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

Optimal. Leaf size=74 \[ \frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d^2 n x^6-\frac{1}{32} b d e n x^8-\frac{1}{100} b e^2 n x^{10} \]

[Out]

-(b*d^2*n*x^6)/36 - (b*d*e*n*x^8)/32 - (b*e^2*n*x^10)/100 + ((10*d^2*x^6 + 15*d*e*x^8 + 6*e^2*x^10)*(a + b*Log
[c*x^n]))/60

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Rubi [A]  time = 0.0865796, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 12, 14} \[ \frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d^2 n x^6-\frac{1}{32} b d e n x^8-\frac{1}{100} b e^2 n x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x^6)/36 - (b*d*e*n*x^8)/32 - (b*e^2*n*x^10)/100 + ((10*d^2*x^6 + 15*d*e*x^8 + 6*e^2*x^10)*(a + b*Log
[c*x^n]))/60

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]]

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]]

Rubi steps

\begin{align*} \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{60} x^5 \left (10 d^2+15 d e x^2+6 e^2 x^4\right ) \, dx\\ &=\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int x^5 \left (10 d^2+15 d e x^2+6 e^2 x^4\right ) \, dx\\ &=\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \left (10 d^2 x^5+15 d e x^7+6 e^2 x^9\right ) \, dx\\ &=-\frac{1}{36} b d^2 n x^6-\frac{1}{32} b d e n x^8-\frac{1}{100} b e^2 n x^{10}+\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0398215, size = 84, normalized size = 1.14 \[ \frac{x^6 \left (1200 d^2 \left (a+b \log \left (c x^n\right )\right )+1800 d e x^2 \left (a+b \log \left (c x^n\right )\right )+720 e^2 x^4 \left (a+b \log \left (c x^n\right )\right )-200 b d^2 n-225 b d e n x^2-72 b e^2 n x^4\right )}{7200} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^6*(-200*b*d^2*n - 225*b*d*e*n*x^2 - 72*b*e^2*n*x^4 + 1200*d^2*(a + b*Log[c*x^n]) + 1800*d*e*x^2*(a + b*Log[
c*x^n]) + 720*e^2*x^4*(a + b*Log[c*x^n])))/7200

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Maple [C]  time = 0.195, size = 434, normalized size = 5.9 \begin{align*}{\frac{b{x}^{6} \left ( 6\,{e}^{2}{x}^{4}+15\,de{x}^{2}+10\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{60}}-{\frac{i}{8}}\pi \,bde{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{8}}\pi \,bde{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{8}}\pi \,bde{x}^{8}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{10}}{10}}-{\frac{b{e}^{2}n{x}^{10}}{100}}+{\frac{a{e}^{2}{x}^{10}}{10}}-{\frac{i}{8}}\pi \,bde{x}^{8}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) bde{x}^{8}}{4}}-{\frac{bden{x}^{8}}{32}}+{\frac{ade{x}^{8}}{4}}+{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{d}^{2}{x}^{6}}{6}}-{\frac{b{d}^{2}n{x}^{6}}{36}}+{\frac{a{d}^{2}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*b*x^6*(6*e^2*x^4+15*d*e*x^2+10*d^2)*ln(x^n)-1/8*I*Pi*b*d*e*x^8*csgn(I*c*x^n)^3+1/12*I*Pi*b*d^2*x^6*csgn(I
*x^n)*csgn(I*c*x^n)^2+1/8*I*Pi*b*d*e*x^8*csgn(I*c*x^n)^2*csgn(I*c)+1/8*I*Pi*b*d*e*x^8*csgn(I*x^n)*csgn(I*c*x^n
)^2+1/10*ln(c)*b*e^2*x^10-1/100*b*e^2*n*x^10+1/10*a*e^2*x^10-1/8*I*Pi*b*d*e*x^8*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)-1/20*I*Pi*b*e^2*x^10*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/20*I*Pi*b*e^2*x^10*csgn(I*c*x^n)^2*csgn(I*c)+
1/20*I*Pi*b*e^2*x^10*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*ln(c)*b*d*e*x^8-1/32*b*d*e*n*x^8+1/4*a*d*e*x^8+1/12*I*Pi*
b*d^2*x^6*csgn(I*c*x^n)^2*csgn(I*c)-1/12*I*Pi*b*d^2*x^6*csgn(I*c*x^n)^3-1/12*I*Pi*b*d^2*x^6*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)-1/20*I*Pi*b*e^2*x^10*csgn(I*c*x^n)^3+1/6*ln(c)*b*d^2*x^6-1/36*b*d^2*n*x^6+1/6*a*d^2*x^6

