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

Optimal. Leaf size=26 \[ \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2544} \[ \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1} \]

Antiderivative was successfully verified.

[In]

Int[((a*m*x^m + b*n*q*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)/(1 + p)

Rule 2544

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c,
d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 26, normalized size = 1.00 \[ \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[((a*m*x^m + b*n*q*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)/(1 + p)

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fricas [A]  time = 0.44, size = 42, normalized size = 1.62 \[ \frac {{\left ({\left (n \log \relax (x) + \log \relax (c)\right )}^{q} b + a x^{m}\right )} {\left ({\left (n \log \relax (x) + \log \relax (c)\right )}^{q} b + a x^{m}\right )}^{p}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a*m*x^m+b*n*q*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,[0,0,2,5,2,0,5,0,3,1,2,3]%%%}+%%%{-2,[0,0,2,4,2,1,5,0,2,1,2,3]%%%}+%%%{5,[
0,0,2,4,2,0,4,1,3,1,2,3]%%%}+%%%{1,[0,0,2,3,2,2,5,0,1,1,2,3]%%%}+%%%{-8,[0,0,2,3,2,1,4,1,2,1,2,3]%%%}+%%%{10,[
0,0,2,3,2,0,3,2,3,1,2,3]%%%}+%%%{3,[0,0,2,2,2,2,4,1,1,1,2,3]%%%}+%%%{-12,[0,0,2,2,2,1,3,2,2,1,2,3]%%%}+%%%{10,
[0,0,2,2,2,0,2,3,3,1,2,3]%%%}+%%%{3,[0,0,2,1,2,2,3,2,1,1,2,3]%%%}+%%%{-8,[0,0,2,1,2,1,2,3,2,1,2,3]%%%}+%%%{5,[
0,0,2,1,2,0,1,4,3,1,2,3]%%%}+%%%{1,[0,0,2,0,2,2,2,3,1,1,2,3]%%%}+%%%{-2,[0,0,2,0,2,1,1,4,2,1,2,3]%%%}+%%%{1,[0
,0,2,0,2,0,0,5,3,1,2,3]%%%} / %%%{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 [F]  time = 0.97, size = 0, normalized size = 0.00 \[ \int \frac {\left (b n q \ln \left (c \,x^{n}\right )^{q -1}+a m \,x^{m}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*m*x^m+b*n*q*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((a*m*x^m+b*n*q*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 [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\left (a\,m\,x^m+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((a*m*x^m + b*n*q*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((a*m*x**m+b*n*q*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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