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

Optimal. Leaf size=53 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b d n}{9 x^3}-\frac{b e n}{x} \]

[Out]

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

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Rubi [A]  time = 0.0466648, antiderivative size = 45, normalized size of antiderivative = 0.85, 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} \[ -\frac{1}{3} \left (\frac{d}{x^3}+\frac{3 e}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{9 x^3}-\frac{b e n}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0022991, size = 63, normalized size = 1.19 \[ -\frac{a d}{3 x^3}-\frac{a e}{x}-\frac{b d \log \left (c x^n\right )}{3 x^3}-\frac{b e \log \left (c x^n\right )}{x}-\frac{b d n}{9 x^3}-\frac{b e n}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.103, size = 249, normalized size = 4.7 \begin{align*} -{\frac{b \left ( 3\,e{x}^{2}+d \right ) \ln \left ({x}^{n} \right ) }{3\,{x}^{3}}}-{\frac{9\,i\pi \,be{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-9\,i\pi \,be{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -9\,i\pi \,be{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+9\,i\pi \,be{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +18\,\ln \left ( c \right ) be{x}^{2}+18\,ben{x}^{2}+18\,ae{x}^{2}+3\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -3\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+3\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +6\,\ln \left ( c \right ) bd+2\,bdn+6\,ad}{18\,{x}^{3}}} \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^4,x)

[Out]

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

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Maxima [A]  time = 1.12883, size = 77, normalized size = 1.45 \begin{align*} -\frac{b e n}{x} - \frac{b e \log \left (c x^{n}\right )}{x} - \frac{a e}{x} - \frac{b d n}{9 \, x^{3}} - \frac{b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a d}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.31323, size = 149, normalized size = 2.81 \begin{align*} -\frac{b d n + 9 \,{\left (b e n + a e\right )} x^{2} + 3 \, a d + 3 \,{\left (3 \, b e x^{2} + b d\right )} \log \left (c\right ) + 3 \,{\left (3 \, b e n x^{2} + b d n\right )} \log \left (x\right )}{9 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.00741, size = 75, normalized size = 1.42 \begin{align*} - \frac{a d}{3 x^{3}} - \frac{a e}{x} - \frac{b d n \log{\left (x \right )}}{3 x^{3}} - \frac{b d n}{9 x^{3}} - \frac{b d \log{\left (c \right )}}{3 x^{3}} - \frac{b e n \log{\left (x \right )}}{x} - \frac{b e n}{x} - \frac{b e \log{\left (c \right )}}{x} \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**4,x)

[Out]

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

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Giac [A]  time = 1.28694, size = 88, normalized size = 1.66 \begin{align*} -\frac{9 \, b n x^{2} e \log \left (x\right ) + 9 \, b n x^{2} e + 9 \, b x^{2} e \log \left (c\right ) + 9 \, a x^{2} e + 3 \, b d n \log \left (x\right ) + b d n + 3 \, b d \log \left (c\right ) + 3 \, a d}{9 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

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