3.367 \(\int x^5 (d+e x^r) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{6} \left (d x^6+\frac{6 e x^{r+6}}{r+6}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d n x^6-\frac{b e n x^{r+6}}{(r+6)^2} \]

[Out]

-(b*d*n*x^6)/36 - (b*e*n*x^(6 + r))/(6 + r)^2 + ((d*x^6 + (6*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6

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Rubi [A]  time = 0.0790428, antiderivative size = 59, normalized size of antiderivative = 1., 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} \[ \frac{1}{6} \left (d x^6+\frac{6 e x^{r+6}}{r+6}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d n x^6-\frac{b e n x^{r+6}}{(r+6)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x^6)/36 - (b*e*n*x^(6 + r))/(6 + r)^2 + ((d*x^6 + (6*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6

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])

Rule 12

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

Rubi steps

\begin{align*} \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{6} \left (d x^6+\frac{6 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{6} x^5 \left (d+\frac{6 e x^r}{6+r}\right ) \, dx\\ &=\frac{1}{6} \left (d x^6+\frac{6 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int x^5 \left (d+\frac{6 e x^r}{6+r}\right ) \, dx\\ &=\frac{1}{6} \left (d x^6+\frac{6 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \left (d x^5+\frac{6 e x^{5+r}}{6+r}\right ) \, dx\\ &=-\frac{1}{36} b d n x^6-\frac{b e n x^{6+r}}{(6+r)^2}+\frac{1}{6} \left (d x^6+\frac{6 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0985877, size = 73, normalized size = 1.24 \[ \frac{x^6 \left (6 a (r+6) \left (d (r+6)+6 e x^r\right )+6 b (r+6) \log \left (c x^n\right ) \left (d (r+6)+6 e x^r\right )-b n \left (d (r+6)^2+36 e x^r\right )\right )}{36 (r+6)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

(x^6*(6*a*(6 + r)*(d*(6 + r) + 6*e*x^r) - b*n*(d*(6 + r)^2 + 36*e*x^r) + 6*b*(6 + r)*(d*(6 + r) + 6*e*x^r)*Log
[c*x^n]))/(36*(6 + r)^2)

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Maple [C]  time = 0.24, size = 613, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d+e*x^r)*(a+b*ln(c*x^n)),x)

[Out]

1/6*x^6*b*(d*r+6*e*x^r+6*d)/(6+r)*ln(x^n)-1/36*x^6*(-216*a*d-36*x^r*a*e*r+36*x^r*b*e*n+18*I*Pi*b*e*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)*x^r*r+12*b*d*n*r+36*b*d*n-216*x^r*a*e-72*ln(c)*b*d*r-6*ln(c)*b*d*r^2-36*ln(c)*b*e*x^r
*r+108*I*Pi*b*d*csgn(I*c*x^n)^3-18*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r+108*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)*x^r-216*ln(c)*b*e*x^r-6*a*d*r^2-18*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*r+36*I*Pi*b*d*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)*r+3*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-216*ln(c)*b*d-72*a*d*r+b*d
*n*r^2+36*I*Pi*b*d*csgn(I*c*x^n)^3*r-108*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)-108*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*
x^n)^2+3*I*Pi*b*d*r^2*csgn(I*c*x^n)^3+108*I*Pi*b*e*csgn(I*c*x^n)^3*x^r+108*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*c)-36*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2*r-36*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)*r-3*I*Pi*b*d*r^2*csg
n(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csgn(I*c)+18*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r-108*I*Pi*b
*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-108*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r)/(6+r)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.32415, size = 396, normalized size = 6.71 \begin{align*} \frac{6 \,{\left (b d r^{2} + 12 \, b d r + 36 \, b d\right )} x^{6} \log \left (c\right ) + 6 \,{\left (b d n r^{2} + 12 \, b d n r + 36 \, b d n\right )} x^{6} \log \left (x\right ) -{\left (36 \, b d n +{\left (b d n - 6 \, a d\right )} r^{2} - 216 \, a d + 12 \,{\left (b d n - 6 \, a d\right )} r\right )} x^{6} + 36 \,{\left ({\left (b e r + 6 \, b e\right )} x^{6} \log \left (c\right ) +{\left (b e n r + 6 \, b e n\right )} x^{6} \log \left (x\right ) -{\left (b e n - a e r - 6 \, a e\right )} x^{6}\right )} x^{r}}{36 \,{\left (r^{2} + 12 \, r + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/36*(6*(b*d*r^2 + 12*b*d*r + 36*b*d)*x^6*log(c) + 6*(b*d*n*r^2 + 12*b*d*n*r + 36*b*d*n)*x^6*log(x) - (36*b*d*
n + (b*d*n - 6*a*d)*r^2 - 216*a*d + 12*(b*d*n - 6*a*d)*r)*x^6 + 36*((b*e*r + 6*b*e)*x^6*log(c) + (b*e*n*r + 6*
b*e*n)*x^6*log(x) - (b*e*n - a*e*r - 6*a*e)*x^6)*x^r)/(r^2 + 12*r + 36)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(d+e*x**r)*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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Giac [B]  time = 1.31339, size = 185, normalized size = 3.14 \begin{align*} \frac{b n r x^{6} x^{r} e \log \left (x\right )}{r^{2} + 12 \, r + 36} + \frac{1}{6} \, b d n x^{6} \log \left (x\right ) + \frac{6 \, b n x^{6} x^{r} e \log \left (x\right )}{r^{2} + 12 \, r + 36} - \frac{1}{36} \, b d n x^{6} - \frac{b n x^{6} x^{r} e}{r^{2} + 12 \, r + 36} + \frac{1}{6} \, b d x^{6} \log \left (c\right ) + \frac{b x^{6} x^{r} e \log \left (c\right )}{r + 6} + \frac{1}{6} \, a d x^{6} + \frac{a x^{6} x^{r} e}{r + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*n*r*x^6*x^r*e*log(x)/(r^2 + 12*r + 36) + 1/6*b*d*n*x^6*log(x) + 6*b*n*x^6*x^r*e*log(x)/(r^2 + 12*r + 36) - 1
/36*b*d*n*x^6 - b*n*x^6*x^r*e/(r^2 + 12*r + 36) + 1/6*b*d*x^6*log(c) + b*x^6*x^r*e*log(c)/(r + 6) + 1/6*a*d*x^
6 + a*x^6*x^r*e/(r + 6)