3.4.72 \(\int \frac {(d+e x^r) (a+b \log (c x^n))}{x^5} \, dx\) [372]

Optimal. Leaf size=71 \[ -\frac {b d n}{16 x^4}-\frac {b e n x^{-4+r}}{(4-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e x^{-4+r} \left (a+b \log \left (c x^n\right )\right )}{4-r} \]

[Out]

-1/16*b*d*n/x^4-b*e*n*x^(-4+r)/(4-r)^2-1/4*d*(a+b*ln(c*x^n))/x^4-e*x^(-4+r)*(a+b*ln(c*x^n))/(4-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 )}{4 x^4}-\frac {e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {b d n}{16 x^4}-\frac {b e n x^{r-4}}{(4-r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)*(a + b*Log[c*x^n]))/x^5,x]

[Out]

-1/16*(b*d*n)/x^4 - (b*e*n*x^(-4 + r))/(4 - r)^2 - (d*(a + b*Log[c*x^n]))/(4*x^4) - (e*x^(-4 + r)*(a + b*Log[c
*x^n]))/(4 - 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^5} \, dx &=-\frac {1}{4} \left (\frac {d}{x^4}+\frac {4 e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d}{4 x^5}+\frac {e x^{-5+r}}{-4+r}\right ) \, dx\\ &=-\frac {b d n}{16 x^4}-\frac {b e n x^{-4+r}}{(4-r)^2}-\frac {1}{4} \left (\frac {d}{x^4}+\frac {4 e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 72, normalized size = 1.01 \begin {gather*} -\frac {4 a (-4+r) \left (d (-4+r)-4 e x^r\right )+b n \left (d (-4+r)^2+16 e x^r\right )+4 b (-4+r) \left (d (-4+r)-4 e x^r\right ) \log \left (c x^n\right )}{16 (-4+r)^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)*(a + b*Log[c*x^n]))/x^5,x]

[Out]

-1/16*(4*a*(-4 + r)*(d*(-4 + r) - 4*e*x^r) + b*n*(d*(-4 + r)^2 + 16*e*x^r) + 4*b*(-4 + r)*(d*(-4 + r) - 4*e*x^
r)*Log[c*x^n])/((-4 + r)^2*x^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 613, normalized size = 8.63

method	result	size
risch	\(-\frac {b \left (d r -4 e \,x^{r}-4 d \right ) \ln \left (x^{n}\right )}{4 \left (-4+r \right ) x^{4}}-\frac {64 x^{r} a e +16 b d n +16 x^{r} b e n -16 x^{r} a e r +64 a d -8 b d n r -32 \ln \left (c \right ) b d r +4 \ln \left (c \right ) b d \,r^{2}+4 a d \,r^{2}+64 d b \ln \left (c \right )-16 \ln \left (c \right ) b e \,x^{r} r -32 a d r +2 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+32 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+32 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+2 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-16 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -16 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -32 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b d n \,r^{2}-32 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}-8 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -8 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -2 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 )+16 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 +64 \ln \left (c \right ) b e \,x^{r}+8 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 +16 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} r -2 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-32 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}-32 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 )+32 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+32 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r}{16 \left (-4+r \right )^{2} x^{4}}\)	\(613\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)*(a+b*ln(c*x^n))/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*b*(d*r-4*e*x^r-4*d)/(-4+r)/x^4*ln(x^n)-1/16*(64*x^r*a*e+16*b*d*n+16*x^r*b*e*n-16*x^r*a*e*r-16*I*Pi*b*d*cs
gn(I*x^n)*csgn(I*c*x^n)^2*r+2*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+64*a*d+32*I*Pi*b*e*csgn(I*c)*csgn(I*c*x
^n)^2*x^r+2*I*Pi*b*d*r^2*csgn(I*c)*csgn(I*c*x^n)^2-32*I*Pi*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-8*b*d*n*r-3
2*ln(c)*b*d*r+4*ln(c)*b*d*r^2+16*I*Pi*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*r+4*a*d*r^2+64*d*b*ln(c)-16*ln(c
)*b*e*x^r*r+8*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r-16*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2*r-32*a*d*r-32*I*Pi*b*d*csgn
(I*c*x^n)^3-8*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*r-8*I*Pi*b*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r*r+b*d*n*r^2-
32*I*Pi*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-2*I*Pi*b*d*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+64*ln(c
)*b*e*x^r+8*I*Pi*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r*r+16*I*Pi*b*d*csgn(I*c*x^n)^3*r+32*I*Pi*b*e*csgn(
I*x^n)*csgn(I*c*x^n)^2*x^r-2*I*Pi*b*d*r^2*csgn(I*c*x^n)^3+32*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2+32*I*Pi*b*d*cs
gn(I*x^n)*csgn(I*c*x^n)^2-32*I*Pi*b*e*csgn(I*c*x^n)^3*x^r)/(-4+r)^2/x^4

