3.171 \(\int x^5 (d+e x^2) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{24} \left (4 d x^6+3 e x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d n x^6-\frac{1}{64} b e n x^8 \]

[Out]

-(b*d*n*x^6)/36 - (b*e*n*x^8)/64 + ((4*d*x^6 + 3*e*x^8)*(a + b*Log[c*x^n]))/24

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Rubi [A]  time = 0.0430605, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {14, 2334} \[ \frac{1}{24} \left (4 d x^6+3 e x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d n x^6-\frac{1}{64} b e n x^8 \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^2)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x^6)/36 - (b*e*n*x^8)/64 + ((4*d*x^6 + 3*e*x^8)*(a + b*Log[c*x^n]))/24

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^5 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{24} \left (4 d x^6+3 e x^8\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{d x^5}{6}+\frac{e x^7}{8}\right ) \, dx\\ &=-\frac{1}{36} b d n x^6-\frac{1}{64} b e n x^8+\frac{1}{24} \left (4 d x^6+3 e x^8\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0027855, size = 69, normalized size = 1.44 \[ \frac{1}{6} a d x^6+\frac{1}{8} a e x^8+\frac{1}{6} b d x^6 \log \left (c x^n\right )+\frac{1}{8} b e x^8 \log \left (c x^n\right )-\frac{1}{36} b d n x^6-\frac{1}{64} b e n x^8 \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^2)*(a + b*Log[c*x^n]),x]

[Out]

(a*d*x^6)/6 - (b*d*n*x^6)/36 + (a*e*x^8)/8 - (b*e*n*x^8)/64 + (b*d*x^6*Log[c*x^n])/6 + (b*e*x^8*Log[c*x^n])/8

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Maple [C]  time = 0.203, size = 266, normalized size = 5.5 \begin{align*}{\frac{b{x}^{6} \left ( 3\,e{x}^{2}+4\,d \right ) \ln \left ({x}^{n} \right ) }{24}}+{\frac{i}{16}}\pi \,be{x}^{8}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{16}}\pi \,be{x}^{8}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{16}}\pi \,be{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{16}}\pi \,be{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{8}}{8}}-{\frac{ben{x}^{8}}{64}}+{\frac{ae{x}^{8}}{8}}+{\frac{i}{12}}\pi \,bd{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{12}}\pi \,bd{x}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,bd{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,bd{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{6}}{6}}-{\frac{bdn{x}^{6}}{36}}+{\frac{ad{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x^2+d)*(a+b*ln(c*x^n)),x)

[Out]

1/24*b*x^6*(3*e*x^2+4*d)*ln(x^n)+1/16*I*Pi*b*e*x^8*csgn(I*x^n)*csgn(I*c*x^n)^2-1/16*I*Pi*b*e*x^8*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)-1/16*I*Pi*b*e*x^8*csgn(I*c*x^n)^3+1/16*I*Pi*b*e*x^8*csgn(I*c*x^n)^2*csgn(I*c)+1/8*ln(c)
*b*e*x^8-1/64*b*e*n*x^8+1/8*a*e*x^8+1/12*I*Pi*b*d*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2-1/12*I*Pi*b*d*x^6*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)-1/12*I*Pi*b*d*x^6*csgn(I*c*x^n)^3+1/12*I*Pi*b*d*x^6*csgn(I*c*x^n)^2*csgn(I*c)+1/6*ln
(c)*b*d*x^6-1/36*b*d*n*x^6+1/6*a*d*x^6

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Maxima [A]  time = 1.16571, size = 77, normalized size = 1.6 \begin{align*} -\frac{1}{64} \, b e n x^{8} + \frac{1}{8} \, b e x^{8} \log \left (c x^{n}\right ) + \frac{1}{8} \, a e x^{8} - \frac{1}{36} \, b d n x^{6} + \frac{1}{6} \, b d x^{6} \log \left (c x^{n}\right ) + \frac{1}{6} \, a d x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/64*b*e*n*x^8 + 1/8*b*e*x^8*log(c*x^n) + 1/8*a*e*x^8 - 1/36*b*d*n*x^6 + 1/6*b*d*x^6*log(c*x^n) + 1/6*a*d*x^6

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Fricas [A]  time = 1.30038, size = 181, normalized size = 3.77 \begin{align*} -\frac{1}{64} \,{\left (b e n - 8 \, a e\right )} x^{8} - \frac{1}{36} \,{\left (b d n - 6 \, a d\right )} x^{6} + \frac{1}{24} \,{\left (3 \, b e x^{8} + 4 \, b d x^{6}\right )} \log \left (c\right ) + \frac{1}{24} \,{\left (3 \, b e n x^{8} + 4 \, b d n x^{6}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/64*(b*e*n - 8*a*e)*x^8 - 1/36*(b*d*n - 6*a*d)*x^6 + 1/24*(3*b*e*x^8 + 4*b*d*x^6)*log(c) + 1/24*(3*b*e*n*x^8
+ 4*b*d*n*x^6)*log(x)

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Sympy [B]  time = 13.618, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{6}}{6} + \frac{a e x^{8}}{8} + \frac{b d n x^{6} \log{\left (x \right )}}{6} - \frac{b d n x^{6}}{36} + \frac{b d x^{6} \log{\left (c \right )}}{6} + \frac{b e n x^{8} \log{\left (x \right )}}{8} - \frac{b e n x^{8}}{64} + \frac{b e x^{8} \log{\left (c \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x**2+d)*(a+b*ln(c*x**n)),x)

[Out]

a*d*x**6/6 + a*e*x**8/8 + b*d*n*x**6*log(x)/6 - b*d*n*x**6/36 + b*d*x**6*log(c)/6 + b*e*n*x**8*log(x)/8 - b*e*
n*x**8/64 + b*e*x**8*log(c)/8

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Giac [A]  time = 1.32868, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{8} \, b n x^{8} e \log \left (x\right ) - \frac{1}{64} \, b n x^{8} e + \frac{1}{8} \, b x^{8} e \log \left (c\right ) + \frac{1}{8} \, a x^{8} e + \frac{1}{6} \, b d n x^{6} \log \left (x\right ) - \frac{1}{36} \, b d n x^{6} + \frac{1}{6} \, b d x^{6} \log \left (c\right ) + \frac{1}{6} \, a d x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/8*b*n*x^8*e*log(x) - 1/64*b*n*x^8*e + 1/8*b*x^8*e*log(c) + 1/8*a*x^8*e + 1/6*b*d*n*x^6*log(x) - 1/36*b*d*n*x
^6 + 1/6*b*d*x^6*log(c) + 1/6*a*d*x^6