3.4.78 \(\int \frac {(d+e x^r) (a+b \log (c x^n))}{x^6} \, dx\) [378]

Optimal. Leaf size=71 \[ -\frac {b d n}{25 x^5}-\frac {b e n x^{-5+r}}{(5-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r} \]

[Out]

-1/25*b*d*n/x^5-b*e*n*x^(-5+r)/(5-r)^2-1/5*d*(a+b*ln(c*x^n))/x^5-e*x^(-5+r)*(a+b*ln(c*x^n))/(5-r)

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Rubi [A]
time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, 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, 2372} \begin {gather*} -\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {b d n}{25 x^5}-\frac {b e n x^{r-5}}{(5-r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/25*(b*d*n)/x^5 - (b*e*n*x^(-5 + r))/(5 - r)^2 - (d*(a + b*Log[c*x^n]))/(5*x^5) - (e*x^(-5 + r)*(a + b*Log[c
*x^n]))/(5 - r)

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 2372

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]}, Dist[a + b*Log[c*x^n], u, 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^6} \, dx &=-\frac {1}{5} \left (\frac {d}{x^5}+\frac {5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d}{5 x^6}+\frac {e x^{-6+r}}{-5+r}\right ) \, dx\\ &=-\frac {b d n}{25 x^5}-\frac {b e n x^{-5+r}}{(5-r)^2}-\frac {1}{5} \left (\frac {d}{x^5}+\frac {5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 72, normalized size = 1.01 \begin {gather*} -\frac {5 a (-5+r) \left (d (-5+r)-5 e x^r\right )+b n \left (d (-5+r)^2+25 e x^r\right )+5 b (-5+r) \left (d (-5+r)-5 e x^r\right ) \log \left (c x^n\right )}{25 (-5+r)^2 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/25*(5*a*(-5 + r)*(d*(-5 + r) - 5*e*x^r) + b*n*(d*(-5 + r)^2 + 25*e*x^r) + 5*b*(-5 + r)*(d*(-5 + r) - 5*e*x^
r)*Log[c*x^n])/((-5 + r)^2*x^5)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 614, normalized size = 8.65

method	result	size
risch	\(-\frac {b \left (d r -5 e \,x^{r}-5 d \right ) \ln \left (x^{n}\right )}{5 \left (-5+r \right ) x^{5}}-\frac {250 x^{r} a e +50 b d n +50 x^{r} b e n -50 x^{r} a e r +250 a d -20 b d n r -100 \ln \left (c \right ) b d r +10 \ln \left (c \right ) b d \,r^{2}+10 a d \,r^{2}+250 d b \ln \left (c \right )-50 \ln \left (c \right ) b e \,x^{r} r -100 a d r +5 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+125 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+125 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-50 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -50 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -125 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b d n \,r^{2}-125 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r}-25 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -25 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -5 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 )+50 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 ) r +250 \ln \left (c \right ) b e \,x^{r}+25 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r} r +50 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} r -5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-125 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}-125 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 )+125 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+125 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+25 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r}{50 \left (-5+r \right )^{2} x^{5}}\)	\(614\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)*(a+b*ln(c*x^n))/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*b*(d*r-5*e*x^r-5*d)/(-5+r)/x^5*ln(x^n)-1/50*(250*x^r*a*e+50*b*d*n+50*x^r*b*e*n-50*x^r*a*e*r+250*a*d+125*I
*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+5*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-50*I*Pi*b*d*csgn(I*x^n)*csg
n(I*c*x^n)^2*r-20*b*d*n*r-125*I*Pi*b*d*csgn(I*c*x^n)^3-100*ln(c)*b*d*r+10*ln(c)*b*d*r^2-25*I*Pi*b*e*csgn(I*c)*
csgn(I*c*x^n)^2*x^r*r+10*a*d*r^2+250*d*b*ln(c)-50*ln(c)*b*e*x^r*r-25*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*
r-100*a*d*r-125*I*Pi*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*b*d*n*r^2+250*ln(c)*b*e*x^r-125*I*Pi*b*e*csgn(I
*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+50*I*Pi*b*d*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-5*I*Pi*b*d*r^2*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)-125*I*Pi*b*e*csgn(I*c*x^n)^3*x^r+50*I*Pi*b*d*csgn(I*c*x^n)^3*r+5*I*Pi*b*d*r^2*csgn(I*
c)*csgn(I*c*x^n)^2+125*I*Pi*b*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r+125*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2+25*I*Pi
*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r*r+25*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r-50*I*Pi*b*d*csgn(I*c)*csgn(I*
c*x^n)^2*r+125*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2-5*I*Pi*b*d*r^2*csgn(I*c*x^n)^3)/(-5+r)^2/x^5

