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

Optimal. Leaf size=67 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {b d n}{x}-\frac {b e n x^{r-1}}{(1-r)^2} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 58, normalized size of antiderivative = 0.87, 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} \[ -\left (\frac {d}{x}+\frac {e x^{r-1}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{x}-\frac {b e n x^{r-1}}{(1-r)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.11, size = 67, normalized size = 1.00 \[ -\frac {a (r-1) \left (d (r-1)-e x^r\right )+b (r-1) \log \left (c x^n\right ) \left (d (r-1)-e x^r\right )+b n \left (d (r-1)^2+e x^r\right )}{(r-1)^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.50, size = 130, normalized size = 1.94 \[ -\frac {b d n + {\left (b d n + a d\right )} r^{2} + a d - 2 \, {\left (b d n + a d\right )} r + {\left (b e n - a e r + a e - {\left (b e r - b e\right )} \log \relax (c) - {\left (b e n r - b e n\right )} \log \relax (x)\right )} x^{r} + {\left (b d r^{2} - 2 \, b d r + b d\right )} \log \relax (c) + {\left (b d n r^{2} - 2 \, b d n r + b d n\right )} \log \relax (x)}{{\left (r^{2} - 2 \, r + 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.42, size = 193, normalized size = 2.88 \[ \frac {b n r x^{r} e \log \relax (x)}{{\left (r^{2} - 2 \, r + 1\right )} x} + \frac {b r x^{r} e \log \relax (c)}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac {b d n \log \relax (x)}{x} - \frac {b n x^{r} e \log \relax (x)}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac {b d n}{x} - \frac {b n x^{r} e}{{\left (r^{2} - 2 \, r + 1\right )} x} + \frac {a r x^{r} e}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac {b d \log \relax (c)}{x} - \frac {b x^{r} e \log \relax (c)}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac {a d}{x} - \frac {a x^{r} e}{{\left (r^{2} - 2 \, r + 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.22, size = 614, normalized size = 9.16 \[ -\frac {\left (d r -e \,x^{r}-d \right ) b \ln \left (x^{n}\right )}{\left (r -1\right ) x}-\frac {2 b d n +2 a e \,x^{r}-2 a e r \,x^{r}+2 b e n \,x^{r}+2 b d \,r^{2} \ln \relax (c )-4 b d r \ln \relax (c )+2 b e \,x^{r} \ln \relax (c )-4 a d r +2 a d +2 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}+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 +2 b d \ln \relax (c )-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 )-i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 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 )-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 )-2 b e r \,x^{r} \ln \relax (c )+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 )+i \pi b d \,\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 \,x^{n}\right )^{3}-i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b e \,x^{r} \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 \left (r -1\right )^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2,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-2>0)', see `assume?` for mor
e details)Is r-2 equal to -1?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.86, size = 449, normalized size = 6.70 \[ \begin {cases} - \frac {a d r^{2}}{r^{2} x - 2 r x + x} + \frac {2 a d r}{r^{2} x - 2 r x + x} - \frac {a d}{r^{2} x - 2 r x + x} + \frac {a e r x^{r}}{r^{2} x - 2 r x + x} - \frac {a e x^{r}}{r^{2} x - 2 r x + x} - \frac {b d n r^{2} \log {\relax (x )}}{r^{2} x - 2 r x + x} - \frac {b d n r^{2}}{r^{2} x - 2 r x + x} + \frac {2 b d n r \log {\relax (x )}}{r^{2} x - 2 r x + x} + \frac {2 b d n r}{r^{2} x - 2 r x + x} - \frac {b d n \log {\relax (x )}}{r^{2} x - 2 r x + x} - \frac {b d n}{r^{2} x - 2 r x + x} - \frac {b d r^{2} \log {\relax (c )}}{r^{2} x - 2 r x + x} + \frac {2 b d r \log {\relax (c )}}{r^{2} x - 2 r x + x} - \frac {b d \log {\relax (c )}}{r^{2} x - 2 r x + x} + \frac {b e n r x^{r} \log {\relax (x )}}{r^{2} x - 2 r x + x} - \frac {b e n x^{r} \log {\relax (x )}}{r^{2} x - 2 r x + x} - \frac {b e n x^{r}}{r^{2} x - 2 r x + x} + \frac {b e r x^{r} \log {\relax (c )}}{r^{2} x - 2 r x + x} - \frac {b e x^{r} \log {\relax (c )}}{r^{2} x - 2 r x + x} & \text {for}\: r \neq 1 \\- \frac {a d}{x} + a e \log {\relax (x )} + b d \left (- \frac {n}{x} - \frac {\log {\left (c x^{n} \right )}}{x}\right ) - b e \left (\begin {cases} - \log {\relax (c )} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*d*r**2/(r**2*x - 2*r*x + x) + 2*a*d*r/(r**2*x - 2*r*x + x) - a*d/(r**2*x - 2*r*x + x) + a*e*r*x*
*r/(r**2*x - 2*r*x + x) - a*e*x**r/(r**2*x - 2*r*x + x) - b*d*n*r**2*log(x)/(r**2*x - 2*r*x + x) - b*d*n*r**2/
(r**2*x - 2*r*x + x) + 2*b*d*n*r*log(x)/(r**2*x - 2*r*x + x) + 2*b*d*n*r/(r**2*x - 2*r*x + x) - b*d*n*log(x)/(
r**2*x - 2*r*x + x) - b*d*n/(r**2*x - 2*r*x + x) - b*d*r**2*log(c)/(r**2*x - 2*r*x + x) + 2*b*d*r*log(c)/(r**2
*x - 2*r*x + x) - b*d*log(c)/(r**2*x - 2*r*x + x) + b*e*n*r*x**r*log(x)/(r**2*x - 2*r*x + x) - b*e*n*x**r*log(
x)/(r**2*x - 2*r*x + x) - b*e*n*x**r/(r**2*x - 2*r*x + x) + b*e*r*x**r*log(c)/(r**2*x - 2*r*x + x) - b*e*x**r*
log(c)/(r**2*x - 2*r*x + x), Ne(r, 1)), (-a*d/x + a*e*log(x) + b*d*(-n/x - log(c*x**n)/x) - b*e*Piecewise((-lo
g(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), True))

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