3.2.49 \(\int \frac {(a+b \log (c x^n)) \log (d (e+f x^m)^k)}{x^3} \, dx\) [149]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^3},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/x^3,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^3} \, 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^3,x]

[Out]

Defer[Int][((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3, 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^3} \, 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^3} \, dx\\ \end {align*}

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(292\) vs. \(2(29)=58\).
time = 0.12, size = 292, normalized size = 10.07 \begin {gather*} \frac {4 b e k m n-2 b e k m^2 n+4 a f k m x^m \, _2F_1\left (1,\frac {-2+m}{m};2-\frac {2}{m};-\frac {f x^m}{e}\right )+b e k (-2+m) m n \, _3F_2\left (1,-\frac {2}{m},-\frac {2}{m};1-\frac {2}{m},1-\frac {2}{m};-\frac {f x^m}{e}\right )+4 b e k m \log \left (c x^n\right )-2 b e k m^2 \log \left (c x^n\right )+b e k (-2+m) m \, _2F_1\left (1,-\frac {2}{m};\frac {-2+m}{m};-\frac {f x^m}{e}\right ) \left (n+2 \log \left (c x^n\right )\right )+8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )-2 b e m n \log \left (d \left (e+f x^m\right )^k\right )+8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )}{8 e (-2+m) x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3,x]

[Out]

(4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, (-2 + m)/m, 2 - 2/m, -((f*x^m)/e)] + b*e*k*(
-2 + m)*m*n*HypergeometricPFQ[{1, -2/m, -2/m}, {1 - 2/m, 1 - 2/m}, -((f*x^m)/e)] + 4*b*e*k*m*Log[c*x^n] - 2*b*
e*k*m^2*Log[c*x^n] + b*e*k*(-2 + m)*m*Hypergeometric2F1[1, -2/m, (-2 + m)/m, -((f*x^m)/e)]*(n + 2*Log[c*x^n])
+ 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m)^k] + 4*b*e*n*Log[d*(e + f*x^m)^k] - 2*b*e*m*n*Log[d*(
e + f*x^m)^k] + 8*b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(8*e*(-2 + m)
*x^2)

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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )}{x^{3}}\, dx\]

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^3,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/x^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((a+b*log(c*x^n))*log(d*(e+f*x^m)^k)/x^3,x, algorithm="maxima")

[Out]

-1/4*(b*(n + 2*log(c)) + 2*b*log(x^n) + 2*a)*log((f*x^m + e)^k)/x^2 + integrate(1/4*((2*(f*k*m + 2*f*log(d))*a
+ (f*k*m*n + 2*(f*k*m + 2*f*log(d))*log(c))*b)*x^m + 4*(b*log(c)*log(d) + a*log(d))*e + 2*((f*k*m + 2*f*log(d
))*b*x^m + 2*b*e*log(d))*log(x^n))/(f*x^3*x^m + x^3*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^3,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^3, 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((a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k)/x**3,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((a+b*log(c*x^n))*log(d*(e+f*x^m)^k)/x^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^3, 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^3} \,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^3,x)

[Out]

int((log(d*(e + f*x^m)^k)*(a + b*log(c*x^n)))/x^3, x)

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