\(\int (\text {d1} x^{\text {q1}})^{\text {m1}} (\text {d2} x^{\text {q2}})^{\text {m2}} (a+b \log (c x^n))^p \, dx\) [193]

Optimal result

Integrand size = 27, antiderivative size = 136 \[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {a (1+\text {m1} \text {q1}+\text {m2} \text {q2})}{b n}} x \left (c x^n\right )^{-\frac {1+\text {m1} \text {q1}+\text {m2} \text {q2}}{n}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \Gamma \left (1+p,-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+\text {m1} \text {q1}+\text {m2} \text {q2}} \]

[Out]

x*(d1*x^q1)^m1*(d2*x^q2)^m2*GAMMA(p+1,-(m1*q1+m2*q2+1)*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p/exp(a*(m1*q1+m2*
q2+1)/b/n)/(m1*q1+m2*q2+1)/((c*x^n)^((m1*q1+m2*q2+1)/n))/((-(m1*q1+m2*q2+1)*(a+b*ln(c*x^n))/b/n)^p)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 2347, 2212} \[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {x \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p e^{-\frac {a (\text {m1} \text {q1}+\text {m2} \text {q2}+1)}{b n}} \left (c x^n\right )^{-\frac {\text {m1} \text {q1}+\text {m2} \text {q2}+1}{n}} \left (-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text {m1} \text {q1}+\text {m2} \text {q2}+1} \]

[In]

Int[(d1*x^q1)^m1*(d2*x^q2)^m2*(a + b*Log[c*x^n])^p,x]

[Out]

(x*(d1*x^q1)^m1*(d2*x^q2)^m2*Gamma[1 + p, -(((1 + m1*q1 + m2*q2)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n]
)^p)/(E^((a*(1 + m1*q1 + m2*q2))/(b*n))*(1 + m1*q1 + m2*q2)*(c*x^n)^((1 + m1*q1 + m2*q2)/n)*(-(((1 + m1*q1 + m
2*q2)*(a + b*Log[c*x^n]))/(b*n)))^p)

Rule 15

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

Rule 2212

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

Rule 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps \begin{align*} \text {integral}& = \left (x^{-\text {m1} \text {q1}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}}\right ) \int x^{\text {m1} \text {q1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx \\ & = \left (x^{-\text {m1} \text {q1}-\text {m2} \text {q2}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}}\right ) \int x^{\text {m1} \text {q1}+\text {m2} \text {q2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {1+\text {m1} \text {q1}+\text {m2} \text {q2}}{n}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}}\right ) \text {Subst}\left (\int e^{\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {e^{-\frac {a (1+\text {m1} \text {q1}+\text {m2} \text {q2})}{b n}} x \left (c x^n\right )^{-\frac {1+\text {m1} \text {q1}+\text {m2} \text {q2}}{n}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \Gamma \left (1+p,-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \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+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+\text {m1} \text {q1}+\text {m2} \text {q2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04 \[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} x^{-\text {m1} \text {q1}-\text {m2} \text {q2}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \Gamma \left (1+p,-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+\text {m1} \text {q1}+\text {m2} \text {q2}} \]

[In]

Integrate[(d1*x^q1)^m1*(d2*x^q2)^m2*(a + b*Log[c*x^n])^p,x]

[Out]

(x^(-(m1*q1) - m2*q2)*(d1*x^q1)^m1*(d2*x^q2)^m2*Gamma[1 + p, -(((1 + m1*q1 + m2*q2)*(a + b*Log[c*x^n]))/(b*n))
]*(a + b*Log[c*x^n])^p)/(E^(((1 + m1*q1 + m2*q2)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m1*q1 + m2*q2
)*(-(((1 + m1*q1 + m2*q2)*(a + b*Log[c*x^n]))/(b*n)))^p)

Maple [F]

\[\int \left (\operatorname {d1} \,x^{\operatorname {q1}}\right )^{\operatorname {m1}} \left (\operatorname {d2} \,x^{\operatorname {q2}}\right )^{\operatorname {m2}} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]

[In]

int((d1*x^q1)^m1*(d2*x^q2)^m2*(a+b*ln(c*x^n))^p,x)

[Out]

int((d1*x^q1)^m1*(d2*x^q2)^m2*(a+b*ln(c*x^n))^p,x)

Fricas [F]

\[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((d1*x^q1)^m1*(d2*x^q2)^m2*(a+b*log(c*x^n))^p,x, algorithm="fricas")

[Out]

integral((d1*x^q1)^m1*(d2*x^q2)^m2*(b*log(c*x^n) + a)^p, x)

Sympy [F(-1)]

Timed out. \[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\text {Timed out} \]

[In]

integrate((d1*x**q1)**m1*(d2*x**q2)**m2*(a+b*ln(c*x**n))**p,x)

[Out]

Timed out

Maxima [F]

\[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((d1*x^q1)^m1*(d2*x^q2)^m2*(a+b*log(c*x^n))^p,x, algorithm="maxima")

[Out]

integrate((d1*x^q1)^m1*(d2*x^q2)^m2*(b*log(c*x^n) + a)^p, x)

Giac [F]

\[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((d1*x^q1)^m1*(d2*x^q2)^m2*(a+b*log(c*x^n))^p,x, algorithm="giac")

[Out]

integrate((d1*x^q1)^m1*(d2*x^q2)^m2*(b*log(c*x^n) + a)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int {\left (d_{1}\,x^{q_{1}}\right )}^{m_{1}}\,{\left (d_{2}\,x^{q_{2}}\right )}^{m_{2}}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]

[In]

int((d1*x^q1)^m1*(d2*x^q2)^m2*(a + b*log(c*x^n))^p,x)

[Out]

int((d1*x^q1)^m1*(d2*x^q2)^m2*(a + b*log(c*x^n))^p, x)