\(\int x^2 (d+e x^r) (a+b \log (c x^n)) \, dx\) [374]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} b d n x^3-\frac {b e n x^{3+r}}{(3+r)^2}+\frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/9*b*d*n*x^3-b*e*n*x^(3+r)/(3+r)^2+1/3*(d*x^3+3*e*x^(3+r)/(3+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} \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{3} \left (d x^3+\frac {3 e x^{r+3}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d n x^3-\frac {b e n x^{r+3}}{(r+3)^2} \]

[In]

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

[Out]

-1/9*(b*d*n*x^3) - (b*e*n*x^(3 + r))/(3 + r)^2 + ((d*x^3 + (3*e*x^(3 + r))/(3 + r))*(a + b*Log[c*x^n]))/3

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}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{3} x^2 \left (d+\frac {3 e x^r}{3+r}\right ) \, dx \\ & = \frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int x^2 \left (d+\frac {3 e x^r}{3+r}\right ) \, dx \\ & = \frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (d x^2+\frac {3 e x^{2+r}}{3+r}\right ) \, dx \\ & = -\frac {1}{9} b d n x^3-\frac {b e n x^{3+r}}{(3+r)^2}+\frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+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 \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^3 \left (3 a (3+r) \left (d (3+r)+3 e x^r\right )-b n \left (d (3+r)^2+9 e x^r\right )+3 b (3+r) \left (d (3+r)+3 e x^r\right ) \log \left (c x^n\right )\right )}{9 (3+r)^2} \]

[In]

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

[Out]

(x^3*(3*a*(3 + r)*(d*(3 + r) + 3*e*x^r) - b*n*(d*(3 + r)^2 + 9*e*x^r) + 3*b*(3 + r)*(d*(3 + r) + 3*e*x^r)*Log[
c*x^n]))/(9*(3 + r)^2)

Maple [B] (verified)

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

Time = 0.70 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.86

method	result	size
parallelrisch	\(-\frac {-9 x^{3} x^{r} \ln \left (c \,x^{n}\right ) b e r -3 x^{3} \ln \left (c \,x^{n}\right ) b d \,r^{2}+x^{3} b d n \,r^{2}-27 x^{3} x^{r} \ln \left (c \,x^{n}\right ) b e -9 x^{3} x^{r} a e r +9 x^{3} x^{r} b e n -18 x^{3} \ln \left (c \,x^{n}\right ) b d r -3 x^{3} a d \,r^{2}+6 x^{3} b d n r -27 x^{3} x^{r} a e -27 x^{3} \ln \left (c \,x^{n}\right ) b d -18 x^{3} a d r +9 b d n \,x^{3}-27 x^{3} a d}{9 \left (3+r \right )^{2}}\)	\(169\)
risch	\(\frac {b \,x^{3} \left (d r +3 e \,x^{r}+3 d \right ) \ln \left (x^{n}\right )}{9+3 r}-\frac {x^{3} \left (-54 x^{r} a e +18 b d n -54 a d -18 x^{r} a e r +18 x^{r} b e n -27 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-36 a d r +2 b d n \,r^{2}-18 \ln \left (c \right ) b e \,x^{r} r +9 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 -54 d b \ln \left (c \right )+27 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+12 b d n r -54 \ln \left (c \right ) b e \,x^{r}-6 \ln \left (c \right ) b d \,r^{2}-36 \ln \left (c \right ) b d r +27 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 )-6 a d \,r^{2}-27 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +27 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}-9 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +3 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 )+18 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 -27 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+9 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -18 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -27 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-18 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+27 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+18 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r \right )}{18 \left (3+r \right )^{2}}\)	\(614\)

[In]

int(x^2*(d+e*x^r)*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

-1/9*(-9*x^3*x^r*ln(c*x^n)*b*e*r-3*x^3*ln(c*x^n)*b*d*r^2+x^3*b*d*n*r^2-27*x^3*x^r*ln(c*x^n)*b*e-9*x^3*x^r*a*e*
r+9*x^3*x^r*b*e*n-18*x^3*ln(c*x^n)*b*d*r-3*x^3*a*d*r^2+6*x^3*b*d*n*r-27*x^3*x^r*a*e-27*x^3*ln(c*x^n)*b*d-18*x^
3*a*d*r+9*b*d*n*x^3-27*x^3*a*d)/(3+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 = 0.31 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, {\left (b d r^{2} + 6 \, b d r + 9 \, b d\right )} x^{3} \log \left (c\right ) + 3 \, {\left (b d n r^{2} + 6 \, b d n r + 9 \, b d n\right )} x^{3} \log \left (x\right ) - {\left (9 \, b d n + {\left (b d n - 3 \, a d\right )} r^{2} - 27 \, a d + 6 \, {\left (b d n - 3 \, a d\right )} r\right )} x^{3} + 9 \, {\left ({\left (b e r + 3 \, b e\right )} x^{3} \log \left (c\right ) + {\left (b e n r + 3 \, b e n\right )} x^{3} \log \left (x\right ) - {\left (b e n - a e r - 3 \, a e\right )} x^{3}\right )} x^{r}}{9 \, {\left (r^{2} + 6 \, r + 9\right )}} \]

[In]

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

[Out]

1/9*(3*(b*d*r^2 + 6*b*d*r + 9*b*d)*x^3*log(c) + 3*(b*d*n*r^2 + 6*b*d*n*r + 9*b*d*n)*x^3*log(x) - (9*b*d*n + (b
*d*n - 3*a*d)*r^2 - 27*a*d + 6*(b*d*n - 3*a*d)*r)*x^3 + 9*((b*e*r + 3*b*e)*x^3*log(c) + (b*e*n*r + 3*b*e*n)*x^
3*log(x) - (b*e*n - a*e*r - 3*a*e)*x^3)*x^r)/(r^2 + 6*r + 9)

