3.1.36 \(\int \frac {d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^2} \, dx\) [36]

Optimal. Leaf size=75 \[ -\frac {e}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}+\left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (\frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2},x\right ) \]

[Out]

(d-a*e*m/b/n/q)*CannotIntegrate(x^(-1+m)/(a*x^m+b*ln(c*x^n)^q)^2,x)-e/b/n/q/(a*x^m+b*ln(c*x^n)^q)

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Rubi [A]
time = 0.19, 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 = {} \begin {gather*} \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(d*x^m + e*Log[c*x^n]^(-1 + q))/(x*(a*x^m + b*Log[c*x^n]^q)^2),x]

[Out]

-(e/(b*n*q*(a*x^m + b*Log[c*x^n]^q))) + (d - (a*e*m)/(b*n*q))*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^2
, x]

Rubi steps

\begin {align*} \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx &=-\frac {e}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}-\left (-d+\frac {a e m}{b n q}\right ) \int \frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 22.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(d*x^m + e*Log[c*x^n]^(-1 + q))/(x*(a*x^m + b*Log[c*x^n]^q)^2),x]

[Out]

Integrate[(d*x^m + e*Log[c*x^n]^(-1 + q))/(x*(a*x^m + b*Log[c*x^n]^q)^2), x]

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Maple [A]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}}{x \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^m+e*ln(c*x^n)^(-1+q))/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

[Out]

int((d*x^m+e*ln(c*x^n)^(-1+q))/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="maxima")

[Out]

-(b*d*log(c) + b*d*log(x^n) - a*e)/(a^2*b*m*x^m*log(x^n) - (n*q - m*log(c))*a^2*b*x^m + (a*b^2*m*log(x^n) - (n
*q - m*log(c))*a*b^2)*(log(c) + log(x^n))^q) + integrate(-((m*n*(q - 1) - m^2*log(c))*a*e + (d*m*n*q*log(c) -
(q^2 - q)*d*n^2)*b + (b*d*m*n*q - a*m^2*e)*log(x^n))/(a^2*b*m^2*x*x^m*log(x^n)^2 - 2*(m*n*q - m^2*log(c))*a^2*
b*x*x^m*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*a^2*b*x*x^m + (a*b^2*m^2*x*log(x^n)^2 - 2*(m*n*q
- m^2*log(c))*a*b^2*x*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*a*b^2*x)*(log(c) + log(x^n))^q), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="fricas")

[Out]

integral((d*x^m + log(c*x^n)^(q - 1)*e)/(2*a*b*x*x^m*log(c*x^n)^q + a^2*x*x^(2*m) + b^2*x*log(c*x^n)^(2*q)), x
)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**m+e*ln(c*x**n)**(-1+q))/x/(a*x**m+b*ln(c*x**n)**q)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="giac")

[Out]

integrate((d*x^m + log(c*x^n)^(q - 1)*e)/((a*x^m + b*log(c*x^n)^q)^2*x), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}}{x\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^m + e*log(c*x^n)^(q - 1))/(x*(a*x^m + b*log(c*x^n)^q)^2),x)

[Out]

int((d*x^m + e*log(c*x^n)^(q - 1))/(x*(a*x^m + b*log(c*x^n)^q)^2), x)

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