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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 12

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

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])

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 = 0.09, 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)

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 159, normalized size = 2.69 \[ \frac {5 \, {\left (b d r^{2} + 10 \, b d r + 25 \, b d\right )} x^{5} \log \relax (c) + 5 \, {\left (b d n r^{2} + 10 \, b d n r + 25 \, b d n\right )} x^{5} \log \relax (x) - {\left (25 \, b d n + {\left (b d n - 5 \, a d\right )} r^{2} - 125 \, a d + 10 \, {\left (b d n - 5 \, a d\right )} r\right )} x^{5} + 25 \, {\left ({\left (b e r + 5 \, b e\right )} x^{5} \log \relax (c) + {\left (b e n r + 5 \, b e n\right )} x^{5} \log \relax (x) - {\left (b e n - a e r - 5 \, a e\right )} x^{5}\right )} x^{r}}{25 \, {\left (r^{2} + 10 \, r + 25\right )}} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.41, size = 137, normalized size = 2.32 \[ \frac {b n r x^{5} x^{r} e \log \relax (x)}{r^{2} + 10 \, r + 25} + \frac {1}{5} \, b d n x^{5} \log \relax (x) + \frac {5 \, b n x^{5} x^{r} e \log \relax (x)}{r^{2} + 10 \, r + 25} - \frac {1}{25} \, b d n x^{5} - \frac {b n x^{5} x^{r} e}{r^{2} + 10 \, r + 25} + \frac {1}{5} \, b d x^{5} \log \relax (c) + \frac {b x^{5} x^{r} e \log \relax (c)}{r + 5} + \frac {1}{5} \, a d x^{5} + \frac {a x^{5} x^{r} e}{r + 5} \]

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)

________________________________________________________________________________________

maple [C]  time = 0.27, size = 614, normalized size = 10.41 \[ \frac {\left (d r +5 e \,x^{r}+5 d \right ) b \,x^{5} \ln \left (x^{n}\right )}{5 r +25}-\frac {\left (50 b d n -250 a e \,x^{r}-50 a e r \,x^{r}+50 b e n \,x^{r}-10 b d \,r^{2} \ln \relax (c )-100 b d r \ln \relax (c )-250 b e \,x^{r} \ln \relax (c )-100 a d r -250 a d +2 b d n \,r^{2}-5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-5 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+25 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-10 a d \,r^{2}+20 b d n r -250 b d \ln \relax (c )+125 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-25 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-25 i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+50 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+125 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-50 b e r \,x^{r} \ln \relax (c )+25 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-125 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+125 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+50 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+125 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-50 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-50 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-125 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-125 i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-125 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}\right ) x^{5}}{50 \left (r +5\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d+e*x^r)*(b*ln(c*x^n)+a),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.12, size = 76, normalized size = 1.29 \[ -\frac {1}{25} \, b d n x^{5} + \frac {1}{5} \, b d x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d x^{5} + \frac {b e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {a e x^{r + 5}}{r + 5} \]

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]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^4\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 69.73, size = 525, normalized size = 8.90 \[ \begin {cases} \frac {5 a d r^{2} x^{5}}{25 r^{2} + 250 r + 625} + \frac {50 a d r x^{5}}{25 r^{2} + 250 r + 625} + \frac {125 a d x^{5}}{25 r^{2} + 250 r + 625} + \frac {25 a e r x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac {125 a e x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac {5 b d n r^{2} x^{5} \log {\relax (x )}}{25 r^{2} + 250 r + 625} - \frac {b d n r^{2} x^{5}}{25 r^{2} + 250 r + 625} + \frac {50 b d n r x^{5} \log {\relax (x )}}{25 r^{2} + 250 r + 625} - \frac {10 b d n r x^{5}}{25 r^{2} + 250 r + 625} + \frac {125 b d n x^{5} \log {\relax (x )}}{25 r^{2} + 250 r + 625} - \frac {25 b d n x^{5}}{25 r^{2} + 250 r + 625} + \frac {5 b d r^{2} x^{5} \log {\relax (c )}}{25 r^{2} + 250 r + 625} + \frac {50 b d r x^{5} \log {\relax (c )}}{25 r^{2} + 250 r + 625} + \frac {125 b d x^{5} \log {\relax (c )}}{25 r^{2} + 250 r + 625} + \frac {25 b e n r x^{5} x^{r} \log {\relax (x )}}{25 r^{2} + 250 r + 625} + \frac {125 b e n x^{5} x^{r} \log {\relax (x )}}{25 r^{2} + 250 r + 625} - \frac {25 b e n x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac {25 b e r x^{5} x^{r} \log {\relax (c )}}{25 r^{2} + 250 r + 625} + \frac {125 b e x^{5} x^{r} \log {\relax (c )}}{25 r^{2} + 250 r + 625} & \text {for}\: r \neq -5 \\\frac {a d x^{5}}{5} + a e \log {\relax (x )} + \frac {b d n x^{5} \log {\relax (x )}}{5} - \frac {b d n x^{5}}{25} + \frac {b d x^{5} \log {\relax (c )}}{5} + \frac {b e n \log {\relax (x )}^{2}}{2} + b e \log {\relax (c )} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________