3.2.48 \(\int \frac {(a+b \log (c x^n)) \log (d (e+f x^m)^k)}{x^2} \, dx\) [148]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/x^2,x)

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Rubi [A]
time = 0.01, 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 {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(282\) vs. \(2(29)=58\).
time = 0.12, size = 282, normalized size = 9.72 \begin {gather*} \frac {2 b e k m n-2 b e k m^2 n+a f k m x^m \, _2F_1\left (1,\frac {-1+m}{m};2-\frac {1}{m};-\frac {f x^m}{e}\right )+b e k (-1+m) m n \, _3F_2\left (1,-\frac {1}{m},-\frac {1}{m};1-\frac {1}{m},1-\frac {1}{m};-\frac {f x^m}{e}\right )+b e k m \log \left (c x^n\right )-b e k m^2 \log \left (c x^n\right )+b e k (-1+m) m \, _2F_1\left (1,-\frac {1}{m};\frac {-1+m}{m};-\frac {f x^m}{e}\right ) \left (n+\log \left (c x^n\right )\right )+a e \log \left (d \left (e+f x^m\right )^k\right )-a e m \log \left (d \left (e+f x^m\right )^k\right )+b e n \log \left (d \left (e+f x^m\right )^k\right )-b e m n \log \left (d \left (e+f x^m\right )^k\right )+b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )}{e (-1+m) x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*e*k*m*n - 2*b*e*k*m^2*n + a*f*k*m*x^m*Hypergeometric2F1[1, (-1 + m)/m, 2 - m^(-1), -((f*x^m)/e)] + b*e*k*
(-1 + m)*m*n*HypergeometricPFQ[{1, -m^(-1), -m^(-1)}, {1 - m^(-1), 1 - m^(-1)}, -((f*x^m)/e)] + b*e*k*m*Log[c*
x^n] - b*e*k*m^2*Log[c*x^n] + b*e*k*(-1 + m)*m*Hypergeometric2F1[1, -m^(-1), (-1 + m)/m, -((f*x^m)/e)]*(n + Lo
g[c*x^n]) + a*e*Log[d*(e + f*x^m)^k] - a*e*m*Log[d*(e + f*x^m)^k] + b*e*n*Log[d*(e + f*x^m)^k] - b*e*m*n*Log[d
*(e + f*x^m)^k] + b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(e*(-1 + m)*x)

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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/x^2,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/x^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((a+b*log(c*x^n))*log(d*(e+f*x^m)^k)/x^2,x, algorithm="maxima")

[Out]

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

[Out]

integral((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k)/x**2,x)

[Out]

Integral((a + b*log(c*x**n))*log(d*(e + f*x**m)**k)/x**2, x)

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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((a+b*log(c*x^n))*log(d*(e+f*x^m)^k)/x^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(e + f*x^m)^k)*(a + b*log(c*x^n)))/x^2,x)

[Out]

int((log(d*(e + f*x^m)^k)*(a + b*log(c*x^n)))/x^2, x)

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