3.206 \(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=121 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n}{x}-3 b d e^2 n x-\frac {1}{9} b e^3 n x^3 \]

[Out]

-1/9*b*d^3*n/x^3-3*b*d^2*e*n/x-3*b*d*e^2*n*x-1/9*b*e^3*n*x^3-1/3*d^3*(a+b*ln(c*x^n))/x^3-3*d^2*e*(a+b*ln(c*x^n
))/x+3*d*e^2*x*(a+b*ln(c*x^n))+1/3*e^3*x^3*(a+b*ln(c*x^n))

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Rubi [A]  time = 0.09, antiderivative size = 91, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {270, 2334, 12} \[ -\frac {1}{3} \left (\frac {9 d^2 e}{x}+\frac {d^3}{x^3}-9 d e^2 x-e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n}{x}-\frac {b d^3 n}{9 x^3}-3 b d e^2 n x-\frac {1}{9} b e^3 n x^3 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-(b*d^3*n)/(9*x^3) - (3*b*d^2*e*n)/x - 3*b*d*e^2*n*x - (b*e^3*n*x^3)/9 - ((d^3/x^3 + (9*d^2*e)/x - 9*d*e^2*x -
e^3*x^3)*(a + b*Log[c*x^n]))/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {9 d^2 e}{x}-9 d e^2 x-e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{3} \left (9 d e^2-\frac {d^3}{x^4}-\frac {9 d^2 e}{x^2}+e^3 x^2\right ) \, dx\\ &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {9 d^2 e}{x}-9 d e^2 x-e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (9 d e^2-\frac {d^3}{x^4}-\frac {9 d^2 e}{x^2}+e^3 x^2\right ) \, dx\\ &=-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n}{x}-3 b d e^2 n x-\frac {1}{9} b e^3 n x^3-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {9 d^2 e}{x}-9 d e^2 x-e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 112, normalized size = 0.93 \[ -\frac {3 a \left (d^3+9 d^2 e x^2-9 d e^2 x^4-e^3 x^6\right )+3 b \left (d^3+9 d^2 e x^2-9 d e^2 x^4-e^3 x^6\right ) \log \left (c x^n\right )+b n \left (d^3+27 d^2 e x^2+27 d e^2 x^4+e^3 x^6\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-1/9*(3*a*(d^3 + 9*d^2*e*x^2 - 9*d*e^2*x^4 - e^3*x^6) + b*n*(d^3 + 27*d^2*e*x^2 + 27*d*e^2*x^4 + e^3*x^6) + 3*
b*(d^3 + 9*d^2*e*x^2 - 9*d*e^2*x^4 - e^3*x^6)*Log[c*x^n])/x^3

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fricas [A]  time = 0.47, size = 156, normalized size = 1.29 \[ -\frac {{\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} + b d^{3} n + 27 \, {\left (b d e^{2} n - a d e^{2}\right )} x^{4} + 3 \, a d^{3} + 27 \, {\left (b d^{2} e n + a d^{2} e\right )} x^{2} - 3 \, {\left (b e^{3} x^{6} + 9 \, b d e^{2} x^{4} - 9 \, b d^{2} e x^{2} - b d^{3}\right )} \log \relax (c) - 3 \, {\left (b e^{3} n x^{6} + 9 \, b d e^{2} n x^{4} - 9 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \relax (x)}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")

[Out]

-1/9*((b*e^3*n - 3*a*e^3)*x^6 + b*d^3*n + 27*(b*d*e^2*n - a*d*e^2)*x^4 + 3*a*d^3 + 27*(b*d^2*e*n + a*d^2*e)*x^
2 - 3*(b*e^3*x^6 + 9*b*d*e^2*x^4 - 9*b*d^2*e*x^2 - b*d^3)*log(c) - 3*(b*e^3*n*x^6 + 9*b*d*e^2*n*x^4 - 9*b*d^2*
e*n*x^2 - b*d^3*n)*log(x))/x^3

