3.203 \(\int x^2 (d+e x^2)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=100 \[ \frac {1}{315} \left (105 d^3 x^3+189 d^2 e x^5+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^3 n x^3-\frac {3}{25} b d^2 e n x^5-\frac {3}{49} b d e^2 n x^7-\frac {1}{81} b e^3 n x^9 \]

[Out]

-1/9*b*d^3*n*x^3-3/25*b*d^2*e*n*x^5-3/49*b*d*e^2*n*x^7-1/81*b*e^3*n*x^9+1/315*(35*e^3*x^9+135*d*e^2*x^7+189*d^
2*e*x^5+105*d^3*x^3)*(a+b*ln(c*x^n))

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Rubi [A]  time = 0.09, antiderivative size = 100, 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} \[ \frac {1}{315} \left (189 d^2 e x^5+105 d^3 x^3+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{25} b d^2 e n x^5-\frac {1}{9} b d^3 n x^3-\frac {3}{49} b d e^2 n x^7-\frac {1}{81} b e^3 n x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^3)/9 - (3*b*d^2*e*n*x^5)/25 - (3*b*d*e^2*n*x^7)/49 - (b*e^3*n*x^9)/81 + ((105*d^3*x^3 + 189*d^2*e*
x^5 + 135*d*e^2*x^7 + 35*e^3*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^2 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{315} \left (105 d^3 x^3+189 d^2 e x^5+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {d^3 x^2}{3}+\frac {3}{5} d^2 e x^4+\frac {3}{7} d e^2 x^6+\frac {e^3 x^8}{9}\right ) \, dx\\ &=-\frac {1}{9} b d^3 n x^3-\frac {3}{25} b d^2 e n x^5-\frac {3}{49} b d e^2 n x^7-\frac {1}{81} b e^3 n x^9+\frac {1}{315} \left (105 d^3 x^3+189 d^2 e x^5+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 133, normalized size = 1.33 \[ \frac {1}{3} d^3 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d^2 e x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{7} d e^2 x^7 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} e^3 x^9 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^3 n x^3-\frac {3}{25} b d^2 e n x^5-\frac {3}{49} b d e^2 n x^7-\frac {1}{81} b e^3 n x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-1/9*(b*d^3*n*x^3) - (3*b*d^2*e*n*x^5)/25 - (3*b*d*e^2*n*x^7)/49 - (b*e^3*n*x^9)/81 + (d^3*x^3*(a + b*Log[c*x^
n]))/3 + (3*d^2*e*x^5*(a + b*Log[c*x^n]))/5 + (3*d*e^2*x^7*(a + b*Log[c*x^n]))/7 + (e^3*x^9*(a + b*Log[c*x^n])
)/9

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fricas [A]  time = 0.87, size = 167, normalized size = 1.67 \[ -\frac {1}{81} \, {\left (b e^{3} n - 9 \, a e^{3}\right )} x^{9} - \frac {3}{49} \, {\left (b d e^{2} n - 7 \, a d e^{2}\right )} x^{7} - \frac {3}{25} \, {\left (b d^{2} e n - 5 \, a d^{2} e\right )} x^{5} - \frac {1}{9} \, {\left (b d^{3} n - 3 \, a d^{3}\right )} x^{3} + \frac {1}{315} \, {\left (35 \, b e^{3} x^{9} + 135 \, b d e^{2} x^{7} + 189 \, b d^{2} e x^{5} + 105 \, b d^{3} x^{3}\right )} \log \relax (c) + \frac {1}{315} \, {\left (35 \, b e^{3} n x^{9} + 135 \, b d e^{2} n x^{7} + 189 \, b d^{2} e n x^{5} + 105 \, b d^{3} n x^{3}\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/81*(b*e^3*n - 9*a*e^3)*x^9 - 3/49*(b*d*e^2*n - 7*a*d*e^2)*x^7 - 3/25*(b*d^2*e*n - 5*a*d^2*e)*x^5 - 1/9*(b*d
^3*n - 3*a*d^3)*x^3 + 1/315*(35*b*e^3*x^9 + 135*b*d*e^2*x^7 + 189*b*d^2*e*x^5 + 105*b*d^3*x^3)*log(c) + 1/315*
(35*b*e^3*n*x^9 + 135*b*d*e^2*n*x^7 + 189*b*d^2*e*n*x^5 + 105*b*d^3*n*x^3)*log(x)