Sympy [B] (verification not implemented)

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

Time = 1.51 (sec) , antiderivative size = 398, normalized size of antiderivative = 6.75 \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {3 a d r^{2} x^{3}}{9 r^{2} + 54 r + 81} + \frac {18 a d r x^{3}}{9 r^{2} + 54 r + 81} + \frac {27 a d x^{3}}{9 r^{2} + 54 r + 81} + \frac {9 a e r x^{3} x^{r}}{9 r^{2} + 54 r + 81} + \frac {27 a e x^{3} x^{r}}{9 r^{2} + 54 r + 81} - \frac {b d n r^{2} x^{3}}{9 r^{2} + 54 r + 81} - \frac {6 b d n r x^{3}}{9 r^{2} + 54 r + 81} - \frac {9 b d n x^{3}}{9 r^{2} + 54 r + 81} + \frac {3 b d r^{2} x^{3} \log {\left (c x^{n} \right )}}{9 r^{2} + 54 r + 81} + \frac {18 b d r x^{3} \log {\left (c x^{n} \right )}}{9 r^{2} + 54 r + 81} + \frac {27 b d x^{3} \log {\left (c x^{n} \right )}}{9 r^{2} + 54 r + 81} - \frac {9 b e n x^{3} x^{r}}{9 r^{2} + 54 r + 81} + \frac {9 b e r x^{3} x^{r} \log {\left (c x^{n} \right )}}{9 r^{2} + 54 r + 81} + \frac {27 b e x^{3} x^{r} \log {\left (c x^{n} \right )}}{9 r^{2} + 54 r + 81} & \text {for}\: r \neq -3 \\\frac {a d x^{3}}{3} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{3}}{9} + \frac {b d x^{3} \log {\left (c x^{n} \right )}}{3} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*(d+e*x**r)*(a+b*ln(c*x**n)),x)

[Out]

Piecewise((3*a*d*r**2*x**3/(9*r**2 + 54*r + 81) + 18*a*d*r*x**3/(9*r**2 + 54*r + 81) + 27*a*d*x**3/(9*r**2 + 5
4*r + 81) + 9*a*e*r*x**3*x**r/(9*r**2 + 54*r + 81) + 27*a*e*x**3*x**r/(9*r**2 + 54*r + 81) - b*d*n*r**2*x**3/(
9*r**2 + 54*r + 81) - 6*b*d*n*r*x**3/(9*r**2 + 54*r + 81) - 9*b*d*n*x**3/(9*r**2 + 54*r + 81) + 3*b*d*r**2*x**
3*log(c*x**n)/(9*r**2 + 54*r + 81) + 18*b*d*r*x**3*log(c*x**n)/(9*r**2 + 54*r + 81) + 27*b*d*x**3*log(c*x**n)/
(9*r**2 + 54*r + 81) - 9*b*e*n*x**3*x**r/(9*r**2 + 54*r + 81) + 9*b*e*r*x**3*x**r*log(c*x**n)/(9*r**2 + 54*r +
81) + 27*b*e*x**3*x**r*log(c*x**n)/(9*r**2 + 54*r + 81), Ne(r, -3)), (a*d*x**3/3 + a*e*log(c*x**n)/n - b*d*n*
x**3/9 + b*d*x**3*log(c*x**n)/3 + 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 \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} \, b d n x^{3} + \frac {1}{3} \, b d x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d x^{3} + \frac {b e x^{r + 3} \log \left (c x^{n}\right )}{r + 3} - \frac {b e n x^{r + 3}}{{\left (r + 3\right )}^{2}} + \frac {a e x^{r + 3}}{r + 3} \]

[In]

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

[Out]

-1/9*b*d*n*x^3 + 1/3*b*d*x^3*log(c*x^n) + 1/3*a*d*x^3 + b*e*x^(r + 3)*log(c*x^n)/(r + 3) - b*e*n*x^(r + 3)/(r
+ 3)^2 + a*e*x^(r + 3)/(r + 3)

Giac [B] (verification not implemented)

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

Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.24 \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e n r x^{3} x^{r} \log \left (x\right )}{r^{2} + 6 \, r + 9} + \frac {3 \, b e n x^{3} x^{r} \log \left (x\right )}{r^{2} + 6 \, r + 9} + \frac {1}{3} \, b d n x^{3} \log \left (x\right ) - \frac {b e n x^{3} x^{r}}{r^{2} + 6 \, r + 9} - \frac {1}{9} \, b d n x^{3} + \frac {b e x^{3} x^{r} \log \left (c\right )}{r + 3} + \frac {1}{3} \, b d x^{3} \log \left (c\right ) + \frac {a e x^{3} x^{r}}{r + 3} + \frac {1}{3} \, a d x^{3} \]

[In]

integrate(x^2*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*e*n*r*x^3*x^r*log(x)/(r^2 + 6*r + 9) + 3*b*e*n*x^3*x^r*log(x)/(r^2 + 6*r + 9) + 1/3*b*d*n*x^3*log(x) - b*e*n
*x^3*x^r/(r^2 + 6*r + 9) - 1/9*b*d*n*x^3 + b*e*x^3*x^r*log(c)/(r + 3) + 1/3*b*d*x^3*log(c) + a*e*x^3*x^r/(r +
3) + 1/3*a*d*x^3

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

int(x^2*(d + e*x^r)*(a + b*log(c*x^n)),x)

[Out]

int(x^2*(d + e*x^r)*(a + b*log(c*x^n)), x)