3.176 \(\int \frac {(d+e x^2) (a+b \log (c x^n))}{x^5} \, dx\)

Optimal. Leaf size=57 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b d n}{16 x^4}-\frac {b e n}{4 x^2} \]

[Out]

-1/16*b*d*n/x^4-1/4*b*e*n/x^2-1/4*d*(a+b*ln(c*x^n))/x^4-1/2*e*(a+b*ln(c*x^n))/x^2

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {14, 2334, 12} \[ -\frac {1}{4} \left (\frac {d}{x^4}+\frac {2 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{16 x^4}-\frac {b e n}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n)/(16*x^4) - (b*e*n)/(4*x^2) - ((d/x^4 + (2*e)/x^2)*(a + b*Log[c*x^n]))/4

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 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 ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac {1}{4} \left (\frac {d}{x^4}+\frac {2 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d-2 e x^2}{4 x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d}{x^4}+\frac {2 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \frac {-d-2 e x^2}{x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d}{x^4}+\frac {2 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (-\frac {d}{x^5}-\frac {2 e}{x^3}\right ) \, dx\\ &=-\frac {b d n}{16 x^4}-\frac {b e n}{4 x^2}-\frac {1}{4} \left (\frac {d}{x^4}+\frac {2 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.21 \[ -\frac {a d}{4 x^4}-\frac {a e}{2 x^2}-\frac {b d \log \left (c x^n\right )}{4 x^4}-\frac {b e \log \left (c x^n\right )}{2 x^2}-\frac {b d n}{16 x^4}-\frac {b e n}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a*d)/x^4 - (b*d*n)/(16*x^4) - (a*e)/(2*x^2) - (b*e*n)/(4*x^2) - (b*d*Log[c*x^n])/(4*x^4) - (b*e*Log[c*x^
n])/(2*x^2)

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fricas [A]  time = 0.44, size = 60, normalized size = 1.05 \[ -\frac {b d n + 4 \, {\left (b e n + 2 \, a e\right )} x^{2} + 4 \, a d + 4 \, {\left (2 \, b e x^{2} + b d\right )} \log \relax (c) + 4 \, {\left (2 \, b e n x^{2} + b d n\right )} \log \relax (x)}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*log(c*x^n))/x^5,x, algorithm="fricas")

[Out]

-1/16*(b*d*n + 4*(b*e*n + 2*a*e)*x^2 + 4*a*d + 4*(2*b*e*x^2 + b*d)*log(c) + 4*(2*b*e*n*x^2 + b*d*n)*log(x))/x^
4

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giac [A]  time = 0.28, size = 65, normalized size = 1.14 \[ -\frac {8 \, b n x^{2} e \log \relax (x) + 4 \, b n x^{2} e + 8 \, b x^{2} e \log \relax (c) + 8 \, a x^{2} e + 4 \, b d n \log \relax (x) + b d n + 4 \, b d \log \relax (c) + 4 \, a d}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*log(c*x^n))/x^5,x, algorithm="giac")

[Out]

-1/16*(8*b*n*x^2*e*log(x) + 4*b*n*x^2*e + 8*b*x^2*e*log(c) + 8*a*x^2*e + 4*b*d*n*log(x) + b*d*n + 4*b*d*log(c)
+ 4*a*d)/x^4

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maple [C]  time = 0.16, size = 248, normalized size = 4.35 \[ -\frac {\left (2 e \,x^{2}+d \right ) b \ln \left (x^{n}\right )}{4 x^{4}}-\frac {-4 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+4 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b e n \,x^{2}+8 b e \,x^{2} \ln \relax (c )+8 a e \,x^{2}+b d n +4 b d \ln \relax (c )+4 a d}{16 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*(2*e*x^2+d)/x^4*ln(x^n)-1/16*(4*I*Pi*b*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b*e*x^2*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)-4*I*Pi*b*e*x^2*csgn(I*c*x^n)^3+4*I*Pi*b*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+8*b*e*x^2*ln(c)+4
*b*e*n*x^2+8*a*e*x^2+2*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2*I
*Pi*b*d*csgn(I*c*x^n)^3+2*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)+4*b*d*ln(c)+b*d*n+4*a*d)/x^4

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maxima [A]  time = 0.47, size = 57, normalized size = 1.00 \[ -\frac {b e n}{4 \, x^{2}} - \frac {b e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a e}{2 \, x^{2}} - \frac {b d n}{16 \, x^{4}} - \frac {b d \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*log(c*x^n))/x^5,x, algorithm="maxima")

[Out]

-1/4*b*e*n/x^2 - 1/2*b*e*log(c*x^n)/x^2 - 1/2*a*e/x^2 - 1/16*b*d*n/x^4 - 1/4*b*d*log(c*x^n)/x^4 - 1/4*a*d/x^4

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mupad [B]  time = 3.38, size = 51, normalized size = 0.89 \[ -\frac {\left (2\,a\,e+b\,e\,n\right )\,x^2+a\,d+\frac {b\,d\,n}{4}}{4\,x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^2}{2}+\frac {b\,d}{4}\right )}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.57, size = 88, normalized size = 1.54 \[ - \frac {a d}{4 x^{4}} - \frac {a e}{2 x^{2}} - \frac {b d n \log {\relax (x )}}{4 x^{4}} - \frac {b d n}{16 x^{4}} - \frac {b d \log {\relax (c )}}{4 x^{4}} - \frac {b e n \log {\relax (x )}}{2 x^{2}} - \frac {b e n}{4 x^{2}} - \frac {b e \log {\relax (c )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*ln(c*x**n))/x**5,x)

[Out]

-a*d/(4*x**4) - a*e/(2*x**2) - b*d*n*log(x)/(4*x**4) - b*d*n/(16*x**4) - b*d*log(c)/(4*x**4) - b*e*n*log(x)/(2
*x**2) - b*e*n/(4*x**2) - b*e*log(c)/(2*x**2)

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