3.4.69 \(\int x (d+e x^r) (a+b \log (c x^n)) \, dx\) [369]

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

[Out]

-1/4*b*d*n*x^2-b*e*n*x^(2+r)/(2+r)^2+1/2*(d*x^2+2*e*x^(2+r)/(2+r))*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {14, 2371, 12} \begin {gather*} \frac {1}{2} \left (d x^2+\frac {2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d n x^2-\frac {b e n x^{r+2}}{(r+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(b*d*n*x^2) - (b*e*n*x^(2 + r))/(2 + r)^2 + ((d*x^2 + (2*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2

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

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Mathematica [A]
time = 0.07, size = 73, normalized size = 1.24 \begin {gather*} \frac {x^2 \left (2 a (2+r) \left (d (2+r)+2 e x^r\right )-b n \left (d (2+r)^2+4 e x^r\right )+2 b (2+r) \left (d (2+r)+2 e x^r\right ) \log \left (c x^n\right )\right )}{4 (2+r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.08, size = 613, normalized size = 10.39

method	result	size
risch	\(\frac {b \,x^{2} \left (d r +2 e \,x^{r}+2 d \right ) \ln \left (x^{n}\right )}{4+2 r}-\frac {x^{2} \left (-8 x^{r} a e +4 b d n +4 x^{r} b e n -4 x^{r} a e r -8 a d +4 b d n r -8 \ln \left (c \right ) b d r -2 \ln \left (c \right ) b d \,r^{2}-2 a d \,r^{2}-8 d b \ln \left (c \right )-4 \ln \left (c \right ) b e \,x^{r} r -8 a d r -i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-4 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -4 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r +4 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b d n \,r^{2}+4 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r}-2 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -2 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +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 )+4 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 ) r -8 \ln \left (c \right ) b e \,x^{r}+2 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r} r +4 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} r +i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+4 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 )-4 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r \right )}{4 \left (2+r \right )^{2}}\)	\(613\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 76, normalized size = 1.29 \begin {gather*} -\frac {1}{4} \, b d n x^{2} + \frac {1}{2} \, b d x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d x^{2} + \frac {b e x^{r + 2} \log \left (c x^{n}\right )}{r + 2} - \frac {b e n x^{r + 2}}{{\left (r + 2\right )}^{2}} + \frac {a e x^{r + 2}}{r + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (57) = 114\).
time = 0.37, size = 158, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left (b d r^{2} + 4 \, b d r + 4 \, b d\right )} x^{2} \log \left (c\right ) + 2 \, {\left (b d n r^{2} + 4 \, b d n r + 4 \, b d n\right )} x^{2} \log \left (x\right ) - {\left (4 \, b d n + {\left (b d n - 2 \, a d\right )} r^{2} - 8 \, a d + 4 \, {\left (b d n - 2 \, a d\right )} r\right )} x^{2} + 4 \, {\left ({\left (b r + 2 \, b\right )} x^{2} e \log \left (c\right ) + {\left (b n r + 2 \, b n\right )} x^{2} e \log \left (x\right ) - {\left (b n - a r - 2 \, a\right )} x^{2} e\right )} x^{r}}{4 \, {\left (r^{2} + 4 \, r + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (51) = 102\).
time = 0.98, size = 398, normalized size = 6.75 \begin {gather*} \begin {cases} \frac {2 a d r^{2} x^{2}}{4 r^{2} + 16 r + 16} + \frac {8 a d r x^{2}}{4 r^{2} + 16 r + 16} + \frac {8 a d x^{2}}{4 r^{2} + 16 r + 16} + \frac {4 a e r x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac {8 a e x^{2} x^{r}}{4 r^{2} + 16 r + 16} - \frac {b d n r^{2} x^{2}}{4 r^{2} + 16 r + 16} - \frac {4 b d n r x^{2}}{4 r^{2} + 16 r + 16} - \frac {4 b d n x^{2}}{4 r^{2} + 16 r + 16} + \frac {2 b d r^{2} x^{2} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} + \frac {8 b d r x^{2} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} + \frac {8 b d x^{2} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} - \frac {4 b e n x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac {4 b e r x^{2} x^{r} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} + \frac {8 b e x^{2} x^{r} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} & \text {for}\: r \neq -2 \\\frac {a d x^{2}}{2} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{2}}{4} + \frac {b d x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*d*r**2*x**2/(4*r**2 + 16*r + 16) + 8*a*d*r*x**2/(4*r**2 + 16*r + 16) + 8*a*d*x**2/(4*r**2 + 16*
r + 16) + 4*a*e*r*x**2*x**r/(4*r**2 + 16*r + 16) + 8*a*e*x**2*x**r/(4*r**2 + 16*r + 16) - b*d*n*r**2*x**2/(4*r
**2 + 16*r + 16) - 4*b*d*n*r*x**2/(4*r**2 + 16*r + 16) - 4*b*d*n*x**2/(4*r**2 + 16*r + 16) + 2*b*d*r**2*x**2*l
og(c*x**n)/(4*r**2 + 16*r + 16) + 8*b*d*r*x**2*log(c*x**n)/(4*r**2 + 16*r + 16) + 8*b*d*x**2*log(c*x**n)/(4*r*
*2 + 16*r + 16) - 4*b*e*n*x**2*x**r/(4*r**2 + 16*r + 16) + 4*b*e*r*x**2*x**r*log(c*x**n)/(4*r**2 + 16*r + 16)
+ 8*b*e*x**2*x**r*log(c*x**n)/(4*r**2 + 16*r + 16), Ne(r, -2)), (a*d*x**2/2 + a*e*log(c*x**n)/n - b*d*n*x**2/4
+ b*d*x**2*log(c*x**n)/2 + b*e*log(c*x**n)**2/(2*n), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (57) = 114\).
time = 1.78, size = 137, normalized size = 2.32 \begin {gather*} \frac {b n r x^{2} x^{r} e \log \left (x\right )}{r^{2} + 4 \, r + 4} + \frac {1}{2} \, b d n x^{2} \log \left (x\right ) + \frac {2 \, b n x^{2} x^{r} e \log \left (x\right )}{r^{2} + 4 \, r + 4} - \frac {1}{4} \, b d n x^{2} - \frac {b n x^{2} x^{r} e}{r^{2} + 4 \, r + 4} + \frac {1}{2} \, b d x^{2} \log \left (c\right ) + \frac {b x^{2} x^{r} e \log \left (c\right )}{r + 2} + \frac {1}{2} \, a d x^{2} + \frac {a x^{2} x^{r} e}{r + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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