3.373 \(\int x^4 (d+e x^r) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{5} \left (d x^5+\frac{5 e x^{r+5}}{r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d n x^5-\frac{b e n x^{r+5}}{(r+5)^2} \]

[Out]

-(b*d*n*x^5)/25 - (b*e*n*x^(5 + r))/(5 + r)^2 + ((d*x^5 + (5*e*x^(5 + r))/(5 + r))*(a + b*Log[c*x^n]))/5

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Rubi [A]  time = 0.0802939, 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} \[ \frac{1}{5} \left (d x^5+\frac{5 e x^{r+5}}{r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d n x^5-\frac{b e n x^{r+5}}{(r+5)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x^5)/25 - (b*e*n*x^(5 + r))/(5 + r)^2 + ((d*x^5 + (5*e*x^(5 + r))/(5 + r))*(a + b*Log[c*x^n]))/5

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^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{5} \left (d x^5+\frac{5 e x^{5+r}}{5+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{5} x^4 \left (d+\frac{5 e x^r}{5+r}\right ) \, dx\\ &=\frac{1}{5} \left (d x^5+\frac{5 e x^{5+r}}{5+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{5} (b n) \int x^4 \left (d+\frac{5 e x^r}{5+r}\right ) \, dx\\ &=\frac{1}{5} \left (d x^5+\frac{5 e x^{5+r}}{5+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{5} (b n) \int \left (d x^4+\frac{5 e x^{4+r}}{5+r}\right ) \, dx\\ &=-\frac{1}{25} b d n x^5-\frac{b e n x^{5+r}}{(5+r)^2}+\frac{1}{5} \left (d x^5+\frac{5 e x^{5+r}}{5+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0890465, size = 73, normalized size = 1.24 \[ \frac{x^5 \left (5 a (r+5) \left (d (r+5)+5 e x^r\right )+5 b (r+5) \log \left (c x^n\right ) \left (d (r+5)+5 e x^r\right )-b n \left (d (r+5)^2+25 e x^r\right )\right )}{25 (r+5)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

(x^5*(5*a*(5 + r)*(d*(5 + r) + 5*e*x^r) - b*n*(d*(5 + r)^2 + 25*e*x^r) + 5*b*(5 + r)*(d*(5 + r) + 5*e*x^r)*Log
[c*x^n]))/(25*(5 + r)^2)

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Maple [C]  time = 0.233, size = 614, 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^4*(d+e*x^r)*(a+b*ln(c*x^n)),x)

[Out]

1/5*x^5*b*(d*r+5*e*x^r+5*d)/(5+r)*ln(x^n)-1/50*x^5*(-250*a*d+125*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*
x^r-50*x^r*a*e*r+50*x^r*b*e*n-25*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r-25*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n
)^2*x^r*r+5*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+50*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
+20*b*d*n*r+50*b*d*n-250*x^r*a*e-100*ln(c)*b*d*r-10*ln(c)*b*d*r^2-50*ln(c)*b*e*x^r*r+125*I*Pi*b*d*csgn(I*c*x^n
)^3-250*ln(c)*b*e*x^r-10*a*d*r^2-250*ln(c)*b*d-100*a*d*r+2*b*d*n*r^2+50*I*Pi*b*d*csgn(I*c*x^n)^3*r-125*I*Pi*b*
d*csgn(I*x^n)*csgn(I*c*x^n)^2+125*I*Pi*b*e*csgn(I*c*x^n)^3*x^r-125*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)+5*I*Pi*b
*d*r^2*csgn(I*c*x^n)^3+25*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r*r-50*I*Pi*b*d*r*csgn(I*c*x^n)^2*csg
n(I*c)+125*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-50*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)^2-125*I*Pi*b*e
*csgn(I*c*x^n)^2*csgn(I*c)*x^r-125*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-5*I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csg
n(I*c)+25*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r-5*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2)/(5+r)^2

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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^4*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.36311, size = 396, normalized size = 6.71 \begin{align*} \frac{5 \,{\left (b d r^{2} + 10 \, b d r + 25 \, b d\right )} x^{5} \log \left (c\right ) + 5 \,{\left (b d n r^{2} + 10 \, b d n r + 25 \, b d n\right )} x^{5} \log \left (x\right ) -{\left (25 \, b d n +{\left (b d n - 5 \, a d\right )} r^{2} - 125 \, a d + 10 \,{\left (b d n - 5 \, a d\right )} r\right )} x^{5} + 25 \,{\left ({\left (b e r + 5 \, b e\right )} x^{5} \log \left (c\right ) +{\left (b e n r + 5 \, b e n\right )} x^{5} \log \left (x\right ) -{\left (b e n - a e r - 5 \, a e\right )} x^{5}\right )} x^{r}}{25 \,{\left (r^{2} + 10 \, r + 25\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/25*(5*(b*d*r^2 + 10*b*d*r + 25*b*d)*x^5*log(c) + 5*(b*d*n*r^2 + 10*b*d*n*r + 25*b*d*n)*x^5*log(x) - (25*b*d*
n + (b*d*n - 5*a*d)*r^2 - 125*a*d + 10*(b*d*n - 5*a*d)*r)*x^5 + 25*((b*e*r + 5*b*e)*x^5*log(c) + (b*e*n*r + 5*
b*e*n)*x^5*log(x) - (b*e*n - a*e*r - 5*a*e)*x^5)*x^r)/(r^2 + 10*r + 25)

