3.162 \(\int (f+g x)^m (a+b \log (c (d+e x)^n)) \, dx\)

Optimal. Leaf size=94 \[ \frac{(f+g x)^{m+1} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (m+1)}+\frac{b e n (f+g x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{e (f+g x)}{e f-d g}\right )}{g (m+1) (m+2) (e f-d g)} \]

[Out]

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

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Rubi [A]  time = 0.0499173, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2395, 68} \[ \frac{(f+g x)^{m+1} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (m+1)}+\frac{b e n (f+g x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{e (f+g x)}{e f-d g}\right )}{g (m+1) (m+2) (e f-d g)} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^m*(a + b*Log[c*(d + e*x)^n]),x]

[Out]

(b*e*n*(f + g*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)*(1 + m)
*(2 + m)) + ((f + g*x)^(1 + m)*(a + b*Log[c*(d + e*x)^n]))/(g*(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 (f+g x)^m \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=\frac{(f+g x)^{1+m} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (1+m)}-\frac{(b e n) \int \frac{(f+g x)^{1+m}}{d+e x} \, dx}{g (1+m)}\\ &=\frac{b e n (f+g x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{e (f+g x)}{e f-d g}\right )}{g (e f-d g) (1+m) (2+m)}+\frac{(f+g x)^{1+m} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.084991, size = 81, normalized size = 0.86 \[ \frac{(f+g x)^{m+1} \left (a+b \log \left (c (d+e x)^n\right )+\frac{b e n (f+g x) \, _2F_1\left (1,m+2;m+3;\frac{e (f+g x)}{e f-d g}\right )}{(m+2) (e f-d g)}\right )}{g (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.049, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{m} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^m*(a+b*ln(c*(e*x+d)^n)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((g*x + f)^m*b*log((e*x + d)^n*c) + (g*x + f)^m*a, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**m*(a+b*ln(c*(e*x+d)**n)),x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))*(f + g*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log((e*x + d)^n*c) + a)*(g*x + f)^m, x)