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]

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

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Rubi [A]  time = 0.0474705, 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 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])

Rule 12

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

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*}

Mathematica [A]  time = 0.0024789, 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]

-(a*d)/(4*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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Maple [C]  time = 0.098, size = 248, normalized size = 4.4 \begin{align*} -{\frac{b \left ( 2\,e{x}^{2}+d \right ) \ln \left ({x}^{n} \right ) }{4\,{x}^{4}}}-{\frac{4\,i\pi \,be{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-4\,i\pi \,be{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -4\,i\pi \,be{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+4\,i\pi \,be{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +8\,\ln \left ( c \right ) be{x}^{2}+4\,ben{x}^{2}+8\,ae{x}^{2}+2\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -2\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4\,\ln \left ( c \right ) bd+bdn+4\,ad}{16\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.1218, size = 77, normalized size = 1.35 \begin{align*} -\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}} \end{align*}

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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Fricas [A]  time = 1.21931, size = 153, normalized size = 2.68 \begin{align*} -\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 \left (c\right ) + 4 \,{\left (2 \, b e n x^{2} + b d n\right )} \log \left (x\right )}{16 \, x^{4}} \end{align*}

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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Sympy [A]  time = 3.37803, size = 88, normalized size = 1.54 \begin{align*} - \frac{a d}{4 x^{4}} - \frac{a e}{2 x^{2}} - \frac{b d n \log{\left (x \right )}}{4 x^{4}} - \frac{b d n}{16 x^{4}} - \frac{b d \log{\left (c \right )}}{4 x^{4}} - \frac{b e n \log{\left (x \right )}}{2 x^{2}} - \frac{b e n}{4 x^{2}} - \frac{b e \log{\left (c \right )}}{2 x^{2}} \end{align*}

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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Giac [A]  time = 1.27367, size = 88, normalized size = 1.54 \begin{align*} -\frac{8 \, b n x^{2} e \log \left (x\right ) + 4 \, b n x^{2} e + 8 \, b x^{2} e \log \left (c\right ) + 8 \, a x^{2} e + 4 \, b d n \log \left (x\right ) + b d n + 4 \, b d \log \left (c\right ) + 4 \, a d}{16 \, x^{4}} \end{align*}

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