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Sympy [A]  time = 97.2254, size = 525, normalized size = 8.9 \begin{align*} \begin{cases} \frac{5 a d r^{2} x^{5}}{25 r^{2} + 250 r + 625} + \frac{50 a d r x^{5}}{25 r^{2} + 250 r + 625} + \frac{125 a d x^{5}}{25 r^{2} + 250 r + 625} + \frac{25 a e r x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac{125 a e x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac{5 b d n r^{2} x^{5} \log{\left (x \right )}}{25 r^{2} + 250 r + 625} - \frac{b d n r^{2} x^{5}}{25 r^{2} + 250 r + 625} + \frac{50 b d n r x^{5} \log{\left (x \right )}}{25 r^{2} + 250 r + 625} - \frac{10 b d n r x^{5}}{25 r^{2} + 250 r + 625} + \frac{125 b d n x^{5} \log{\left (x \right )}}{25 r^{2} + 250 r + 625} - \frac{25 b d n x^{5}}{25 r^{2} + 250 r + 625} + \frac{5 b d r^{2} x^{5} \log{\left (c \right )}}{25 r^{2} + 250 r + 625} + \frac{50 b d r x^{5} \log{\left (c \right )}}{25 r^{2} + 250 r + 625} + \frac{125 b d x^{5} \log{\left (c \right )}}{25 r^{2} + 250 r + 625} + \frac{25 b e n r x^{5} x^{r} \log{\left (x \right )}}{25 r^{2} + 250 r + 625} + \frac{125 b e n x^{5} x^{r} \log{\left (x \right )}}{25 r^{2} + 250 r + 625} - \frac{25 b e n x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac{25 b e r x^{5} x^{r} \log{\left (c \right )}}{25 r^{2} + 250 r + 625} + \frac{125 b e x^{5} x^{r} \log{\left (c \right )}}{25 r^{2} + 250 r + 625} & \text{for}\: r \neq -5 \\\frac{a d x^{5}}{5} + a e \log{\left (x \right )} + \frac{b d n x^{5} \log{\left (x \right )}}{5} - \frac{b d n x^{5}}{25} + \frac{b d x^{5} \log{\left (c \right )}}{5} + \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**4*(d+e*x**r)*(a+b*ln(c*x**n)),x)

[Out]

Piecewise((5*a*d*r**2*x**5/(25*r**2 + 250*r + 625) + 50*a*d*r*x**5/(25*r**2 + 250*r + 625) + 125*a*d*x**5/(25*
r**2 + 250*r + 625) + 25*a*e*r*x**5*x**r/(25*r**2 + 250*r + 625) + 125*a*e*x**5*x**r/(25*r**2 + 250*r + 625) +
5*b*d*n*r**2*x**5*log(x)/(25*r**2 + 250*r + 625) - b*d*n*r**2*x**5/(25*r**2 + 250*r + 625) + 50*b*d*n*r*x**5*
log(x)/(25*r**2 + 250*r + 625) - 10*b*d*n*r*x**5/(25*r**2 + 250*r + 625) + 125*b*d*n*x**5*log(x)/(25*r**2 + 25
0*r + 625) - 25*b*d*n*x**5/(25*r**2 + 250*r + 625) + 5*b*d*r**2*x**5*log(c)/(25*r**2 + 250*r + 625) + 50*b*d*r
*x**5*log(c)/(25*r**2 + 250*r + 625) + 125*b*d*x**5*log(c)/(25*r**2 + 250*r + 625) + 25*b*e*n*r*x**5*x**r*log(
x)/(25*r**2 + 250*r + 625) + 125*b*e*n*x**5*x**r*log(x)/(25*r**2 + 250*r + 625) - 25*b*e*n*x**5*x**r/(25*r**2
+ 250*r + 625) + 25*b*e*r*x**5*x**r*log(c)/(25*r**2 + 250*r + 625) + 125*b*e*x**5*x**r*log(c)/(25*r**2 + 250*r
+ 625), Ne(r, -5)), (a*d*x**5/5 + a*e*log(x) + b*d*n*x**5*log(x)/5 - b*d*n*x**5/25 + b*d*x**5*log(c)/5 + b*e*
n*log(x)**2/2 + b*e*log(c)*log(x), True))

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Giac [B]  time = 1.31001, size = 185, normalized size = 3.14 \begin{align*} \frac{b n r x^{5} x^{r} e \log \left (x\right )}{r^{2} + 10 \, r + 25} + \frac{1}{5} \, b d n x^{5} \log \left (x\right ) + \frac{5 \, b n x^{5} x^{r} e \log \left (x\right )}{r^{2} + 10 \, r + 25} - \frac{1}{25} \, b d n x^{5} - \frac{b n x^{5} x^{r} e}{r^{2} + 10 \, r + 25} + \frac{1}{5} \, b d x^{5} \log \left (c\right ) + \frac{b x^{5} x^{r} e \log \left (c\right )}{r + 5} + \frac{1}{5} \, a d x^{5} + \frac{a x^{5} x^{r} e}{r + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*n*r*x^5*x^r*e*log(x)/(r^2 + 10*r + 25) + 1/5*b*d*n*x^5*log(x) + 5*b*n*x^5*x^r*e*log(x)/(r^2 + 10*r + 25) - 1
/25*b*d*n*x^5 - b*n*x^5*x^r*e/(r^2 + 10*r + 25) + 1/5*b*d*x^5*log(c) + b*x^5*x^r*e*log(c)/(r + 5) + 1/5*a*d*x^
5 + a*x^5*x^r*e/(r + 5)