\(\int x^4 (d+e x^r) (a+b \log (c x^n)) \, dx\) [373]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d n x^5-\frac {b e n x^{5+r}}{(5+r)^2}+\frac {1}{5} \left (d x^5+\frac {5 e x^{5+r}}{5+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[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] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {14, 2371, 12} \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{5} \left (d x^5+\frac {5 e x^{r+5}}{r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d n x^5-\frac {b e n x^{r+5}}{(r+5)^2} \]

[In]

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

[Out]

-1/25*(b*d*n*x^5) - (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 2371

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] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^5 \left (5 a (5+r) \left (d (5+r)+5 e x^r\right )-b n \left (d (5+r)^2+25 e x^r\right )+5 b (5+r) \left (d (5+r)+5 e x^r\right ) \log \left (c x^n\right )\right )}{25 (5+r)^2} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(168\) vs. \(2(55)=110\).

Time = 2.62 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.86

method	result	size
parallelrisch	\(-\frac {-25 x^{5} x^{r} \ln \left (c \,x^{n}\right ) b e r -5 x^{5} \ln \left (c \,x^{n}\right ) b d \,r^{2}+x^{5} b d n \,r^{2}-125 x^{5} x^{r} \ln \left (c \,x^{n}\right ) b e -25 x^{5} x^{r} a e r +25 x^{5} x^{r} b e n -50 x^{5} \ln \left (c \,x^{n}\right ) b d r -5 x^{5} a d \,r^{2}+10 x^{5} b d n r -125 x^{5} x^{r} a e -125 x^{5} b \ln \left (c \,x^{n}\right ) d -50 x^{5} a d r +25 b d n \,x^{5}-125 x^{5} a d}{25 \left (5+r \right )^{2}}\)	\(169\)
risch	\(\frac {b \,x^{5} \left (d r +5 e \,x^{r}+5 d \right ) \ln \left (x^{n}\right )}{25+5 r}-\frac {x^{5} \left (-250 x^{r} a e +50 b d n -250 a d -50 x^{r} a e r +50 x^{r} b e n -125 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-5 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-5 i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-100 a d r +2 b d n \,r^{2}-50 \ln \left (c \right ) b e \,x^{r} r +25 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{r} r -250 d b \ln \left (c \right )+125 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+20 b d n r -250 \ln \left (c \right ) b e \,x^{r}-10 \ln \left (c \right ) b d \,r^{2}-100 \ln \left (c \right ) b d r +125 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-10 a d \,r^{2}-125 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-25 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +125 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{r}-25 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +5 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+50 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) r -125 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+25 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -50 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -125 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-50 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +5 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+125 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+50 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r \right )}{50 \left (5+r \right )^{2}}\)	\(614\)

[In]

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

[Out]

-1/25*(-25*x^5*x^r*ln(c*x^n)*b*e*r-5*x^5*ln(c*x^n)*b*d*r^2+x^5*b*d*n*r^2-125*x^5*x^r*ln(c*x^n)*b*e-25*x^5*x^r*
a*e*r+25*x^5*x^r*b*e*n-50*x^5*ln(c*x^n)*b*d*r-5*x^5*a*d*r^2+10*x^5*b*d*n*r-125*x^5*x^r*a*e-125*x^5*b*ln(c*x^n)
*d-50*x^5*a*d*r+25*b*d*n*x^5-125*x^5*a*d)/(5+r)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (55) = 110\).

Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {5 \, {\left (b d r^{2} + 10 \, b d r + 25 \, b d\right )} x^{5} \log \left (c\right ) + 5 \, {\left (b d n r^{2} + 10 \, b d n r + 25 \, b d n\right )} x^{5} \log \left (x\right ) - {\left (25 \, b d n + {\left (b d n - 5 \, a d\right )} r^{2} - 125 \, a d + 10 \, {\left (b d n - 5 \, a d\right )} r\right )} x^{5} + 25 \, {\left ({\left (b e r + 5 \, b e\right )} x^{5} \log \left (c\right ) + {\left (b e n r + 5 \, b e n\right )} x^{5} \log \left (x\right ) - {\left (b e n - a e r - 5 \, a e\right )} x^{5}\right )} x^{r}}{25 \, {\left (r^{2} + 10 \, r + 25\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (51) = 102\).

Time = 5.03 (sec) , antiderivative size = 398, normalized size of antiderivative = 6.75 \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {5 a d r^{2} x^{5}}{25 r^{2} + 250 r + 625} + \frac {50 a d r x^{5}}{25 r^{2} + 250 r + 625} + \frac {125 a d x^{5}}{25 r^{2} + 250 r + 625} + \frac {25 a e r x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac {125 a e x^{5} x^{r}}{25 r^{2} + 250 r + 625} - \frac {b d n r^{2} x^{5}}{25 r^{2} + 250 r + 625} - \frac {10 b d n r x^{5}}{25 r^{2} + 250 r + 625} - \frac {25 b d n x^{5}}{25 r^{2} + 250 r + 625} + \frac {5 b d r^{2} x^{5} \log {\left (c x^{n} \right )}}{25 r^{2} + 250 r + 625} + \frac {50 b d r x^{5} \log {\left (c x^{n} \right )}}{25 r^{2} + 250 r + 625} + \frac {125 b d x^{5} \log {\left (c x^{n} \right )}}{25 r^{2} + 250 r + 625} - \frac {25 b e n x^{5} x^{r}}{25 r^{2} + 250 r + 625} + \frac {25 b e r x^{5} x^{r} \log {\left (c x^{n} \right )}}{25 r^{2} + 250 r + 625} + \frac {125 b e x^{5} x^{r} \log {\left (c x^{n} \right )}}{25 r^{2} + 250 r + 625} & \text {for}\: r \neq -5 \\\frac {a d x^{5}}{5} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{5}}{25} + \frac {b d x^{5} \log {\left (c x^{n} \right )}}{5} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, b d n x^{5} + \frac {1}{5} \, b d x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d x^{5} + \frac {b e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {a e x^{r + 5}}{r + 5} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (55) = 110\).

Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.24 \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e n r x^{5} x^{r} \log \left (x\right )}{r^{2} + 10 \, r + 25} + \frac {5 \, b e n x^{5} x^{r} \log \left (x\right )}{r^{2} + 10 \, r + 25} + \frac {1}{5} \, b d n x^{5} \log \left (x\right ) - \frac {b e n x^{5} x^{r}}{r^{2} + 10 \, r + 25} - \frac {1}{25} \, b d n x^{5} + \frac {b e x^{5} x^{r} \log \left (c\right )}{r + 5} + \frac {1}{5} \, b d x^{5} \log \left (c\right ) + \frac {a e x^{5} x^{r}}{r + 5} + \frac {1}{5} \, a d x^{5} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^4\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[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)