3.1.37 \(\int \frac {d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^3} \, dx\) [37]
Optimal. Leaf size=77 \[ -\frac {e}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}+\left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (\frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^3},x\right ) \]
[Out]
(d-a*e*m/b/n/q)*CannotIntegrate(x^(-1+m)/(a*x^m+b*ln(c*x^n)^q)^3,x)-1/2*e/b/n/q/(a*x^m+b*ln(c*x^n)^q)^2
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Rubi [A]
time = 0.18, 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 )^3} \, 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)^3),x]
[Out]
-1/2*e/(b*n*q*(a*x^m + b*Log[c*x^n]^q)^2) + (d - (a*e*m)/(b*n*q))*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^
q)^3, 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 )^3} \, dx &=-\frac {e}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}-\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 )^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 63.42, 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 )^3} \, 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)^3),x]
[Out]
Integrate[(d*x^m + e*Log[c*x^n]^(-1 + q))/(x*(a*x^m + b*Log[c*x^n]^q)^3), x]
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Maple [A]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}}{x \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{3}}\, dx\]
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)^3,x)
[Out]
int((d*x^m+e*ln(c*x^n)^(-1+q))/x/(a*x^m+b*ln(c*x^n)^q)^3,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)^3,x, algorithm="maxima")
[Out]
-1/2*(a*b*d*m^2*x^m*log(x^n)^3 + (a^2*m^2*e - (4*d*m*n*q - 3*d*m^2*log(c))*a*b)*x^m*log(x^n)^2 + ((2*m^2*log(c
) + m*n)*a^2*e - (8*d*m*n*q*log(c) - 3*d*m^2*log(c)^2 - (3*q^2 - q)*d*n^2)*a*b)*x^m*log(x^n) - ((n^2*q^2 - m^2
*log(c)^2 - m*n*log(c))*a^2*e + (4*d*m*n*q*log(c)^2 - d*m^2*log(c)^3 - (3*q^2 - q)*d*n^2*log(c))*a*b)*x^m - ((
m*n*(2*q - 1)*log(c) - 2*m^2*log(c)^2)*a*b*e + (2*d*m*n*q*log(c)^2 - (2*q^2 - q)*d*n^2*log(c))*b^2 + 2*(b^2*d*
m*n*q - a*b*m^2*e)*log(x^n)^2 + ((m*n*(2*q - 1) - 4*m^2*log(c))*a*b*e + (4*d*m*n*q*log(c) - (2*q^2 - q)*d*n^2)
*b^2)*log(x^n))*(log(c) + log(x^n))^q)/(a^4*b*m^3*x^(3*m)*log(x^n)^3 - 3*(m^2*n*q - m^3*log(c))*a^4*b*x^(3*m)*
log(x^n)^2 + 3*(m*n^2*q^2 - 2*m^2*n*q*log(c) + m^3*log(c)^2)*a^4*b*x^(3*m)*log(x^n) - (n^3*q^3 - 3*m*n^2*q^2*l
og(c) + 3*m^2*n*q*log(c)^2 - m^3*log(c)^3)*a^4*b*x^(3*m) + (a^2*b^3*m^3*x^m*log(x^n)^3 - 3*(m^2*n*q - m^3*log(
