3.1 \(\int \frac {\log ^{-1+q}(c x^n) (a x^m+b \log ^q(c x^n))^p}{x} \, dx\)

Optimal. Leaf size=76 \[ \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{b n (p+1) q}-\frac {a m \text {Int}\left (x^{m-1} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p,x\right )}{b n q} \]

[Out]

-a*m*CannotIntegrate(x^(-1+m)*(a*x^m+b*ln(c*x^n)^q)^p,x)/b/n/q+(a*x^m+b*ln(c*x^n)^q)^(1+p)/b/n/(1+p)/q

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Rubi [A]  time = 0.25, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Log[c*x^n]^(-1 + q)*(a*x^m + b*Log[c*x^n]^q)^p)/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^(1 + p)/(b*n*(1 + p)*q) - (a*m*Defer[Int][x^(-1 + m)*(a*x^m + b*Log[c*x^n]^q)^p, x])/
(b*n*q)

Rubi steps

\begin {align*} \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx &=\frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{b n (1+p) q}-\frac {(a m) \int x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p \, dx}{b n q}\\ \end {align*}

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Mathematica [A]  time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Log[c*x^n]^(-1 + q)*(a*x^m + b*Log[c*x^n]^q)^p)/x,x]

[Out]

Integrate[(Log[c*x^n]^(-1 + q)*(a*x^m + b*Log[c*x^n]^q)^p)/x, x]

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{p} \log \left (c x^{n}\right )^{q - 1}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)*(a*x^m+b*log(c*x^n)^q)^p/x,x, algorithm="fricas")

[Out]

integral((a*x^m + b*log(c*x^n)^q)^p*log(c*x^n)^(q - 1)/x, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)*(a*x^m+b*log(c*x^n)^q)^p/x,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to check s
ign: (4*pi/(pi*sign(n*t_nostep+ln(abs(c)))*sign(pi*sign(c)-pi)-pi*sign(pi*sign(c)-pi)-2*atan((pi*sign(c)-pi)/(
2*n*t_nostep+2*ln(abs(c)))))/2)>(-4*pi/(pi*sign(n*t_nostep+ln(abs(c)))*sign(pi*sign(c)-pi)-pi*sign(pi*sign(c)-
pi)-2*atan((pi*sign(c)-pi)/(2*n*t_nostep+2*ln(abs(c)))))/2)Unable to check sign: (4*pi/(pi*sign(n*t_nostep+ln(
abs(c)))*sign(pi*sign(c)-pi)-pi*sign(pi*sign(c)-pi)-2*atan((pi*sign(c)-pi)/(2*n*t_nostep+2*ln(abs(c)))))/2)>(-
4*pi/(pi*sign(n*t_nostep+ln(abs(c)))*sign(pi*sign(c)-pi)-pi*sign(pi*sign(c)-pi)-2*atan((pi*sign(c)-pi)/(2*n*t_
nostep+2*ln(abs(c)))))/2)Evaluation time: 2.46Unable to divide, perhaps due to rounding error%%%{1,[0,0,2,5,2,
0,5,0,2,1,2,2]%%%}+%%%{-2,[0,0,2,4,2,1,5,0,1,1,2,2]%%%}+%%%{5,[0,0,2,4,2,0,4,1,2,1,2,2]%%%}+%%%{1,[0,0,2,3,2,2
,5,0,0,1,2,2]%%%}+%%%{-8,[0,0,2,3,2,1,4,1,1,1,2,2]%%%}+%%%{10,[0,0,2,3,2,0,3,2,2,1,2,2]%%%}+%%%{3,[0,0,2,2,2,2
,4,1,0,1,2,2]%%%}+%%%{-12,[0,0,2,2,2,1,3,2,1,1,2,2]%%%}+%%%{10,[0,0,2,2,2,0,2,3,2,1,2,2]%%%}+%%%{3,[0,0,2,1,2,
2,3,2,0,1,2,2]%%%}+%%%{-8,[0,0,2,1,2,1,2,3,1,1,2,2]%%%}+%%%{5,[0,0,2,1,2,0,1,4,2,1,2,2]%%%}+%%%{1,[0,0,2,0,2,2
,2,3,0,1,2,2]%%%}+%%%{-2,[0,0,2,0,2,1,1,4,1,1,2,2]%%%}+%%%{1,[0,0,2,0,2,0,0,5,2,1,2,2]%%%} / %%%{1,[0,0,2,5,3,
0,5,0,2,1,2,2]%%%}+%%%{-2,[0,0,2,4,3,1,5,0,1,1,2,2]%%%}+%%%{5,[0,0,2,4,3,0,4,1,2,1,2,2]%%%}+%%%{1,[0,0,2,3,3,2
,5,0,0,1,2,2]%%%}+%%%{-8,[0,0,2,3,3,1,4,1,1,1,2,2]%%%}+%%%{10,[0,0,2,3,3,0,3,2,2,1,2,2]%%%}+%%%{3,[0,0,2,2,3,2
,4,1,0,1,2,2]%%%}+%%%{-12,[0,0,2,2,3,1,3,2,1,1,2,2]%%%}+%%%{10,[0,0,2,2,3,0,2,3,2,1,2,2]%%%}+%%%{3,[0,0,2,1,3,
2,3,2,0,1,2,2]%%%}+%%%{-8,[0,0,2,1,3,1,2,3,1,1,2,2]%%%}+%%%{5,[0,0,2,1,3,0,1,4,2,1,2,2]%%%}+%%%{1,[0,0,2,0,3,2
,2,3,0,1,2,2]%%%}+%%%{-2,[0,0,2,0,3,1,1,4,1,1,2,2]%%%}+%%%{1,[0,0,2,0,3,0,0,5,2,1,2,2]%%%} Error: Bad Argument
Value

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maple [A]  time = 0.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p} \ln \left (c \,x^{n}\right )^{q -1}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x^n)^(-1+q)*(a*x^m+b*ln(c*x^n)^q)^p/x,x)

[Out]

int(ln(c*x^n)^(-1+q)*(a*x^m+b*ln(c*x^n)^q)^p/x,x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)*(a*x^m+b*log(c*x^n)^q)^p/x,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,x^n\right )}^{q-1}\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(c*x^n)^(q - 1)*(a*x^m + b*log(c*x^n)^q)^p)/x,x)

[Out]

int((log(c*x^n)^(q - 1)*(a*x^m + b*log(c*x^n)^q)^p)/x, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x**n)**(-1+q)*(a*x**m+b*ln(c*x**n)**q)**p/x,x)

[Out]

Timed out

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