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giac [A]  time = 0.26, size = 166, normalized size = 1.37 \[ \frac {3 \, b n x^{6} e^{3} \log \relax (x) - b n x^{6} e^{3} + 3 \, b x^{6} e^{3} \log \relax (c) + 27 \, b d n x^{4} e^{2} \log \relax (x) + 3 \, a x^{6} e^{3} - 27 \, b d n x^{4} e^{2} + 27 \, b d x^{4} e^{2} \log \relax (c) - 27 \, b d^{2} n x^{2} e \log \relax (x) + 27 \, a d x^{4} e^{2} - 27 \, b d^{2} n x^{2} e - 27 \, b d^{2} x^{2} e \log \relax (c) - 27 \, a d^{2} x^{2} e - 3 \, b d^{3} n \log \relax (x) - b d^{3} n - 3 \, b d^{3} \log \relax (c) - 3 \, a d^{3}}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="giac")

[Out]

1/9*(3*b*n*x^6*e^3*log(x) - b*n*x^6*e^3 + 3*b*x^6*e^3*log(c) + 27*b*d*n*x^4*e^2*log(x) + 3*a*x^6*e^3 - 27*b*d*
n*x^4*e^2 + 27*b*d*x^4*e^2*log(c) - 27*b*d^2*n*x^2*e*log(x) + 27*a*d*x^4*e^2 - 27*b*d^2*n*x^2*e - 27*b*d^2*x^2
*e*log(c) - 27*a*d^2*x^2*e - 3*b*d^3*n*log(x) - b*d^3*n - 3*b*d^3*log(c) - 3*a*d^3)/x^3

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maple [C]  time = 0.25, size = 585, normalized size = 4.83 \[ -\frac {\left (-e^{3} x^{6}-9 d \,e^{2} x^{4}+9 d^{2} e \,x^{2}+d^{3}\right ) b \ln \left (x^{n}\right )}{3 x^{3}}-\frac {-6 a \,e^{3} x^{6}-54 b d \,e^{2} x^{4} \ln \relax (c )-54 a d \,e^{2} x^{4}+6 a \,d^{3}-6 b \,e^{3} x^{6} \ln \relax (c )+2 b \,d^{3} n +6 b \,d^{3} \ln \relax (c )+54 a \,d^{2} e \,x^{2}-3 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+54 b \,d^{2} e \,x^{2} \ln \relax (c )-27 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 b \,e^{3} n \,x^{6}+3 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-3 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-27 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-27 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+27 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+27 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+54 b d \,e^{2} n \,x^{4}+54 b \,d^{2} e n \,x^{2}+27 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-27 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+27 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{18 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(b*ln(c*x^n)+a)/x^4,x)

[Out]

-1/3*b*(-e^3*x^6-9*d*e^2*x^4+9*d^2*e*x^2+d^3)/x^3*ln(x^n)-1/18*(-6*a*e^3*x^6-54*b*d*e^2*x^4*ln(c)-54*a*d*e^2*x
^4+6*a*d^3-6*b*e^3*x^6*ln(c)+2*b*d^3*n+6*b*d^3*ln(c)+54*a*d^2*e*x^2+54*b*d^2*e*x^2*ln(c)-3*I*Pi*b*d^3*csgn(I*c
*x^n)^3+2*b*e^3*n*x^6+27*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+3*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+27*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-27*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)-27*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)-27*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+27*
I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^3+3*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*
c*x^n)^2+3*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-27*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3-3*I*Pi*b*d^3*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)+27*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^3-3*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b
*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2+54*b*d*e^2*n*x^4+54*b*d^2*e*n*x^2)/x^3

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maxima [A]  time = 0.50, size = 137, normalized size = 1.13 \[ -\frac {1}{9} \, b e^{3} n x^{3} + \frac {1}{3} \, b e^{3} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a e^{3} x^{3} - 3 \, b d e^{2} n x + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x - \frac {3 \, b d^{2} e n}{x} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{x} - \frac {3 \, a d^{2} e}{x} - \frac {b d^{3} n}{9 \, x^{3}} - \frac {b d^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="maxima")

