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

Optimal. Leaf size=81 \[ \left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (x^{m-1} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p,x\right )+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{b n (p+1) q} \]

[Out]

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

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Rubi [A]  time = 0.23, 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 = {} \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.90, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[((d*x^m + e*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 = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )} {\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{p}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x^m + e*log(c*x^n)^(q - 1))*(a*x^m + b*log(c*x^n)^q)^p/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((d*x^m+e*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 divide,
perhaps due to rounding error%%%{1,[0,0,2,5,2,0,5,0,2,1,2,2,1]%%%}+%%%{-2,[0,0,2,4,2,1,5,0,1,1,2,2,1]%%%}+%%%
{5,[0,0,2,4,2,0,4,1,2,1,2,2,1]%%%}+%%%{1,[0,0,2,3,2,2,5,0,0,1,2,2,1]%%%}+%%%{-8,[0,0,2,3,2,1,4,1,1,1,2,2,1]%%%
}+%%%{10,[0,0,2,3,2,0,3,2,2,1,2,2,1]%%%}+%%%{3,[0,0,2,2,2,2,4,1,0,1,2,2,1]%%%}+%%%{-12,[0,0,2,2,2,1,3,2,1,1,2,
2,1]%%%}+%%%{10,[0,0,2,2,2,0,2,3,2,1,2,2,1]%%%}+%%%{3,[0,0,2,1,2,2,3,2,0,1,2,2,1]%%%}+%%%{-8,[0,0,2,1,2,1,2,3,
1,1,2,2,1]%%%}+%%%{5,[0,0,2,1,2,0,1,4,2,1,2,2,1]%%%}+%%%{1,[0,0,2,0,2,2,2,3,0,1,2,2,1]%%%}+%%%{-2,[0,0,2,0,2,1
,1,4,1,1,2,2,1]%%%}+%%%{1,[0,0,2,0,2,0,0,5,2,1,2,2,1]%%%} / %%%{1,[0,0,2,5,3,0,5,0,2,1,2,2,0]%%%}+%%%{-2,[0,0,
2,4,3,1,5,0,1,1,2,2,0]%%%}+%%%{5,[0,0,2,4,3,0,4,1,2,1,2,2,0]%%%}+%%%{1,[0,0,2,3,3,2,5,0,0,1,2,2,0]%%%}+%%%{-8,
[0,0,2,3,3,1,4,1,1,1,2,2,0]%%%}+%%%{10,[0,0,2,3,3,0,3,2,2,1,2,2,0]%%%}+%%%{3,[0,0,2,2,3,2,4,1,0,1,2,2,0]%%%}+%
%%{-12,[0,0,2,2,3,1,3,2,1,1,2,2,0]%%%}+%%%{10,[0,0,2,2,3,0,2,3,2,1,2,2,0]%%%}+%%%{3,[0,0,2,1,3,2,3,2,0,1,2,2,0
]%%%}+%%%{-8,[0,0,2,1,3,1,2,3,1,1,2,2,0]%%%}+%%%{5,[0,0,2,1,3,0,1,4,2,1,2,2,0]%%%}+%%%{1,[0,0,2,0,3,2,2,3,0,1,
2,2,0]%%%}+%%%{-2,[0,0,2,0,3,1,1,4,1,1,2,2,0]%%%}+%%%{1,[0,0,2,0,3,0,0,5,2,1,2,2,0]%%%} Error: Bad Argument Va
lue

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maple [A]  time = 0.90, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{q -1}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*x^m+e*ln(c*x^n)^(q-1))*(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((d*x^m+e*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 \[ \int \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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