3.368 \(\int x^3 (d+e x^r) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{4} \left (d x^4+\frac{4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d n x^4-\frac{b e n x^{r+4}}{(r+4)^2} \]

[Out]

-(b*d*n*x^4)/16 - (b*e*n*x^(4 + r))/(4 + r)^2 + ((d*x^4 + (4*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4

________________________________________________________________________________________

Rubi [A]  time = 0.0796116, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {14, 2334, 12} \[ \frac{1}{4} \left (d x^4+\frac{4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d n x^4-\frac{b e n x^{r+4}}{(r+4)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x^4)/16 - (b*e*n*x^(4 + r))/(4 + r)^2 + ((d*x^4 + (4*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{4} \left (d x^4+\frac{4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{4} x^3 \left (d+\frac{4 e x^r}{4+r}\right ) \, dx\\ &=\frac{1}{4} \left (d x^4+\frac{4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int x^3 \left (d+\frac{4 e x^r}{4+r}\right ) \, dx\\ &=\frac{1}{4} \left (d x^4+\frac{4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \left (d x^3+\frac{4 e x^{3+r}}{4+r}\right ) \, dx\\ &=-\frac{1}{16} b d n x^4-\frac{b e n x^{4+r}}{(4+r)^2}+\frac{1}{4} \left (d x^4+\frac{4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0899306, size = 73, normalized size = 1.24 \[ \frac{x^4 \left (4 a (r+4) \left (d (r+4)+4 e x^r\right )+4 b (r+4) \log \left (c x^n\right ) \left (d (r+4)+4 e x^r\right )-b n \left (d (r+4)^2+16 e x^r\right )\right )}{16 (r+4)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

(x^4*(4*a*(4 + r)*(d*(4 + r) + 4*e*x^r) - b*n*(d*(4 + r)^2 + 16*e*x^r) + 4*b*(4 + r)*(d*(4 + r) + 4*e*x^r)*Log
[c*x^n]))/(16*(4 + r)^2)

________________________________________________________________________________________

Maple [C]  time = 0.237, size = 613, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d+e*x^r)*(a+b*ln(c*x^n)),x)

[Out]

1/4*b*x^4*(d*r+4*e*x^r+4*d)/(4+r)*ln(x^n)-1/16*x^4*(-64*a*d-16*x^r*a*e*r+16*x^r*b*e*n+8*I*Pi*b*e*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)*x^r*r+8*b*d*n*r+32*I*Pi*b*d*csgn(I*c*x^n)^3+16*b*d*n-64*x^r*a*e-32*ln(c)*b*d*r-4*ln(c)*
b*d*r^2-16*ln(c)*b*e*x^r*r-64*ln(c)*b*e*x^r-8*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r-4*a*d*r^2-8*I*Pi*b*e*cs
gn(I*x^n)*csgn(I*c*x^n)^2*x^r*r+16*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*r-64*ln(c)*b*d+32*I*Pi*b*e*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-32*a*d*r+2*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+b*d*n*r^2+32*
I*Pi*b*e*csgn(I*c*x^n)^3*x^r-32*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)-32*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2+2*I
*Pi*b*d*r^2*csgn(I*c*x^n)^3+16*I*Pi*b*d*csgn(I*c*x^n)^3*r-16*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2*r-16*I*Pi*b*
d*csgn(I*c*x^n)^2*csgn(I*c)*r-2*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csgn(I
*c)-32*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-32*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+8*I*Pi*b*e*csgn(I*c*
x^n)^3*x^r*r+32*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(4+r)^2

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.29177, size = 389, normalized size = 6.59 \begin{align*} \frac{4 \,{\left (b d r^{2} + 8 \, b d r + 16 \, b d\right )} x^{4} \log \left (c\right ) + 4 \,{\left (b d n r^{2} + 8 \, b d n r + 16 \, b d n\right )} x^{4} \log \left (x\right ) -{\left (16 \, b d n +{\left (b d n - 4 \, a d\right )} r^{2} - 64 \, a d + 8 \,{\left (b d n - 4 \, a d\right )} r\right )} x^{4} + 16 \,{\left ({\left (b e r + 4 \, b e\right )} x^{4} \log \left (c\right ) +{\left (b e n r + 4 \, b e n\right )} x^{4} \log \left (x\right ) -{\left (b e n - a e r - 4 \, a e\right )} x^{4}\right )} x^{r}}{16 \,{\left (r^{2} + 8 \, r + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/16*(4*(b*d*r^2 + 8*b*d*r + 16*b*d)*x^4*log(c) + 4*(b*d*n*r^2 + 8*b*d*n*r + 16*b*d*n)*x^4*log(x) - (16*b*d*n
+ (b*d*n - 4*a*d)*r^2 - 64*a*d + 8*(b*d*n - 4*a*d)*r)*x^4 + 16*((b*e*r + 4*b*e)*x^4*log(c) + (b*e*n*r + 4*b*e*
n)*x^4*log(x) - (b*e*n - a*e*r - 4*a*e)*x^4)*x^r)/(r^2 + 8*r + 16)

