3.371 \(\int \frac {(d+e x^r) (a+b \log (c x^n))}{x^3} \, dx\)

Optimal. Leaf size=71 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {b d n}{4 x^2}-\frac {b e n x^{r-2}}{(2-r)^2} \]

[Out]

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

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Rubi [A]  time = 0.07, 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} \[ -\frac {1}{2} \left (\frac {d}{x^2}+\frac {2 e x^{r-2}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{4 x^2}-\frac {b e n x^{r-2}}{(2-r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)*(a + b*Log[c*x^n]))/x^3,x]

[Out]

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

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

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Mathematica [A]  time = 0.11, size = 72, normalized size = 1.01 \[ -\frac {2 a (r-2) \left (d (r-2)-2 e x^r\right )+2 b (r-2) \log \left (c x^n\right ) \left (d (r-2)-2 e x^r\right )+b n \left (d (r-2)^2+4 e x^r\right )}{4 (r-2)^2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)*(a + b*Log[c*x^n]))/x^3,x]

[Out]

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

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fricas [B]  time = 0.46, size = 140, normalized size = 1.97 \[ -\frac {4 \, b d n + {\left (b d n + 2 \, a d\right )} r^{2} + 8 \, a d - 4 \, {\left (b d n + 2 \, a d\right )} r + 4 \, {\left (b e n - a e r + 2 \, a e - {\left (b e r - 2 \, b e\right )} \log \relax (c) - {\left (b e n r - 2 \, b e n\right )} \log \relax (x)\right )} x^{r} + 2 \, {\left (b d r^{2} - 4 \, b d r + 4 \, b d\right )} \log \relax (c) + 2 \, {\left (b d n r^{2} - 4 \, b d n r + 4 \, b d n\right )} \log \relax (x)}{4 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.40, size = 396, normalized size = 5.58 \[ -\frac {b d n r^{2} \log \relax (x)}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b n r x^{r} e \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d n r^{2}}{4 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d r^{2} \log \relax (c)}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b r x^{r} e \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, b d n r \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b n x^{r} e \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b d n r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {a d r^{2}}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b n x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {a r x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, b d r \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b x^{r} e \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b d n \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d n}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, a d r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, a x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b d \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, a d}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^3,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.23, size = 613, normalized size = 8.63 \[ -\frac {\left (d r -2 e \,x^{r}-2 d \right ) b \ln \left (x^{n}\right )}{2 \left (r -2\right ) x^{2}}-\frac {4 b d n +8 a e \,x^{r}-4 a e r \,x^{r}+4 b e n \,x^{r}+2 b d \,r^{2} \ln \relax (c )-8 b d r \ln \relax (c )+8 b e \,x^{r} \ln \relax (c )-8 a d r +8 a d +b d n \,r^{2}+i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a d \,r^{2}-4 b d n r +8 b d \ln \relax (c )-4 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 )-i \pi b d \,r^{2} \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 e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b d r \,\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^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-4 b e r \,x^{r} \ln \relax (c )+2 i \pi b e r \,x^{r} \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 d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (r -2\right )^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)*(b*ln(c*x^n)+a)/x^3,x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(r-3>0)', see `assume?` for mor
e details)Is r-3 equal to -1?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^r)*(a + b*log(c*x^n)))/x^3,x)

[Out]

int(((d + e*x^r)*(a + b*log(c*x^n)))/x^3, x)

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sympy [A]  time = 8.94, size = 644, normalized size = 9.07 \[ \begin {cases} - \frac {2 a d r^{2}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 a d r}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 a d}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 a e r x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 a e x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {2 b d n r^{2} \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {b d n r^{2}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 b d n r \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b d n r}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b d n \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {4 b d n}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {2 b d r^{2} \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 b d r \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b d \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b e n r x^{r} \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b e n x^{r} \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {4 b e n x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b e r x^{r} \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b e x^{r} \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} & \text {for}\: r \neq 2 \\- \frac {a d}{2 x^{2}} + a e \log {\relax (x )} + b d \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e \left (\begin {cases} - \log {\relax (c )} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)*(a+b*ln(c*x**n))/x**3,x)

[Out]

Piecewise((-2*a*d*r**2/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 8*a*d*r/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 8*a
*d/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 4*a*e*r*x**r/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 8*a*e*x**r/(4*r**2
*x**2 - 16*r*x**2 + 16*x**2) - 2*b*d*n*r**2*log(x)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - b*d*n*r**2/(4*r**2*x*
*2 - 16*r*x**2 + 16*x**2) + 8*b*d*n*r*log(x)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 4*b*d*n*r/(4*r**2*x**2 - 16
*r*x**2 + 16*x**2) - 8*b*d*n*log(x)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 4*b*d*n/(4*r**2*x**2 - 16*r*x**2 + 1
6*x**2) - 2*b*d*r**2*log(c)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 8*b*d*r*log(c)/(4*r**2*x**2 - 16*r*x**2 + 16
*x**2) - 8*b*d*log(c)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 4*b*e*n*r*x**r*log(x)/(4*r**2*x**2 - 16*r*x**2 + 1
6*x**2) - 8*b*e*n*x**r*log(x)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 4*b*e*n*x**r/(4*r**2*x**2 - 16*r*x**2 + 16
*x**2) + 4*b*e*r*x**r*log(c)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 8*b*e*x**r*log(c)/(4*r**2*x**2 - 16*r*x**2
+ 16*x**2), Ne(r, 2)), (-a*d/(2*x**2) + a*e*log(x) + b*d*(-n/(4*x**2) - log(c*x**n)/(2*x**2)) - b*e*Piecewise(
(-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), True))

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