[Out]

-1/9*b*e^3*n*x^3 + 1/3*b*e^3*x^3*log(c*x^n) + 1/3*a*e^3*x^3 - 3*b*d*e^2*n*x + 3*b*d*e^2*x*log(c*x^n) + 3*a*d*e
^2*x - 3*b*d^2*e*n/x - 3*b*d^2*e*log(c*x^n)/x - 3*a*d^2*e/x - 1/9*b*d^3*n/x^3 - 1/3*b*d^3*log(c*x^n)/x^3 - 1/3
*a*d^3/x^3

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mupad [B]  time = 3.53, size = 141, normalized size = 1.17 \[ \ln \left (c\,x^n\right )\,\left (\frac {\frac {8\,b\,e^3\,x^6}{3}+8\,b\,d\,e^2\,x^4}{x^3}-\frac {\frac {b\,d^3}{3}+3\,b\,d^2\,e\,x^2+5\,b\,d\,e^2\,x^4+\frac {7\,b\,e^3\,x^6}{3}}{x^3}\right )-\frac {a\,d^3+x^2\,\left (9\,a\,d^2\,e+9\,b\,d^2\,e\,n\right )+\frac {b\,d^3\,n}{3}}{3\,x^3}+\frac {e^3\,x^3\,\left (3\,a-b\,n\right )}{9}+3\,d\,e^2\,x\,\left (a-b\,n\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^3*(a + b*log(c*x^n)))/x^4,x)

[Out]

log(c*x^n)*(((8*b*e^3*x^6)/3 + 8*b*d*e^2*x^4)/x^3 - ((b*d^3)/3 + (7*b*e^3*x^6)/3 + 3*b*d^2*e*x^2 + 5*b*d*e^2*x
^4)/x^3) - (a*d^3 + x^2*(9*a*d^2*e + 9*b*d^2*e*n) + (b*d^3*n)/3)/(3*x^3) + (e^3*x^3*(3*a - b*n))/9 + 3*d*e^2*x
*(a - b*n)

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sympy [A]  time = 6.92, size = 202, normalized size = 1.67 \[ - \frac {a d^{3}}{3 x^{3}} - \frac {3 a d^{2} e}{x} + 3 a d e^{2} x + \frac {a e^{3} x^{3}}{3} - \frac {b d^{3} n \log {\relax (x )}}{3 x^{3}} - \frac {b d^{3} n}{9 x^{3}} - \frac {b d^{3} \log {\relax (c )}}{3 x^{3}} - \frac {3 b d^{2} e n \log {\relax (x )}}{x} - \frac {3 b d^{2} e n}{x} - \frac {3 b d^{2} e \log {\relax (c )}}{x} + 3 b d e^{2} n x \log {\relax (x )} - 3 b d e^{2} n x + 3 b d e^{2} x \log {\relax (c )} + \frac {b e^{3} n x^{3} \log {\relax (x )}}{3} - \frac {b e^{3} n x^{3}}{9} + \frac {b e^{3} x^{3} \log {\relax (c )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3*(a+b*ln(c*x**n))/x**4,x)

[Out]

-a*d**3/(3*x**3) - 3*a*d**2*e/x + 3*a*d*e**2*x + a*e**3*x**3/3 - b*d**3*n*log(x)/(3*x**3) - b*d**3*n/(9*x**3)
- b*d**3*log(c)/(3*x**3) - 3*b*d**2*e*n*log(x)/x - 3*b*d**2*e*n/x - 3*b*d**2*e*log(c)/x + 3*b*d*e**2*n*x*log(x
) - 3*b*d*e**2*n*x + 3*b*d*e**2*x*log(c) + b*e**3*n*x**3*log(x)/3 - b*e**3*n*x**3/9 + b*e**3*x**3*log(c)/3

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