________________________________________________________________________________________

Sympy [A]  time = 43.3819, size = 525, normalized size = 8.9 \begin{align*} \begin{cases} \frac{4 a d r^{2} x^{4}}{16 r^{2} + 128 r + 256} + \frac{32 a d r x^{4}}{16 r^{2} + 128 r + 256} + \frac{64 a d x^{4}}{16 r^{2} + 128 r + 256} + \frac{16 a e r x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac{64 a e x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac{4 b d n r^{2} x^{4} \log{\left (x \right )}}{16 r^{2} + 128 r + 256} - \frac{b d n r^{2} x^{4}}{16 r^{2} + 128 r + 256} + \frac{32 b d n r x^{4} \log{\left (x \right )}}{16 r^{2} + 128 r + 256} - \frac{8 b d n r x^{4}}{16 r^{2} + 128 r + 256} + \frac{64 b d n x^{4} \log{\left (x \right )}}{16 r^{2} + 128 r + 256} - \frac{16 b d n x^{4}}{16 r^{2} + 128 r + 256} + \frac{4 b d r^{2} x^{4} \log{\left (c \right )}}{16 r^{2} + 128 r + 256} + \frac{32 b d r x^{4} \log{\left (c \right )}}{16 r^{2} + 128 r + 256} + \frac{64 b d x^{4} \log{\left (c \right )}}{16 r^{2} + 128 r + 256} + \frac{16 b e n r x^{4} x^{r} \log{\left (x \right )}}{16 r^{2} + 128 r + 256} + \frac{64 b e n x^{4} x^{r} \log{\left (x \right )}}{16 r^{2} + 128 r + 256} - \frac{16 b e n x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac{16 b e r x^{4} x^{r} \log{\left (c \right )}}{16 r^{2} + 128 r + 256} + \frac{64 b e x^{4} x^{r} \log{\left (c \right )}}{16 r^{2} + 128 r + 256} & \text{for}\: r \neq -4 \\\frac{a d x^{4}}{4} + a e \log{\left (x \right )} + \frac{b d n x^{4} \log{\left (x \right )}}{4} - \frac{b d n x^{4}}{16} + \frac{b d x^{4} \log{\left (c \right )}}{4} + \frac{b e n \log{\left (x \right )}^{2}}{2} + b e \log{\left (c \right )} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(d+e*x**r)*(a+b*ln(c*x**n)),x)

[Out]

Piecewise((4*a*d*r**2*x**4/(16*r**2 + 128*r + 256) + 32*a*d*r*x**4/(16*r**2 + 128*r + 256) + 64*a*d*x**4/(16*r
**2 + 128*r + 256) + 16*a*e*r*x**4*x**r/(16*r**2 + 128*r + 256) + 64*a*e*x**4*x**r/(16*r**2 + 128*r + 256) + 4
*b*d*n*r**2*x**4*log(x)/(16*r**2 + 128*r + 256) - b*d*n*r**2*x**4/(16*r**2 + 128*r + 256) + 32*b*d*n*r*x**4*lo
g(x)/(16*r**2 + 128*r + 256) - 8*b*d*n*r*x**4/(16*r**2 + 128*r + 256) + 64*b*d*n*x**4*log(x)/(16*r**2 + 128*r
+ 256) - 16*b*d*n*x**4/(16*r**2 + 128*r + 256) + 4*b*d*r**2*x**4*log(c)/(16*r**2 + 128*r + 256) + 32*b*d*r*x**
4*log(c)/(16*r**2 + 128*r + 256) + 64*b*d*x**4*log(c)/(16*r**2 + 128*r + 256) + 16*b*e*n*r*x**4*x**r*log(x)/(1
6*r**2 + 128*r + 256) + 64*b*e*n*x**4*x**r*log(x)/(16*r**2 + 128*r + 256) - 16*b*e*n*x**4*x**r/(16*r**2 + 128*
r + 256) + 16*b*e*r*x**4*x**r*log(c)/(16*r**2 + 128*r + 256) + 64*b*e*x**4*x**r*log(c)/(16*r**2 + 128*r + 256)
, Ne(r, -4)), (a*d*x**4/4 + a*e*log(x) + b*d*n*x**4*log(x)/4 - b*d*n*x**4/16 + b*d*x**4*log(c)/4 + b*e*n*log(x
)**2/2 + b*e*log(c)*log(x), True))

________________________________________________________________________________________

Giac [B]  time = 1.27836, size = 185, normalized size = 3.14 \begin{align*} \frac{b n r x^{4} x^{r} e \log \left (x\right )}{r^{2} + 8 \, r + 16} + \frac{1}{4} \, b d n x^{4} \log \left (x\right ) + \frac{4 \, b n x^{4} x^{r} e \log \left (x\right )}{r^{2} + 8 \, r + 16} - \frac{1}{16} \, b d n x^{4} - \frac{b n x^{4} x^{r} e}{r^{2} + 8 \, r + 16} + \frac{1}{4} \, b d x^{4} \log \left (c\right ) + \frac{b x^{4} x^{r} e \log \left (c\right )}{r + 4} + \frac{1}{4} \, a d x^{4} + \frac{a x^{4} x^{r} e}{r + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*n*r*x^4*x^r*e*log(x)/(r^2 + 8*r + 16) + 1/4*b*d*n*x^4*log(x) + 4*b*n*x^4*x^r*e*log(x)/(r^2 + 8*r + 16) - 1/1
6*b*d*n*x^4 - b*n*x^4*x^r*e/(r^2 + 8*r + 16) + 1/4*b*d*x^4*log(c) + b*x^4*x^r*e*log(c)/(r + 4) + 1/4*a*d*x^4 +
a*x^4*x^r*e/(r + 4)