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giac [A]  time = 0.32, size = 173, normalized size = 1.73 \[ \frac {1}{9} \, b n x^{9} e^{3} \log \relax (x) - \frac {1}{81} \, b n x^{9} e^{3} + \frac {1}{9} \, b x^{9} e^{3} \log \relax (c) + \frac {3}{7} \, b d n x^{7} e^{2} \log \relax (x) + \frac {1}{9} \, a x^{9} e^{3} - \frac {3}{49} \, b d n x^{7} e^{2} + \frac {3}{7} \, b d x^{7} e^{2} \log \relax (c) + \frac {3}{5} \, b d^{2} n x^{5} e \log \relax (x) + \frac {3}{7} \, a d x^{7} e^{2} - \frac {3}{25} \, b d^{2} n x^{5} e + \frac {3}{5} \, b d^{2} x^{5} e \log \relax (c) + \frac {3}{5} \, a d^{2} x^{5} e + \frac {1}{3} \, b d^{3} n x^{3} \log \relax (x) - \frac {1}{9} \, b d^{3} n x^{3} + \frac {1}{3} \, b d^{3} x^{3} \log \relax (c) + \frac {1}{3} \, a d^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/9*b*n*x^9*e^3*log(x) - 1/81*b*n*x^9*e^3 + 1/9*b*x^9*e^3*log(c) + 3/7*b*d*n*x^7*e^2*log(x) + 1/9*a*x^9*e^3 -
3/49*b*d*n*x^7*e^2 + 3/7*b*d*x^7*e^2*log(c) + 3/5*b*d^2*n*x^5*e*log(x) + 3/7*a*d*x^7*e^2 - 3/25*b*d^2*n*x^5*e
+ 3/5*b*d^2*x^5*e*log(c) + 3/5*a*d^2*x^5*e + 1/3*b*d^3*n*x^3*log(x) - 1/9*b*d^3*n*x^3 + 1/3*b*d^3*x^3*log(c) +
1/3*a*d^3*x^3

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maple [C]  time = 0.22, size = 602, normalized size = 6.02 \[ \frac {a \,d^{3} x^{3}}{3}+\frac {3 a \,d^{2} e \,x^{5}}{5}+\frac {3 a d \,e^{2} x^{7}}{7}+\frac {3 b \,d^{2} e \,x^{5} \ln \relax (c )}{5}+\frac {3 b d \,e^{2} x^{7} \ln \relax (c )}{7}+\frac {a \,e^{3} x^{9}}{9}+\frac {b \,d^{3} x^{3} \ln \relax (c )}{3}+\frac {b \,e^{3} x^{9} \ln \relax (c )}{9}+\frac {\left (35 e^{3} x^{6}+135 d \,e^{2} x^{4}+189 d^{2} e \,x^{2}+105 d^{3}\right ) b \,x^{3} \ln \left (x^{n}\right )}{315}+\frac {3 i \pi b d \,e^{2} x^{7} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}+\frac {3 i \pi b \,d^{2} e \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {b \,d^{3} n \,x^{3}}{9}+\frac {3 i \pi b \,d^{2} e \,x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}-\frac {i \pi b \,e^{3} 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 {3 i \pi b d \,e^{2} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {3 i \pi b d \,e^{2} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{14}-\frac {3 i \pi b \,d^{2} e \,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 {b \,e^{3} n \,x^{9}}{81}-\frac {3 i \pi b d \,e^{2} x^{7} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{14}-\frac {3 i \pi b \,d^{2} e \,x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,e^{3} x^{9} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{18}+\frac {i \pi b \,e^{3} x^{9} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{18}-\frac {i \pi b \,e^{3} x^{9} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{18}-\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {3 b d \,e^{2} n \,x^{7}}{49}-\frac {3 b \,d^{2} e n \,x^{5}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x^2+d)^3*(b*ln(c*x^n)+a),x)

[Out]

1/3*a*d^3*x^3+3/5*a*d^2*e*x^5+3/7*a*d*e^2*x^7+3/5*b*d^2*e*x^5*ln(c)+3/7*ln(c)*b*d*e^2*x^7+1/9*a*e^3*x^9+1/3*b*
d^3*x^3*ln(c)+1/9*ln(c)*b*e^3*x^9+1/315*b*x^3*(35*e^3*x^6+135*d*e^2*x^4+189*d^2*e*x^2+105*d^3)*ln(x^n)+3/14*I*
Pi*b*d*e^2*x^7*csgn(I*x^n)*csgn(I*c*x^n)^2-1/9*b*d^3*n*x^3-3/10*I*Pi*b*d^2*e*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)+3/14*I*Pi*b*d*e^2*x^7*csgn(I*c*x^n)^2*csgn(I*c)-1/6*I*Pi*b*d^3*x^3*csgn(I*c*x^n)^3-1/18*I*Pi*b*e^3*x^9*
csgn(I*c*x^n)^3-3/14*I*Pi*b*d*e^2*x^7*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/81*b*e^3*n*x^9-1/6*I*Pi*b*d^3*x^3*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/10*I*Pi*b*d^2*e*x^5*csgn(I*c*x^n)^2*csgn(I*c)-1/18*I*Pi*b*e^3*x^9*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/10*I*Pi*b*d^2*e*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-3/10*I*Pi*b*d^2*e*x^5*csgn(I*c
*x^n)^3+1/6*I*Pi*b*d^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I*Pi*b*d^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/18*I*Pi*
b*e^3*x^9*csgn(I*c*x^n)^2*csgn(I*c)-3/49*b*d*e^2*n*x^7-3/14*I*Pi*b*d*e^2*x^7*csgn(I*c*x^n)^3+1/18*I*Pi*b*e^3*x
^9*csgn(I*x^n)*csgn(I*c*x^n)^2-3/25*b*d^2*e*n*x^5

