3.189 \(\int x^4 (d+e x^2)^2 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=74 \[ \frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^2 n x^5-\frac {2}{49} b d e n x^7-\frac {1}{81} b e^2 n x^9 \]

[Out]

-1/25*b*d^2*n*x^5-2/49*b*d*e*n*x^7-1/81*b*e^2*n*x^9+1/315*(35*e^2*x^9+90*d*e*x^7+63*d^2*x^5)*(a+b*ln(c*x^n))

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Rubi [A]  time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {270, 2334} \[ \frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^2 n x^5-\frac {2}{49} b d e n x^7-\frac {1}{81} b e^2 n x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)^2*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^7)/49 - (b*e^2*n*x^9)/81 + ((63*d^2*x^5 + 90*d*e*x^7 + 35*e^2*x^9)*(a + b*Log
[c*x^n]))/315

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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])

Rubi steps

\begin {align*} \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {d^2 x^4}{5}+\frac {2}{7} d e x^6+\frac {e^2 x^8}{9}\right ) \, dx\\ &=-\frac {1}{25} b d^2 n x^5-\frac {2}{49} b d e n x^7-\frac {1}{81} b e^2 n x^9+\frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 95, normalized size = 1.28 \[ \frac {1}{5} d^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {2}{7} d e x^7 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} e^2 x^9 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^2 n x^5-\frac {2}{49} b d e n x^7-\frac {1}{81} b e^2 n x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)^2*(a + b*Log[c*x^n]),x]

[Out]

-1/25*(b*d^2*n*x^5) - (2*b*d*e*n*x^7)/49 - (b*e^2*n*x^9)/81 + (d^2*x^5*(a + b*Log[c*x^n]))/5 + (2*d*e*x^7*(a +
b*Log[c*x^n]))/7 + (e^2*x^9*(a + b*Log[c*x^n]))/9

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fricas [A]  time = 0.58, size = 118, normalized size = 1.59 \[ -\frac {1}{81} \, {\left (b e^{2} n - 9 \, a e^{2}\right )} x^{9} - \frac {2}{49} \, {\left (b d e n - 7 \, a d e\right )} x^{7} - \frac {1}{25} \, {\left (b d^{2} n - 5 \, a d^{2}\right )} x^{5} + \frac {1}{315} \, {\left (35 \, b e^{2} x^{9} + 90 \, b d e x^{7} + 63 \, b d^{2} x^{5}\right )} \log \relax (c) + \frac {1}{315} \, {\left (35 \, b e^{2} n x^{9} + 90 \, b d e n x^{7} + 63 \, b d^{2} n x^{5}\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/81*(b*e^2*n - 9*a*e^2)*x^9 - 2/49*(b*d*e*n - 7*a*d*e)*x^7 - 1/25*(b*d^2*n - 5*a*d^2)*x^5 + 1/315*(35*b*e^2*
x^9 + 90*b*d*e*x^7 + 63*b*d^2*x^5)*log(c) + 1/315*(35*b*e^2*n*x^9 + 90*b*d*e*n*x^7 + 63*b*d^2*n*x^5)*log(x)

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giac [A]  time = 0.39, size = 123, normalized size = 1.66 \[ \frac {1}{9} \, b n x^{9} e^{2} \log \relax (x) - \frac {1}{81} \, b n x^{9} e^{2} + \frac {1}{9} \, b x^{9} e^{2} \log \relax (c) + \frac {2}{7} \, b d n x^{7} e \log \relax (x) + \frac {1}{9} \, a x^{9} e^{2} - \frac {2}{49} \, b d n x^{7} e + \frac {2}{7} \, b d x^{7} e \log \relax (c) + \frac {2}{7} \, a d x^{7} e + \frac {1}{5} \, b d^{2} n x^{5} \log \relax (x) - \frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \relax (c) + \frac {1}{5} \, a d^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/9*b*n*x^9*e^2*log(x) - 1/81*b*n*x^9*e^2 + 1/9*b*x^9*e^2*log(c) + 2/7*b*d*n*x^7*e*log(x) + 1/9*a*x^9*e^2 - 2/
49*b*d*n*x^7*e + 2/7*b*d*x^7*e*log(c) + 2/7*a*d*x^7*e + 1/5*b*d^2*n*x^5*log(x) - 1/25*b*d^2*n*x^5 + 1/5*b*d^2*
x^5*log(c) + 1/5*a*d^2*x^5

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maple [C]  time = 0.22, size = 434, normalized size = 5.86 \[ -\frac {i \pi b \,e^{2} x^{9} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{18}+\frac {i \pi b \,e^{2} x^{9} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{18}+\frac {i \pi b \,e^{2} x^{9} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{18}-\frac {i \pi b \,e^{2} x^{9} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{18}-\frac {i \pi b d e \,x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{7}+\frac {i \pi b d e \,x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{7}+\frac {i \pi b d e \,x^{7} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{7}-\frac {i \pi b d e \,x^{7} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{7}-\frac {b \,e^{2} n \,x^{9}}{81}+\frac {b \,e^{2} x^{9} \ln \relax (c )}{9}+\frac {a \,e^{2} x^{9}}{9}-\frac {i \pi b \,d^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{10}+\frac {i \pi b \,d^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b \,d^{2} x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b \,d^{2} x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {2 b d e n \,x^{7}}{49}+\frac {2 b d e \,x^{7} \ln \relax (c )}{7}+\frac {2 a d e \,x^{7}}{7}-\frac {b \,d^{2} n \,x^{5}}{25}+\frac {b \,d^{2} x^{5} \ln \relax (c )}{5}+\frac {a \,d^{2} x^{5}}{5}+\frac {\left (35 e^{2} x^{4}+90 d e \,x^{2}+63 d^{2}\right ) b \,x^{5} \ln \left (x^{n}\right )}{315} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)^2*(b*ln(c*x^n)+a),x)

