3.2.83 \(\int x^5 (d+e x^2)^2 (a+b \log (c x^n)) \, dx\) [183]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, 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 = {272, 45, 2371, 12, 14} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36*(b*d^2*n*x^6) - (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*L
og[c*x^n]))/60

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 45

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 272

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 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^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*}

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Mathematica [A]
time = 0.03, size = 84, normalized size = 1.14 \begin {gather*} \frac {x^6 \left (-200 b d^2 n-225 b d e n x^2-72 b e^2 n x^4+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 )\right )}{7200} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.02, size = 434, normalized size = 5.86

method	result	size
risch	\(\frac {b \,x^{6} \left (6 e^{2} x^{4}+15 d e \,x^{2}+10 d^{2}\right ) \ln \left (x^{n}\right )}{60}+\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{20}-\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{20}+\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}+\frac {\ln \left (c \right ) b \,e^{2} x^{10}}{10}-\frac {b \,e^{2} n \,x^{10}}{100}+\frac {x^{10} a \,e^{2}}{10}+\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{12}-\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{20}+\frac {\ln \left (c \right ) b d e \,x^{8}}{4}-\frac {b d e n \,x^{8}}{32}+\frac {a d e \,x^{8}}{4}-\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{12}+\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{20}+\frac {\ln \left (c \right ) b \,d^{2} x^{6}}{6}-\frac {b \,d^{2} n \,x^{6}}{36}+\frac {x^{6} a \,d^{2}}{6}\)	\(434\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 100, normalized size = 1.35 \begin {gather*} -\frac {1}{100} \, b n x^{10} e^{2} + \frac {1}{10} \, b x^{10} e^{2} \log \left (c x^{n}\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 x^{n}\right ) + \frac {1}{4} \, a d x^{8} e - \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 {gather*}

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*n*x^10*e^2 + 1/10*b*x^10*e^2*log(c*x^n) + 1/10*a*x^10*e^2 - 1/32*b*d*n*x^8*e + 1/4*b*d*x^8*e*log(c*x^
n) + 1/4*a*d*x^8*e - 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 = 0.38, size = 114, normalized size = 1.54 \begin {gather*} -\frac {1}{100} \, {\left (b n - 10 \, a\right )} x^{10} e^{2} - \frac {1}{32} \, {\left (b d n - 8 \, a d\right )} x^{8} e - \frac {1}{36} \, {\left (b d^{2} n - 6 \, a d^{2}\right )} x^{6} + \frac {1}{60} \, {\left (6 \, b x^{10} e^{2} + 15 \, b d x^{8} e + 10 \, b d^{2} x^{6}\right )} \log \left (c\right ) + \frac {1}{60} \, {\left (6 \, b n x^{10} e^{2} + 15 \, b d n x^{8} e + 10 \, b d^{2} n x^{6}\right )} \log \left (x\right ) \end {gather*}

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

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Sympy [A]
time = 2.48, size = 116, normalized size = 1.57 \begin {gather*} \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}}{36} + \frac {b d^{2} x^{6} \log {\left (c x^{n} \right )}}{6} - \frac {b d e n x^{8}}{32} + \frac {b d e x^{8} \log {\left (c x^{n} \right )}}{4} - \frac {b e^{2} n x^{10}}{100} + \frac {b e^{2} x^{10} \log {\left (c x^{n} \right )}}{10} \end {gather*}

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**6/36 + b*d**2*x**6*log(c*x**n)/6 - b*d*e*n*x**8/3
2 + b*d*e*x**8*log(c*x**n)/4 - b*e**2*n*x**10/100 + b*e**2*x**10*log(c*x**n)/10

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Giac [A]
time = 4.84, size = 123, normalized size = 1.66 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 3.70, size = 82, normalized size = 1.11 \begin {gather*} \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^6}{6}+\frac {b\,d\,e\,x^8}{4}+\frac {b\,e^2\,x^{10}}{10}\right )+\frac {d^2\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {e^2\,x^{10}\,\left (10\,a-b\,n\right )}{100}+\frac {d\,e\,x^8\,\left (8\,a-b\,n\right )}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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