) \, dx\)

Optimal. Leaf size=69 \[ \frac{(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac{e p (f x)^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac{e x}{d}\right )}{d f^2 (m+1) (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.0293683, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2395, 64} \[ \frac{(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac{e p (f x)^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac{e x}{d}\right )}{d f^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*Log[c*(d + e*x)^p],x]

[Out]

-((e*p*(f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((e*x)/d)])/(d*f^2*(1 + m)*(2 + m))) + ((f*x)^(1 + m)
*Log[c*(d + e*x)^p])/(f*(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 64

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

Rubi steps

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

Mathematica [A]  time = 0.0246867, size = 56, normalized size = 0.81 \[ \frac{x (f x)^m \left (d (m+2) \log \left (c (d+e x)^p\right )-e p x \, _2F_1\left (1,m+2;m+3;-\frac{e x}{d}\right )\right )}{d (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.851, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m}\ln \left ( c \left ( ex+d \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f*x)**m*log(c*(d + e*x)**p), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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