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maxima [A]  time = 0.54, size = 143, normalized size = 1.43 \[ -\frac {1}{81} \, b e^{3} n x^{9} + \frac {1}{9} \, b e^{3} x^{9} \log \left (c x^{n}\right ) + \frac {1}{9} \, a e^{3} x^{9} - \frac {3}{49} \, b d e^{2} n x^{7} + \frac {3}{7} \, b d e^{2} x^{7} \log \left (c x^{n}\right ) + \frac {3}{7} \, a d e^{2} x^{7} - \frac {3}{25} \, b d^{2} e n x^{5} + \frac {3}{5} \, b d^{2} e x^{5} \log \left (c x^{n}\right ) + \frac {3}{5} \, a d^{2} e x^{5} - \frac {1}{9} \, b d^{3} n x^{3} + \frac {1}{3} \, b d^{3} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/81*b*e^3*n*x^9 + 1/9*b*e^3*x^9*log(c*x^n) + 1/9*a*e^3*x^9 - 3/49*b*d*e^2*n*x^7 + 3/7*b*d*e^2*x^7*log(c*x^n)
+ 3/7*a*d*e^2*x^7 - 3/25*b*d^2*e*n*x^5 + 3/5*b*d^2*e*x^5*log(c*x^n) + 3/5*a*d^2*e*x^5 - 1/9*b*d^3*n*x^3 + 1/3
*b*d^3*x^3*log(c*x^n) + 1/3*a*d^3*x^3

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mupad [B]  time = 3.63, size = 113, normalized size = 1.13 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^3\,x^3}{3}+\frac {3\,b\,d^2\,e\,x^5}{5}+\frac {3\,b\,d\,e^2\,x^7}{7}+\frac {b\,e^3\,x^9}{9}\right )+\frac {d^3\,x^3\,\left (3\,a-b\,n\right )}{9}+\frac {e^3\,x^9\,\left (9\,a-b\,n\right )}{81}+\frac {3\,d^2\,e\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {3\,d\,e^2\,x^7\,\left (7\,a-b\,n\right )}{49} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d + e*x^2)^3*(a + b*log(c*x^n)),x)

[Out]

log(c*x^n)*((b*d^3*x^3)/3 + (b*e^3*x^9)/9 + (3*b*d^2*e*x^5)/5 + (3*b*d*e^2*x^7)/7) + (d^3*x^3*(3*a - b*n))/9 +
(e^3*x^9*(9*a - b*n))/81 + (3*d^2*e*x^5*(5*a - b*n))/25 + (3*d*e^2*x^7*(7*a - b*n))/49

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sympy [B]  time = 15.40, size = 230, normalized size = 2.30 \[ \frac {a d^{3} x^{3}}{3} + \frac {3 a d^{2} e x^{5}}{5} + \frac {3 a d e^{2} x^{7}}{7} + \frac {a e^{3} x^{9}}{9} + \frac {b d^{3} n x^{3} \log {\relax (x )}}{3} - \frac {b d^{3} n x^{3}}{9} + \frac {b d^{3} x^{3} \log {\relax (c )}}{3} + \frac {3 b d^{2} e n x^{5} \log {\relax (x )}}{5} - \frac {3 b d^{2} e n x^{5}}{25} + \frac {3 b d^{2} e x^{5} \log {\relax (c )}}{5} + \frac {3 b d e^{2} n x^{7} \log {\relax (x )}}{7} - \frac {3 b d e^{2} n x^{7}}{49} + \frac {3 b d e^{2} x^{7} \log {\relax (c )}}{7} + \frac {b e^{3} n x^{9} \log {\relax (x )}}{9} - \frac {b e^{3} n x^{9}}{81} + \frac {b e^{3} x^{9} \log {\relax (c )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(e*x**2+d)**3*(a+b*ln(c*x**n)),x)

[Out]

a*d**3*x**3/3 + 3*a*d**2*e*x**5/5 + 3*a*d*e**2*x**7/7 + a*e**3*x**9/9 + b*d**3*n*x**3*log(x)/3 - b*d**3*n*x**3
/9 + b*d**3*x**3*log(c)/3 + 3*b*d**2*e*n*x**5*log(x)/5 - 3*b*d**2*e*n*x**5/25 + 3*b*d**2*e*x**5*log(c)/5 + 3*b
*d*e**2*n*x**7*log(x)/7 - 3*b*d*e**2*n*x**7/49 + 3*b*d*e**2*x**7*log(c)/7 + b*e**3*n*x**9*log(x)/9 - b*e**3*n*
x**9/81 + b*e**3*x**9*log(c)/9

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