[Out]

1/315*b*x^5*(35*e^2*x^4+90*d*e*x^2+63*d^2)*ln(x^n)+1/18*I*Pi*b*e^2*x^9*csgn(I*x^n)*csgn(I*c*x^n)^2+1/10*I*Pi*b
*d^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)+1/7*I*Pi*b*d*e*x^7*csgn(I*x^n)*csgn(I*c*x^n)^2+1/10*I*Pi*b*d^2*x^5*csgn(I*x
^n)*csgn(I*c*x^n)^2+1/9*ln(c)*b*e^2*x^9-1/81*b*e^2*n*x^9+1/9*a*e^2*x^9-1/18*I*Pi*b*e^2*x^9*csgn(I*c*x^n)^3-1/7
*I*Pi*b*d*e*x^7*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/10*I*Pi*b*d^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/
10*I*Pi*b*d^2*x^5*csgn(I*c*x^n)^3+2/7*ln(c)*b*d*e*x^7-2/49*b*d*e*n*x^7+2/7*a*d*e*x^7+1/18*I*Pi*b*e^2*x^9*csgn(
I*c*x^n)^2*csgn(I*c)-1/7*I*Pi*b*d*e*x^7*csgn(I*c*x^n)^3+1/7*I*Pi*b*d*e*x^7*csgn(I*c*x^n)^2*csgn(I*c)-1/18*I*Pi
*b*e^2*x^9*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/5*ln(c)*b*d^2*x^5-1/25*b*d^2*n*x^5+1/5*a*d^2*x^5

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maxima [A]  time = 0.48, size = 100, normalized size = 1.35 \[ -\frac {1}{81} \, b e^{2} n x^{9} + \frac {1}{9} \, b e^{2} x^{9} \log \left (c x^{n}\right ) + \frac {1}{9} \, a e^{2} x^{9} - \frac {2}{49} \, b d e n x^{7} + \frac {2}{7} \, b d e x^{7} \log \left (c x^{n}\right ) + \frac {2}{7} \, a d e x^{7} - \frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/81*b*e^2*n*x^9 + 1/9*b*e^2*x^9*log(c*x^n) + 1/9*a*e^2*x^9 - 2/49*b*d*e*n*x^7 + 2/7*b*d*e*x^7*log(c*x^n) + 2
/7*a*d*e*x^7 - 1/25*b*d^2*n*x^5 + 1/5*b*d^2*x^5*log(c*x^n) + 1/5*a*d^2*x^5

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mupad [B]  time = 3.69, size = 82, normalized size = 1.11 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^5}{5}+\frac {2\,b\,d\,e\,x^7}{7}+\frac {b\,e^2\,x^9}{9}\right )+\frac {d^2\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {e^2\,x^9\,\left (9\,a-b\,n\right )}{81}+\frac {2\,d\,e\,x^7\,\left (7\,a-b\,n\right )}{49} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d + e*x^2)^2*(a + b*log(c*x^n)),x)

[Out]

log(c*x^n)*((b*d^2*x^5)/5 + (b*e^2*x^9)/9 + (2*b*d*e*x^7)/7) + (d^2*x^5*(5*a - b*n))/25 + (e^2*x^9*(9*a - b*n)
)/81 + (2*d*e*x^7*(7*a - b*n))/49

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sympy [B]  time = 14.50, size = 158, normalized size = 2.14 \[ \frac {a d^{2} x^{5}}{5} + \frac {2 a d e x^{7}}{7} + \frac {a e^{2} x^{9}}{9} + \frac {b d^{2} n x^{5} \log {\relax (x )}}{5} - \frac {b d^{2} n x^{5}}{25} + \frac {b d^{2} x^{5} \log {\relax (c )}}{5} + \frac {2 b d e n x^{7} \log {\relax (x )}}{7} - \frac {2 b d e n x^{7}}{49} + \frac {2 b d e x^{7} \log {\relax (c )}}{7} + \frac {b e^{2} n x^{9} \log {\relax (x )}}{9} - \frac {b e^{2} n x^{9}}{81} + \frac {b e^{2} x^{9} \log {\relax (c )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)**2*(a+b*ln(c*x**n)),x)

[Out]

a*d**2*x**5/5 + 2*a*d*e*x**7/7 + a*e**2*x**9/9 + b*d**2*n*x**5*log(x)/5 - b*d**2*n*x**5/25 + b*d**2*x**5*log(c
)/5 + 2*b*d*e*n*x**7*log(x)/7 - 2*b*d*e*n*x**7/49 + 2*b*d*e*x**7*log(c)/7 + b*e**2*n*x**9*log(x)/9 - b*e**2*n*
x**9/81 + b*e**2*x**9*log(c)/9

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