c))*a^2*b^3*x^m*log(x^n)^2 + 3*(m*n^2*q^2 - 2*m^2*n*q*log(c) + m^3*log(c)^2)*a^2*b^3*x^m*log(x^n) - (n^3*q^3 -
3*m*n^2*q^2*log(c) + 3*m^2*n*q*log(c)^2 - m^3*log(c)^3)*a^2*b^3*x^m)*(log(c) + log(x^n))^(2*q) + 2*(a^3*b^2*m
^3*x^(2*m)*log(x^n)^3 - 3*(m^2*n*q - m^3*log(c))*a^3*b^2*x^(2*m)*log(x^n)^2 + 3*(m*n^2*q^2 - 2*m^2*n*q*log(c)
+ m^3*log(c)^2)*a^3*b^2*x^(2*m)*log(x^n) - (n^3*q^3 - 3*m*n^2*q^2*log(c) + 3*m^2*n*q*log(c)^2 - m^3*log(c)^3)*
a^3*b^2*x^(2*m))*(log(c) + log(x^n))^q) - integrate(-1/2*(2*(b*d*m^3*n*q - a*m^4*e)*log(x^n)^3 + (m^3*n*(2*q -
3)*log(c)^2 - 2*m^4*log(c)^3 + 2*(q^2 - 1)*m^2*n^2*log(c) - (2*q^3 - 3*q^2 + q)*m*n^3)*a*e + ((m^3*n*(2*q - 3
) - 6*m^4*log(c))*a*e + (6*d*m^3*n*q*log(c) - (2*q^2 - 3*q)*d*m^2*n^2)*b)*log(x^n)^2 + (2*d*m^3*n*q*log(c)^3 -
(2*q^2 - 3*q)*d*m^2*n^2*log(c)^2 - 2*(q^3 - q)*d*m*n^3*log(c) + (2*q^4 - 3*q^3 + q^2)*d*n^4)*b + 2*((m^3*n*(2
*q - 3)*log(c) - 3*m^4*log(c)^2 + (q^2 - 1)*m^2*n^2)*a*e + (3*d*m^3*n*q*log(c)^2 - (2*q^2 - 3*q)*d*m^2*n^2*log
(c) - (q^3 - q)*d*m*n^3)*b)*log(x^n))/(a^3*b*m^4*x*x^(2*m)*log(x^n)^4 - 4*(m^3*n*q - m^4*log(c))*a^3*b*x*x^(2*
m)*log(x^n)^3 + 6*(m^2*n^2*q^2 - 2*m^3*n*q*log(c) + m^4*log(c)^2)*a^3*b*x*x^(2*m)*log(x^n)^2 - 4*(m*n^3*q^3 -
3*m^2*n^2*q^2*log(c) + 3*m^3*n*q*log(c)^2 - m^4*log(c)^3)*a^3*b*x*x^(2*m)*log(x^n) + (n^4*q^4 - 4*m*n^3*q^3*lo
g(c) + 6*m^2*n^2*q^2*log(c)^2 - 4*m^3*n*q*log(c)^3 + m^4*log(c)^4)*a^3*b*x*x^(2*m) + (a^2*b^2*m^4*x*x^m*log(x^
n)^4 - 4*(m^3*n*q - m^4*log(c))*a^2*b^2*x*x^m*log(x^n)^3 + 6*(m^2*n^2*q^2 - 2*m^3*n*q*log(c) + m^4*log(c)^2)*a
^2*b^2*x*x^m*log(x^n)^2 - 4*(m*n^3*q^3 - 3*m^2*n^2*q^2*log(c) + 3*m^3*n*q*log(c)^2 - m^4*log(c)^3)*a^2*b^2*x*x
^m*log(x^n) + (n^4*q^4 - 4*m*n^3*q^3*log(c) + 6*m^2*n^2*q^2*log(c)^2 - 4*m^3*n*q*log(c)^3 + m^4*log(c)^4)*a^2*
b^2*x*x^m)*(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)^3,x, algorithm="fricas")
[Out]
integral((d*x^m + log(c*x^n)^(q - 1)*e)/(3*a*b^2*x*x^m*log(c*x^n)^(2*q) + 3*a^2*b*x*x^(2*m)*log(c*x^n)^q + a^3
*x*x^(3*m) + b^3*x*log(c*x^n)^(3*q)), x)
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**3,x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 3006 deep
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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)^3,x, algorithm="giac")
[Out]
integrate((d*x^m + log(c*x^n)^(q - 1)*e)/((a*x^m + b*log(c*x^n)^q)^3*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 )}^3} \,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)^3),x)
[Out]
int((d*x^m + e*log(c*x^n)^(q - 1))/(x*(a*x^m + b*log(c*x^n)^q)^3), x)
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