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Maxima [A]  time = 1.05436, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{100} \, b e^{2} n x^{10} + \frac{1}{10} \, b e^{2} x^{10} \log \left (c x^{n}\right ) + \frac{1}{10} \, a e^{2} x^{10} - \frac{1}{32} \, b d e n x^{8} + \frac{1}{4} \, b d e x^{8} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d e x^{8} - \frac{1}{36} \, b d^{2} n x^{6} + \frac{1}{6} \, b d^{2} x^{6} \log \left (c x^{n}\right ) + \frac{1}{6} \, a d^{2} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/100*b*e^2*n*x^10 + 1/10*b*e^2*x^10*log(c*x^n) + 1/10*a*e^2*x^10 - 1/32*b*d*e*n*x^8 + 1/4*b*d*e*x^8*log(c*x^
n) + 1/4*a*d*e*x^8 - 1/36*b*d^2*n*x^6 + 1/6*b*d^2*x^6*log(c*x^n) + 1/6*a*d^2*x^6

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Fricas [A]  time = 1.33028, size = 297, normalized size = 4.01 \begin{align*} -\frac{1}{100} \,{\left (b e^{2} n - 10 \, a e^{2}\right )} x^{10} - \frac{1}{32} \,{\left (b d e n - 8 \, a d e\right )} x^{8} - \frac{1}{36} \,{\left (b d^{2} n - 6 \, a d^{2}\right )} x^{6} + \frac{1}{60} \,{\left (6 \, b e^{2} x^{10} + 15 \, b d e x^{8} + 10 \, b d^{2} x^{6}\right )} \log \left (c\right ) + \frac{1}{60} \,{\left (6 \, b e^{2} n x^{10} + 15 \, b d e n x^{8} + 10 \, b d^{2} 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)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/100*(b*e^2*n - 10*a*e^2)*x^10 - 1/32*(b*d*e*n - 8*a*d*e)*x^8 - 1/36*(b*d^2*n - 6*a*d^2)*x^6 + 1/60*(6*b*e^2
*x^10 + 15*b*d*e*x^8 + 10*b*d^2*x^6)*log(c) + 1/60*(6*b*e^2*n*x^10 + 15*b*d*e*n*x^8 + 10*b*d^2*n*x^6)*log(x)

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Sympy [B]  time = 34.1154, size = 151, normalized size = 2.04 \begin{align*} \frac{a d^{2} x^{6}}{6} + \frac{a d e x^{8}}{4} + \frac{a e^{2} x^{10}}{10} + \frac{b d^{2} n x^{6} \log{\left (x \right )}}{6} - \frac{b d^{2} n x^{6}}{36} + \frac{b d^{2} x^{6} \log{\left (c \right )}}{6} + \frac{b d e n x^{8} \log{\left (x \right )}}{4} - \frac{b d e n x^{8}}{32} + \frac{b d e x^{8} \log{\left (c \right )}}{4} + \frac{b e^{2} n x^{10} \log{\left (x \right )}}{10} - \frac{b e^{2} n x^{10}}{100} + \frac{b e^{2} x^{10} \log{\left (c \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x**6/6 + a*d*e*x**8/4 + a*e**2*x**10/10 + b*d**2*n*x**6*log(x)/6 - b*d**2*n*x**6/36 + b*d**2*x**6*log(c
)/6 + b*d*e*n*x**8*log(x)/4 - b*d*e*n*x**8/32 + b*d*e*x**8*log(c)/4 + b*e**2*n*x**10*log(x)/10 - b*e**2*n*x**1
0/100 + b*e**2*x**10*log(c)/10

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Giac [A]  time = 1.30353, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{10} \, b n x^{10} e^{2} \log \left (x\right ) - \frac{1}{100} \, b n x^{10} e^{2} + \frac{1}{10} \, b x^{10} e^{2} \log \left (c\right ) + \frac{1}{4} \, b d n x^{8} e \log \left (x\right ) + \frac{1}{10} \, a x^{10} e^{2} - \frac{1}{32} \, b d n x^{8} e + \frac{1}{4} \, b d x^{8} e \log \left (c\right ) + \frac{1}{4} \, a d x^{8} e + \frac{1}{6} \, b d^{2} n x^{6} \log \left (x\right ) - \frac{1}{36} \, b d^{2} n x^{6} + \frac{1}{6} \, b d^{2} x^{6} \log \left (c\right ) + \frac{1}{6} \, a d^{2} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b*n*x^10*e^2*log(x) - 1/100*b*n*x^10*e^2 + 1/10*b*x^10*e^2*log(c) + 1/4*b*d*n*x^8*e*log(x) + 1/10*a*x^10*
e^2 - 1/32*b*d*n*x^8*e + 1/4*b*d*x^8*e*log(c) + 1/4*a*d*x^8*e + 1/6*b*d^2*n*x^6*log(x) - 1/36*b*d^2*n*x^6 + 1/
6*b*d^2*x^6*log(c) + 1/6*a*d^2*x^6