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^6,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-6>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (64) = 128\).
time = 0.35, size = 140, normalized size = 1.97 \begin {gather*} -\frac {25 \, b d n + {\left (b d n + 5 \, a d\right )} r^{2} + 125 \, a d - 10 \, {\left (b d n + 5 \, a d\right )} r - 25 \, {\left ({\left (b r - 5 \, b\right )} e \log \left (c\right ) + {\left (b n r - 5 \, b n\right )} e \log \left (x\right ) - {\left (b n - a r + 5 \, a\right )} e\right )} x^{r} + 5 \, {\left (b d r^{2} - 10 \, b d r + 25 \, b d\right )} \log \left (c\right ) + 5 \, {\left (b d n r^{2} - 10 \, b d n r + 25 \, b d n\right )} \log \left (x\right )}{25 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^6,x, algorithm="fricas")

[Out]

-1/25*(25*b*d*n + (b*d*n + 5*a*d)*r^2 + 125*a*d - 10*(b*d*n + 5*a*d)*r - 25*((b*r - 5*b)*e*log(c) + (b*n*r - 5
*b*n)*e*log(x) - (b*n - a*r + 5*a)*e)*x^r + 5*(b*d*r^2 - 10*b*d*r + 25*b*d)*log(c) + 5*(b*d*n*r^2 - 10*b*d*n*r
+ 25*b*d*n)*log(x))/((r^2 - 10*r + 25)*x^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (63) = 126\).
time = 8.00, size = 495, normalized size = 6.97 \begin {gather*} \begin {cases} - \frac {5 a d r^{2}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {50 a d r}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 a d}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {25 a e r x^{r}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 a e x^{r}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {b d n r^{2}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {10 b d n r}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {25 b d n}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {5 b d r^{2} \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {50 b d r \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 b d \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {25 b e n x^{r}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {25 b e r x^{r} \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 b e x^{r} \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} & \text {for}\: r \neq 5 \\- \frac {a d}{5 x^{5}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{25 x^{5}} - \frac {\log {\left (c x^{n} \right )}}{5 x^{5}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)*(a+b*ln(c*x**n))/x**6,x)

[Out]

Piecewise((-5*a*d*r**2/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 50*a*d*r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5
) - 125*a*d/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 25*a*e*r*x**r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 12
5*a*e*x**r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - b*d*n*r**2/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 10*b*d
*n*r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 25*b*d*n/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 5*b*d*r**2*log
(c*x**n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 50*b*d*r*log(c*x**n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5)
- 125*b*d*log(c*x**n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 25*b*e*n*x**r/(25*r**2*x**5 - 250*r*x**5 + 625*
x**5) + 25*b*e*r*x**r*log(c*x**n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 125*b*e*x**r*log(c*x**n)/(25*r**2*x
**5 - 250*r*x**5 + 625*x**5), Ne(r, 5)), (-a*d/(5*x**5) + a*e*log(x) + b*d*(-n/(25*x**5) - log(c*x**n)/(5*x**5
)) - b*e*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (64) = 128\).
time = 3.35, size = 397, normalized size = 5.59 \begin {gather*} -\frac {b d n r^{2} \log \left (x\right )}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b d n r^{2}}{25 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b d r^{2} \log \left (c\right )}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, b d n r}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {a d r^{2}}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b n x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {a r x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, b d r \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b d n \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b d n}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, a d r}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, a x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b d \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, a d}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^6,x, algorithm="giac")

[Out]

-1/5*b*d*n*r^2*log(x)/((r^2 - 10*r + 25)*x^5) + b*n*r*x^r*e*log(x)/((r^2 - 10*r + 25)*x^5) - 1/25*b*d*n*r^2/((
r^2 - 10*r + 25)*x^5) - 1/5*b*d*r^2*log(c)/((r^2 - 10*r + 25)*x^5) + b*r*x^r*e*log(c)/((r^2 - 10*r + 25)*x^5)
+ 2*b*d*n*r*log(x)/((r^2 - 10*r + 25)*x^5) - 5*b*n*x^r*e*log(x)/((r^2 - 10*r + 25)*x^5) + 2/5*b*d*n*r/((r^2 -
10*r + 25)*x^5) - 1/5*a*d*r^2/((r^2 - 10*r + 25)*x^5) - b*n*x^r*e/((r^2 - 10*r + 25)*x^5) + a*r*x^r*e/((r^2 -
10*r + 25)*x^5) + 2*b*d*r*log(c)/((r^2 - 10*r + 25)*x^5) - 5*b*x^r*e*log(c)/((r^2 - 10*r + 25)*x^5) - 5*b*d*n*
log(x)/((r^2 - 10*r + 25)*x^5) - b*d*n/((r^2 - 10*r + 25)*x^5) + 2*a*d*r/((r^2 - 10*r + 25)*x^5) - 5*a*x^r*e/(
(r^2 - 10*r + 25)*x^5) - 5*b*d*log(c)/((r^2 - 10*r + 25)*x^5) - 5*a*d/((r^2 - 10*r + 25)*x^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^r)*(a + b*log(c*x^n)))/x^6,x)

[Out]

int(((d + e*x^r)*(a + b*log(c*x^n)))/x^6, x)

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