3.2.62 \(\int (f+g x)^m (a+b \log (c (d+e x)^n)) \, dx\) [162]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {2442, 70} \begin {gather*} \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)} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.05, size = 81, normalized size = 0.86 \begin {gather*} \frac {(f+g x)^{1+m} \left (a+\frac {b e n (f+g x) \, _2F_1\left (1,2+m;3+m;\frac {e (f+g x)}{e f-d g}\right )}{(e f-d g) (2+m)}+b \log \left (c (d+e x)^n\right )\right )}{g (1+m)} \end {gather*}

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 = 0.16, size = 0, normalized size = 0.00 \[\int \left (g x +f \right )^{m} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c) + (g*x + f)^m*a, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c) + a)*(g*x + f)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (f+g\,x\right )}^m\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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