3.368 \(\int x^3 (d+e x^r) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=59 \[ \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d n x^4-\frac {b e n x^{r+4}}{(r+4)^2} \]

[Out]

-1/16*b*d*n*x^4-b*e*n*x^(4+r)/(4+r)^2+1/4*(d*x^4+4*e*x^(4+r)/(4+r))*(a+b*ln(c*x^n))

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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} \[ \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d n x^4-\frac {b e n x^{r+4}}{(r+4)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x^4)/16 - (b*e*n*x^(4 + r))/(4 + r)^2 + ((d*x^4 + (4*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4

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^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{4} x^3 \left (d+\frac {4 e x^r}{4+r}\right ) \, dx\\ &=\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int x^3 \left (d+\frac {4 e x^r}{4+r}\right ) \, dx\\ &=\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (d x^3+\frac {4 e x^{3+r}}{4+r}\right ) \, dx\\ &=-\frac {1}{16} b d n x^4-\frac {b e n x^{4+r}}{(4+r)^2}+\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 73, normalized size = 1.24 \[ \frac {x^4 \left (4 a (r+4) \left (d (r+4)+4 e x^r\right )+4 b (r+4) \log \left (c x^n\right ) \left (d (r+4)+4 e x^r\right )-b n \left (d (r+4)^2+16 e x^r\right )\right )}{16 (r+4)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

(x^4*(4*a*(4 + r)*(d*(4 + r) + 4*e*x^r) - b*n*(d*(4 + r)^2 + 16*e*x^r) + 4*b*(4 + r)*(d*(4 + r) + 4*e*x^r)*Log
[c*x^n]))/(16*(4 + r)^2)

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fricas [B]  time = 0.44, size = 159, normalized size = 2.69 \[ \frac {4 \, {\left (b d r^{2} + 8 \, b d r + 16 \, b d\right )} x^{4} \log \relax (c) + 4 \, {\left (b d n r^{2} + 8 \, b d n r + 16 \, b d n\right )} x^{4} \log \relax (x) - {\left (16 \, b d n + {\left (b d n - 4 \, a d\right )} r^{2} - 64 \, a d + 8 \, {\left (b d n - 4 \, a d\right )} r\right )} x^{4} + 16 \, {\left ({\left (b e r + 4 \, b e\right )} x^{4} \log \relax (c) + {\left (b e n r + 4 \, b e n\right )} x^{4} \log \relax (x) - {\left (b e n - a e r - 4 \, a e\right )} x^{4}\right )} x^{r}}{16 \, {\left (r^{2} + 8 \, r + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/16*(4*(b*d*r^2 + 8*b*d*r + 16*b*d)*x^4*log(c) + 4*(b*d*n*r^2 + 8*b*d*n*r + 16*b*d*n)*x^4*log(x) - (16*b*d*n
+ (b*d*n - 4*a*d)*r^2 - 64*a*d + 8*(b*d*n - 4*a*d)*r)*x^4 + 16*((b*e*r + 4*b*e)*x^4*log(c) + (b*e*n*r + 4*b*e*
n)*x^4*log(x) - (b*e*n - a*e*r - 4*a*e)*x^4)*x^r)/(r^2 + 8*r + 16)

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giac [B]  time = 0.36, size = 137, normalized size = 2.32 \[ \frac {b n r x^{4} x^{r} e \log \relax (x)}{r^{2} + 8 \, r + 16} + \frac {1}{4} \, b d n x^{4} \log \relax (x) + \frac {4 \, b n x^{4} x^{r} e \log \relax (x)}{r^{2} + 8 \, r + 16} - \frac {1}{16} \, b d n x^{4} - \frac {b n x^{4} x^{r} e}{r^{2} + 8 \, r + 16} + \frac {1}{4} \, b d x^{4} \log \relax (c) + \frac {b x^{4} x^{r} e \log \relax (c)}{r + 4} + \frac {1}{4} \, a d x^{4} + \frac {a x^{4} x^{r} e}{r + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*n*r*x^4*x^r*e*log(x)/(r^2 + 8*r + 16) + 1/4*b*d*n*x^4*log(x) + 4*b*n*x^4*x^r*e*log(x)/(r^2 + 8*r + 16) - 1/1
6*b*d*n*x^4 - b*n*x^4*x^r*e/(r^2 + 8*r + 16) + 1/4*b*d*x^4*log(c) + b*x^4*x^r*e*log(c)/(r + 4) + 1/4*a*d*x^4 +
a*x^4*x^r*e/(r + 4)

