3.2.45 \(\int x (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\) [145]

Optimal. Leaf size=27 \[ \text {Int}\left (x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ),x\right ) \]

[Out]

Unintegrable(x*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),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 x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(292\) vs. \(2(27)=54\).
time = 0.13, size = 292, normalized size = 10.81 \begin {gather*} -\frac {x^2 \left (-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 m (2+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 )+b e k m (2+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 )+4 b e k m \log \left (c x^n\right )+2 b e k m^2 \log \left (c x^n\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 )\right )}{8 e (2+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(x^2*(-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*m*(2 + m)*n*HypergeometricPFQ[{1, 2/m, 2/m}, {1 + 2/m, 1 + 2/m}, -((f*x^m)/e)] + b*e*k*m*(2 + m)*Hype
rgeometric2F1[1, 2/m, (2 + m)/m, -((f*x^m)/e)]*(n - 2*Log[c*x^n]) + 4*b*e*k*m*Log[c*x^n] + 2*b*e*k*m^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]))/(e*(2 +
m))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int x\,\ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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