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]

-1/9*b*d*n/x^3-b*e*n*x^(-3+r)/(3-r)^2-1/3*d*(a+b*ln(c*x^n))/x^3-e*x^(-3+r)*(a+b*ln(c*x^n))/(3-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} \[ -\frac {1}{3} \left (\frac {d}{x^3}+\frac {3 e x^{r-3}}{3-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{9 x^3}-\frac {b e n x^{r-3}}{(3-r)^2} \]

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*}

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Mathematica [A]  time = 0.12, 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]

-1/9*(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])/((-3 + r)^2*x^3)

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fricas [B]  time = 0.44, size = 140, normalized size = 1.97 \[ -\frac {9 \, b d n + {\left (b d n + 3 \, a d\right )} r^{2} + 27 \, a d - 6 \, {\left (b d n + 3 \, a d\right )} r + 9 \, {\left (b e n - a e r + 3 \, a e - {\left (b e r - 3 \, b e\right )} \log \relax (c) - {\left (b e n r - 3 \, b e n\right )} \log \relax (x)\right )} x^{r} + 3 \, {\left (b d r^{2} - 6 \, b d r + 9 \, b d\right )} \log \relax (c) + 3 \, {\left (b d n r^{2} - 6 \, b d n r + 9 \, b d n\right )} \log \relax (x)}{9 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} \]

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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giac [B]  time = 0.42, size = 397, normalized size = 5.59 \[ -\frac {b d n r^{2} \log \relax (x)}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {b n r x^{r} e \log \relax (x)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b d n r^{2}}{9 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b d r^{2} \log \relax (c)}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {b r x^{r} e \log \relax (c)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, b d n r \log \relax (x)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b n x^{r} e \log \relax (x)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, b d n r}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {a d r^{2}}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b n x^{r} e}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {a r x^{r} e}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, b d r \log \relax (c)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b x^{r} e \log \relax (c)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b d n \log \relax (x)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b d n}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, a d r}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, a x^{r} e}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b d \log \relax (c)}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, a d}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} \]

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)

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maple [C]  time = 0.23, size = 614, normalized size = 8.65 \[ -\frac {\left (d r -3 e \,x^{r}-3 d \right ) b \ln \left (x^{n}\right )}{3 \left (r -3\right ) x^{3}}-\frac {18 b d n +54 a e \,x^{r}-18 a e r \,x^{r}+18 b e n \,x^{r}+6 b d \,r^{2} \ln \relax (c )-36 b d r \ln \relax (c )+54 b e \,x^{r} \ln \relax (c )-36 a d r +54 a d +2 b d n \,r^{2}+3 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+6 a d \,r^{2}-12 b d n r +54 b d \ln \relax (c )-27 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 )-3 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 )-9 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+18 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 )-27 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 )-18 b e r \,x^{r} \ln \relax (c )+9 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 )+27 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-27 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-3 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-27 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-18 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+27 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+27 i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+27 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{18 \left (r -3\right )^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^4,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-4>0)', see `assume?` for mor
e details)Is r-4 equal to -1?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 14.95, size = 644, normalized size = 9.07 \[ \begin {cases} - \frac {3 a d r^{2}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {18 a d r}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 a d}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {9 a e r x^{r}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 a e x^{r}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {3 b d n r^{2} \log {\relax (x )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {b d n r^{2}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {18 b d n r \log {\relax (x )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {6 b d n r}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 b d n \log {\relax (x )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {9 b d n}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {3 b d r^{2} \log {\relax (c )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {18 b d r \log {\relax (c )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 b d \log {\relax (c )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {9 b e n r x^{r} \log {\relax (x )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 b e n x^{r} \log {\relax (x )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {9 b e n x^{r}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {9 b e r x^{r} \log {\relax (c )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 b e x^{r} \log {\relax (c )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} & \text {for}\: r \neq 3 \\- \frac {a d}{3 x^{3}} + a e \log {\relax (x )} + b d \left (- \frac {n}{9 x^{3}} - \frac {\log {\left (c x^{n} \right )}}{3 x^{3}}\right ) - b e \left (\begin {cases} - \log {\relax (c )} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-3*a*d*r**2/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 18*a*d*r/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 27
*a*d/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 9*a*e*r*x**r/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 27*a*e*x**r/(9*r
**2*x**3 - 54*r*x**3 + 81*x**3) - 3*b*d*n*r**2*log(x)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - b*d*n*r**2/(9*r**2
*x**3 - 54*r*x**3 + 81*x**3) + 18*b*d*n*r*log(x)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 6*b*d*n*r/(9*r**2*x**3
- 54*r*x**3 + 81*x**3) - 27*b*d*n*log(x)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 9*b*d*n/(9*r**2*x**3 - 54*r*x**
3 + 81*x**3) - 3*b*d*r**2*log(c)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 18*b*d*r*log(c)/(9*r**2*x**3 - 54*r*x**
3 + 81*x**3) - 27*b*d*log(c)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 9*b*e*n*r*x**r*log(x)/(9*r**2*x**3 - 54*r*x
**3 + 81*x**3) - 27*b*e*n*x**r*log(x)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 9*b*e*n*x**r/(9*r**2*x**3 - 54*r*x
**3 + 81*x**3) + 9*b*e*r*x**r*log(c)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 27*b*e*x**r*log(c)/(9*r**2*x**3 - 5
4*r*x**3 + 81*x**3), Ne(r, 3)), (-a*d/(3*x**3) + a*e*log(x) + b*d*(-n/(9*x**3) - log(c*x**n)/(3*x**3)) - b*e*P
iecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), True))

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