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maple [C]  time = 0.27, size = 613, normalized size = 10.39 \[ \frac {\left (d r +4 e \,x^{r}+4 d \right ) b \,x^{4} \ln \left (x^{n}\right )}{16+4 r}-\frac {\left (16 b d n -64 a e \,x^{r}-16 a e r \,x^{r}+16 b e n \,x^{r}-4 b d \,r^{2} \ln \relax (c )-32 b d r \ln \relax (c )-64 b e \,x^{r} \ln \relax (c )-32 a d r -64 a d +b d n \,r^{2}-2 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 a d \,r^{2}+8 b d n r -64 b d \ln \relax (c )+32 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 )+2 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 )-8 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-8 i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+16 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 )+32 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 )-16 b e r \,x^{r} \ln \relax (c )+8 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 )-32 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+32 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+16 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+32 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-16 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-16 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-32 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-32 i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-32 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}\right ) x^{4}}{16 \left (r +4\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d+e*x^r)*(b*ln(c*x^n)+a),x)

[Out]

1/4*b*x^4*(d*r+4*e*x^r+4*d)/(4+r)*ln(x^n)-1/16*x^4*(16*b*d*n-64*a*e*x^r-16*a*e*r*x^r+16*b*e*n*x^r-4*b*d*r^2*ln
(c)-32*b*d*r*ln(c)-64*b*e*x^r*ln(c)-32*a*d*r-64*a*d+b*d*n*r^2-4*a*d*r^2+8*b*d*n*r-64*b*d*ln(c)-2*I*Pi*b*d*r^2*
csgn(I*c*x^n)^2*csgn(I*c)-32*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+2*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+32*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-8*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*r-8*
I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r+32*I*Pi*b*d*csgn(I*c*x^n)^3+8*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*x^r*r+16*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*r-32*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+8*I*Pi
*b*e*csgn(I*c*x^n)^3*x^r*r-16*b*e*r*x^r*ln(c)+2*I*Pi*b*d*r^2*csgn(I*c*x^n)^3+32*I*Pi*b*e*csgn(I*c*x^n)^3*x^r-3
2*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)-32*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2+16*I*Pi*b*d*csgn(I*c*x^n)^3*r-16*
I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2*r-16*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)*r-2*I*Pi*b*d*r^2*csgn(I*x^n)*csgn
(I*c*x^n)^2+32*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(4+r)^2

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maxima [A]  time = 0.99, size = 76, normalized size = 1.29 \[ -\frac {1}{16} \, b d n x^{4} + \frac {1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d x^{4} + \frac {b e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {a e x^{r + 4}}{r + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/16*b*d*n*x^4 + 1/4*b*d*x^4*log(c*x^n) + 1/4*a*d*x^4 + b*e*x^(r + 4)*log(c*x^n)/(r + 4) - b*e*n*x^(r + 4)/(r
+ 4)^2 + a*e*x^(r + 4)/(r + 4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d + e*x^r)*(a + b*log(c*x^n)),x)

[Out]

int(x^3*(d + e*x^r)*(a + b*log(c*x^n)), x)