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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^5,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-5>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 = 0.36, size = 140, normalized size = 1.97 \begin {gather*} -\frac {16 \, b d n + {\left (b d n + 4 \, a d\right )} r^{2} + 64 \, a d - 8 \, {\left (b d n + 4 \, a d\right )} r - 16 \, {\left ({\left (b r - 4 \, b\right )} e \log \left (c\right ) + {\left (b n r - 4 \, b n\right )} e \log \left (x\right ) - {\left (b n - a r + 4 \, a\right )} e\right )} x^{r} + 4 \, {\left (b d r^{2} - 8 \, b d r + 16 \, b d\right )} \log \left (c\right ) + 4 \, {\left (b d n r^{2} - 8 \, b d n r + 16 \, b d n\right )} \log \left (x\right )}{16 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} \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^5,x, algorithm="fricas")

[Out]

-1/16*(16*b*d*n + (b*d*n + 4*a*d)*r^2 + 64*a*d - 8*(b*d*n + 4*a*d)*r - 16*((b*r - 4*b)*e*log(c) + (b*n*r - 4*b
*n)*e*log(x) - (b*n - a*r + 4*a)*e)*x^r + 4*(b*d*r^2 - 8*b*d*r + 16*b*d)*log(c) + 4*(b*d*n*r^2 - 8*b*d*n*r + 1
6*b*d*n)*log(x))/((r^2 - 8*r + 16)*x^4)

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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 = 5.56, size = 495, normalized size = 6.97 \begin {gather*} \begin {cases} - \frac {4 a d r^{2}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {32 a d r}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 a d}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {16 a e r x^{r}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 a e x^{r}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {b d n r^{2}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {8 b d n r}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {16 b d n}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {4 b d r^{2} \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {32 b d r \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 b d \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {16 b e n x^{r}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {16 b e r x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 b e x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} & \text {for}\: r \neq 4 \\- \frac {a d}{4 x^{4}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{16 x^{4}} - \frac {\log {\left (c x^{n} \right )}}{4 x^{4}}\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**5,x)

[Out]

Piecewise((-4*a*d*r**2/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 32*a*d*r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4
) - 64*a*d/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 16*a*e*r*x**r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 64*
a*e*x**r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - b*d*n*r**2/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 8*b*d*n*
r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 16*b*d*n/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 4*b*d*r**2*log(c*
x**n)/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 32*b*d*r*log(c*x**n)/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 6
4*b*d*log(c*x**n)/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 16*b*e*n*x**r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4
) + 16*b*e*r*x**r*log(c*x**n)/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 64*b*e*x**r*log(c*x**n)/(16*r**2*x**4 -
128*r*x**4 + 256*x**4), Ne(r, 4)), (-a*d/(4*x**4) + a*e*log(x) + b*d*(-n/(16*x**4) - log(c*x**n)/(4*x**4)) -
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 = 3.44, size = 397, normalized size = 5.59 \begin {gather*} -\frac {b d n r^{2} \log \left (x\right )}{4 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b d n r^{2}}{16 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b d r^{2} \log \left (c\right )}{4 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {b d n r}{2 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {a d r^{2}}{4 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b n x^{r} e}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {a r x^{r} e}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {2 \, b d r \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b d n \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b d n}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {2 \, a d r}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, a x^{r} e}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b d \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, a d}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} \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^5,x, algorithm="giac")

[Out]

-1/4*b*d*n*r^2*log(x)/((r^2 - 8*r + 16)*x^4) + b*n*r*x^r*e*log(x)/((r^2 - 8*r + 16)*x^4) - 1/16*b*d*n*r^2/((r^
2 - 8*r + 16)*x^4) - 1/4*b*d*r^2*log(c)/((r^2 - 8*r + 16)*x^4) + b*r*x^r*e*log(c)/((r^2 - 8*r + 16)*x^4) + 2*b
*d*n*r*log(x)/((r^2 - 8*r + 16)*x^4) - 4*b*n*x^r*e*log(x)/((r^2 - 8*r + 16)*x^4) + 1/2*b*d*n*r/((r^2 - 8*r + 1
6)*x^4) - 1/4*a*d*r^2/((r^2 - 8*r + 16)*x^4) - b*n*x^r*e/((r^2 - 8*r + 16)*x^4) + a*r*x^r*e/((r^2 - 8*r + 16)*
x^4) + 2*b*d*r*log(c)/((r^2 - 8*r + 16)*x^4) - 4*b*x^r*e*log(c)/((r^2 - 8*r + 16)*x^4) - 4*b*d*n*log(x)/((r^2
- 8*r + 16)*x^4) - b*d*n/((r^2 - 8*r + 16)*x^4) + 2*a*d*r/((r^2 - 8*r + 16)*x^4) - 4*a*x^r*e/((r^2 - 8*r + 16)
*x^4) - 4*b*d*log(c)/((r^2 - 8*r + 16)*x^4) - 4*a*d/((r^2 - 8*r + 16)*x^4)

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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^5} \,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^5,x)

[Out]

int(((d + e*x^r)*(a + b*log(c*x^n)))/x^5, x)

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