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

Optimal. Leaf size=82 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 65, normalized size of antiderivative = 0.79, 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} \[ -\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^2-\frac {d^2}{3 x^4}-\frac {2 d e}{x^2}\right ) \, dx\\ &=-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 80, normalized size = 0.98 \[ -\frac {3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )+3 b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \log \left (c x^n\right )+b n \left (d^2+18 d e x^2+9 e^2 x^4\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4) + b*n*(d^2 + 18*d*e*x^2 + 9*e^2*x^4) + 3*b*(d^2 + 6*d*e*x^2 - 3*e^2*x^
4)*Log[c*x^n])/x^3

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 110, normalized size = 1.34 \[ -\frac {9 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + b d^{2} n + 3 \, a d^{2} + 18 \, {\left (b d e n + a d e\right )} x^{2} - 3 \, {\left (3 \, b e^{2} x^{4} - 6 \, b d e x^{2} - b d^{2}\right )} \log \relax (c) - 3 \, {\left (3 \, b e^{2} n x^{4} - 6 \, b d e n x^{2} - b d^{2} n\right )} \log \relax (x)}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(9*(b*e^2*n - a*e^2)*x^4 + b*d^2*n + 3*a*d^2 + 18*(b*d*e*n + a*d*e)*x^2 - 3*(3*b*e^2*x^4 - 6*b*d*e*x^2 -
b*d^2)*log(c) - 3*(3*b*e^2*n*x^4 - 6*b*d*e*n*x^2 - b*d^2*n)*log(x))/x^3

________________________________________________________________________________________

giac [A]  time = 0.28, size = 116, normalized size = 1.41 \[ \frac {9 \, b n x^{4} e^{2} \log \relax (x) - 9 \, b n x^{4} e^{2} + 9 \, b x^{4} e^{2} \log \relax (c) - 18 \, b d n x^{2} e \log \relax (x) + 9 \, a x^{4} e^{2} - 18 \, b d n x^{2} e - 18 \, b d x^{2} e \log \relax (c) - 18 \, a d x^{2} e - 3 \, b d^{2} n \log \relax (x) - b d^{2} n - 3 \, b d^{2} \log \relax (c) - 3 \, a d^{2}}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(9*b*n*x^4*e^2*log(x) - 9*b*n*x^4*e^2 + 9*b*x^4*e^2*log(c) - 18*b*d*n*x^2*e*log(x) + 9*a*x^4*e^2 - 18*b*d*
n*x^2*e - 18*b*d*x^2*e*log(c) - 18*a*d*x^2*e - 3*b*d^2*n*log(x) - b*d^2*n - 3*b*d^2*log(c) - 3*a*d^2)/x^3

________________________________________________________________________________________

maple [C]  time = 0.23, size = 417, normalized size = 5.09 \[ -\frac {\left (-3 e^{2} x^{4}+6 d e \,x^{2}+d^{2}\right ) b \ln \left (x^{n}\right )}{3 x^{3}}-\frac {9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 b \,e^{2} n \,x^{4}-18 b \,e^{2} x^{4} \ln \relax (c )-18 a \,e^{2} x^{4}-3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+36 b d e n \,x^{2}+36 b d e \,x^{2} \ln \relax (c )+36 a d e \,x^{2}+2 b \,d^{2} n +6 b \,d^{2} \ln \relax (c )+6 a \,d^{2}}{18 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*(-3*e^2*x^4+6*d*e*x^2+d^2)/x^3*ln(x^n)-1/18*(18*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+9*I*Pi*b*e^2
*x^4*csgn(I*c*x^n)^3+9*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-3*I*Pi*b*d^2*csgn(I*c*x^n)^3-9*I*Pi*
b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+18*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b*d^2*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)-18*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-18*b*e^2*x^4*ln(c)-9*I*Pi*b*e^2*x^4
*csgn(I*c*x^n)^2*csgn(I*c)+3*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^3+3*I*Pi*b*d
^2*csgn(I*x^n)*csgn(I*c*x^n)^2+18*b*e^2*n*x^4-18*a*e^2*x^4+36*b*d*e*x^2*ln(c)+36*b*d*e*n*x^2+36*a*d*e*x^2+6*b*
d^2*ln(c)+2*b*d^2*n+6*a*d^2)/x^3

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 92, normalized size = 1.12 \[ -b e^{2} n x + b e^{2} x \log \left (c x^{n}\right ) + a e^{2} x - \frac {2 \, b d e n}{x} - \frac {2 \, b d e \log \left (c x^{n}\right )}{x} - \frac {2 \, a d e}{x} - \frac {b d^{2} n}{9 \, x^{3}} - \frac {b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.48, size = 90, normalized size = 1.10 \[ e^2\,x\,\left (a-b\,n\right )-\frac {x^2\,\left (6\,a\,d\,e+6\,b\,d\,e\,n\right )+a\,d^2+\frac {b\,d^2\,n}{3}}{3\,x^3}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^2}{3}+2\,b\,d\,e\,x^2+\frac {5\,b\,e^2\,x^4}{3}}{x^3}-\frac {8\,b\,e^2\,x}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 2.76, size = 131, normalized size = 1.60 \[ - \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - \frac {b d^{2} n \log {\relax (x )}}{3 x^{3}} - \frac {b d^{2} n}{9 x^{3}} - \frac {b d^{2} \log {\relax (c )}}{3 x^{3}} - \frac {2 b d e n \log {\relax (x )}}{x} - \frac {2 b d e n}{x} - \frac {2 b d e \log {\relax (c )}}{x} + b e^{2} n x \log {\relax (x )} - b e^{2} n x + b e^{2} x \log {\relax (c )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________