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sympy [A]  time = 30.32, size = 525, normalized size = 8.90 \[ \begin {cases} \frac {4 a d r^{2} x^{4}}{16 r^{2} + 128 r + 256} + \frac {32 a d r x^{4}}{16 r^{2} + 128 r + 256} + \frac {64 a d x^{4}}{16 r^{2} + 128 r + 256} + \frac {16 a e r x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {64 a e x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {4 b d n r^{2} x^{4} \log {\relax (x )}}{16 r^{2} + 128 r + 256} - \frac {b d n r^{2} x^{4}}{16 r^{2} + 128 r + 256} + \frac {32 b d n r x^{4} \log {\relax (x )}}{16 r^{2} + 128 r + 256} - \frac {8 b d n r x^{4}}{16 r^{2} + 128 r + 256} + \frac {64 b d n x^{4} \log {\relax (x )}}{16 r^{2} + 128 r + 256} - \frac {16 b d n x^{4}}{16 r^{2} + 128 r + 256} + \frac {4 b d r^{2} x^{4} \log {\relax (c )}}{16 r^{2} + 128 r + 256} + \frac {32 b d r x^{4} \log {\relax (c )}}{16 r^{2} + 128 r + 256} + \frac {64 b d x^{4} \log {\relax (c )}}{16 r^{2} + 128 r + 256} + \frac {16 b e n r x^{4} x^{r} \log {\relax (x )}}{16 r^{2} + 128 r + 256} + \frac {64 b e n x^{4} x^{r} \log {\relax (x )}}{16 r^{2} + 128 r + 256} - \frac {16 b e n x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {16 b e r x^{4} x^{r} \log {\relax (c )}}{16 r^{2} + 128 r + 256} + \frac {64 b e x^{4} x^{r} \log {\relax (c )}}{16 r^{2} + 128 r + 256} & \text {for}\: r \neq -4 \\\frac {a d x^{4}}{4} + a e \log {\relax (x )} + \frac {b d n x^{4} \log {\relax (x )}}{4} - \frac {b d n x^{4}}{16} + \frac {b d x^{4} \log {\relax (c )}}{4} + \frac {b e n \log {\relax (x )}^{2}}{2} + b e \log {\relax (c )} \log {\relax (x )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(d+e*x**r)*(a+b*ln(c*x**n)),x)

[Out]

Piecewise((4*a*d*r**2*x**4/(16*r**2 + 128*r + 256) + 32*a*d*r*x**4/(16*r**2 + 128*r + 256) + 64*a*d*x**4/(16*r
**2 + 128*r + 256) + 16*a*e*r*x**4*x**r/(16*r**2 + 128*r + 256) + 64*a*e*x**4*x**r/(16*r**2 + 128*r + 256) + 4
*b*d*n*r**2*x**4*log(x)/(16*r**2 + 128*r + 256) - b*d*n*r**2*x**4/(16*r**2 + 128*r + 256) + 32*b*d*n*r*x**4*lo
g(x)/(16*r**2 + 128*r + 256) - 8*b*d*n*r*x**4/(16*r**2 + 128*r + 256) + 64*b*d*n*x**4*log(x)/(16*r**2 + 128*r
+ 256) - 16*b*d*n*x**4/(16*r**2 + 128*r + 256) + 4*b*d*r**2*x**4*log(c)/(16*r**2 + 128*r + 256) + 32*b*d*r*x**
4*log(c)/(16*r**2 + 128*r + 256) + 64*b*d*x**4*log(c)/(16*r**2 + 128*r + 256) + 16*b*e*n*r*x**4*x**r*log(x)/(1
6*r**2 + 128*r + 256) + 64*b*e*n*x**4*x**r*log(x)/(16*r**2 + 128*r + 256) - 16*b*e*n*x**4*x**r/(16*r**2 + 128*
r + 256) + 16*b*e*r*x**4*x**r*log(c)/(16*r**2 + 128*r + 256) + 64*b*e*x**4*x**r*log(c)/(16*r**2 + 128*r + 256)
, Ne(r, -4)), (a*d*x**4/4 + a*e*log(x) + b*d*n*x**4*log(x)/4 - b*d*n*x**4/16 + b*d*x**4*log(c)/4 + b*e*n*log(x
)**2/2 + b*e*log(c)*log(x), True))

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