3.450 \(\int (f x)^m (d+e x^r) (a+b \log (c x^n))^p \, dx\)

Optimal. Leaf size=220 \[ \frac{d (f x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{f (m+1)}+\frac{e x^{r+1} (f x)^m e^{-\frac{a (m+r+1)}{b n}} \left (c x^n\right )^{-\frac{m+r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+r+1} \]

[Out]

(d*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*
n))*f*(1 + m)*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + (e*x^(1 + r)*(f*x)^m*Gamma[1 +
p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + r))/(b*n))*(1 + m + r)*(c*
x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p)

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Rubi [A]  time = 0.256212, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2353, 2310, 2181, 20} \[ \frac{d (f x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{f (m+1)}+\frac{e x^{r+1} (f x)^m e^{-\frac{a (m+r+1)}{b n}} \left (c x^n\right )^{-\frac{m+r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+r+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*
n))*f*(1 + m)*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + (e*x^(1 + r)*(f*x)^m*Gamma[1 +
p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + r))/(b*n))*(1 + m + r)*(c*
x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p)

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\int \left (d (f x)^m \left (a+b \log \left (c x^n\right )\right )^p+e x^r (f x)^m \left (a+b \log \left (c x^n\right )\right )^p\right ) \, dx\\ &=d \int (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx+e \int x^r (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\left (e x^{-m} (f x)^m\right ) \int x^{m+r} \left (a+b \log \left (c x^n\right )\right )^p \, dx+\frac{\left (d (f x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+m) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{f n}\\ &=\frac{d e^{-\frac{a (1+m)}{b n}} (f x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{f (1+m)}+\frac{\left (e x^{1+r} (f x)^m \left (c x^n\right )^{-\frac{1+m+r}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+m+r) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{d e^{-\frac{a (1+m)}{b n}} (f x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{f (1+m)}+\frac{e e^{-\frac{a (1+m+r)}{b n}} x^{1+r} (f x)^m \left (c x^n\right )^{-\frac{1+m+r}{n}} \Gamma \left (1+p,-\frac{(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+r}\\ \end{align*}

Mathematica [A]  time = 0.428966, size = 200, normalized size = 0.91 \[ x^{-m} (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{d \exp \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+1}+\frac{e \exp \left (-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+r+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((f*x)^m*(a + b*Log[c*x^n])^p*((d*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m)*(a - b*n*L
og[x] + b*Log[c*x^n]))/(b*n))*(1 + m)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + (e*Gamma[1 + p, -(((1 + m +
r)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m + r)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m + r)*(-(((1 +
m + r)*(a + b*Log[c*x^n]))/(b*n)))^p)))/x^m

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Maple [F]  time = 1.322, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( d+e{x}^{r} \right ) \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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