3.144 \(\int x^2 (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ),x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0194123, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.178429, size = 292, normalized size = 10.43 \[ -\frac{x^3 \left (b e k m (m+3) n \, _3F_2\left (1,\frac{3}{m},\frac{3}{m};1+\frac{3}{m},1+\frac{3}{m};-\frac{f x^m}{e}\right )-27 a e \log \left (d \left (e+f x^m\right )^k\right )-9 a e m \log \left (d \left (e+f x^m\right )^k\right )+9 a f k m x^m \, _2F_1\left (1,\frac{m+3}{m};2+\frac{3}{m};-\frac{f x^m}{e}\right )-27 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-9 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k m (m+3) \left (n-3 \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac{3}{m};\frac{m+3}{m};-\frac{f x^m}{e}\right )+3 b e k m^2 \log \left (c x^n\right )+9 b e k m \log \left (c x^n\right )+9 b e n \log \left (d \left (e+f x^m\right )^k\right )+3 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n-6 b e k m n\right )}{27 e (m+3)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^3*(-6*b*e*k*m*n - 2*b*e*k*m^2*n + 9*a*f*k*m*x^m*Hypergeometric2F1[1, (3 + m)/m, 2 + 3/m, -((f*x^m)/e)] + b
*e*k*m*(3 + m)*n*HypergeometricPFQ[{1, 3/m, 3/m}, {1 + 3/m, 1 + 3/m}, -((f*x^m)/e)] + b*e*k*m*(3 + m)*Hypergeo
metric2F1[1, 3/m, (3 + m)/m, -((f*x^m)/e)]*(n - 3*Log[c*x^n]) + 9*b*e*k*m*Log[c*x^n] + 3*b*e*k*m^2*Log[c*x^n]
- 27*a*e*Log[d*(e + f*x^m)^k] - 9*a*e*m*Log[d*(e + f*x^m)^k] + 9*b*e*n*Log[d*(e + f*x^m)^k] + 3*b*e*m*n*Log[d*
(e + f*x^m)^k] - 27*b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 9*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/(27*e*(3 +
m))

________________________________________________________________________________________

Maple [A]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{k} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{9} \,{\left (3 \, b x^{3} \log \left (x^{n}\right ) -{\left (b{\left (n - 3 \, \log \left (c\right )\right )} - 3 \, a\right )} x^{3}\right )} \log \left ({\left (f x^{m} + e\right )}^{k}\right ) + \int -\frac{{\left (3 \,{\left (f k m - 3 \, f \log \left (d\right )\right )} a -{\left (f k m n - 3 \,{\left (f k m - 3 \, f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{2} x^{m} - 9 \,{\left (b e \log \left (c\right ) \log \left (d\right ) + a e \log \left (d\right )\right )} x^{2} + 3 \,{\left ({\left (f k m - 3 \, f \log \left (d\right )\right )} b x^{2} x^{m} - 3 \, b e x^{2} \log \left (d\right )\right )} \log \left (x^{n}\right )}{9 \,{\left (f x^{m} + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*b*x^3*log(x^n) - (b*(n - 3*log(c)) - 3*a)*x^3)*log((f*x^m + e)^k) + integrate(-1/9*((3*(f*k*m - 3*f*log
(d))*a - (f*k*m*n - 3*(f*k*m - 3*f*log(d))*log(c))*b)*x^2*x^m - 9*(b*e*log(c)*log(d) + a*e*log(d))*x^2 + 3*((f
*k*m - 3*f*log(d))*b*x^2*x^m - 3*b*e*x^2*log(d))*log(x^n))/(f*x^m + e), x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} \log \left (c x^{n}\right ) + a x^{2}\right )} \log \left ({\left (f x^{m} + e\right )}^{k} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(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^2*log((f*x^m + e)^k*d), x)