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

Optimal. Leaf size=71 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac{b d n}{9 x^3}-\frac{b e n x^{r-3}}{(3-r)^2} \]

[Out]

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

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Rubi [A]  time = 0.0732404, antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {14, 2334} \[ -\frac{1}{3} \left (\frac{d}{x^3}+\frac{3 e x^{r-3}}{3-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{9 x^3}-\frac{b e n x^{r-3}}{(3-r)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n)/(9*x^3) - (b*e*n*x^(-3 + r))/(3 - r)^2 - ((d/x^3 + (3*e*x^(-3 + r))/(3 - r))*(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])

Rubi steps

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

Mathematica [A]  time = 0.113142, size = 72, normalized size = 1.01 \[ -\frac{3 a (r-3) \left (d (r-3)-3 e x^r\right )+3 b (r-3) \log \left (c x^n\right ) \left (d (r-3)-3 e x^r\right )+b n \left (d (r-3)^2+9 e x^r\right )}{9 (r-3)^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.159, size = 614, normalized size = 8.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*(d*r-3*e*x^r-3*d)/(-3+r)/x^3*ln(x^n)-1/18*(54*a*d+18*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-18*
x^r*a*e*r+18*x^r*b*e*n-9*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r-9*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*r
-12*b*d*n*r+18*b*d*n+54*x^r*a*e-36*ln(c)*b*d*r+6*ln(c)*b*d*r^2-18*ln(c)*b*e*x^r*r+54*ln(c)*b*e*x^r+6*a*d*r^2+5
4*ln(c)*b*d-36*a*d*r+2*b*d*n*r^2+18*I*Pi*b*d*csgn(I*c*x^n)^3*r-3*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)-27*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+9*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r*r+2
7*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-27*I*Pi*b*e*csgn(I*c*x^n)^3*x^r+27*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)-3
*I*Pi*b*d*r^2*csgn(I*c*x^n)^3+3*I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)
^2-18*I*Pi*b*d*r*csgn(I*c*x^n)^2*csgn(I*c)-27*I*Pi*b*d*csgn(I*c*x^n)^3+9*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r+3*I*Pi
*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+27*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+27*I*Pi*b*e*csgn(I*x^n)*csgn(I*
c*x^n)^2*x^r-27*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(-3+r)^2/x^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.40798, size = 352, normalized size = 4.96 \begin{align*} -\frac{9 \, b d n +{\left (b d n + 3 \, a d\right )} r^{2} + 27 \, a d - 6 \,{\left (b d n + 3 \, a d\right )} r + 9 \,{\left (b e n - a e r + 3 \, a e -{\left (b e r - 3 \, b e\right )} \log \left (c\right ) -{\left (b e n r - 3 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 3 \,{\left (b d r^{2} - 6 \, b d r + 9 \, b d\right )} \log \left (c\right ) + 3 \,{\left (b d n r^{2} - 6 \, b d n r + 9 \, b d n\right )} \log \left (x\right )}{9 \,{\left (r^{2} - 6 \, r + 9\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.25464, size = 536, normalized size = 7.55 \begin{align*} -\frac{b d n r^{2} \log \left (x\right )}{3 \,{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{b d n r^{2}}{9 \,{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{b d r^{2} \log \left (c\right )}{3 \,{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{3 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{2 \, b d n r}{3 \,{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{a d r^{2}}{3 \,{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{b n x^{r} e}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{a r x^{r} e}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{2 \, b d r \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{3 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{3 \, b d n \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{b d n}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac{2 \, a d r}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{3 \, a x^{r} e}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{3 \, b d \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac{3 \, a d}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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