3.57 \(\int \frac {x (a+b \log (c x^n))}{(d+e x)^4} \, dx\)

Optimal. Leaf size=117 \[ -\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}+\frac {b n \log (x)}{6 d^2 e^2}-\frac {b n \log (d+e x)}{6 d^2 e^2}+\frac {b n}{6 d e^2 (d+e x)}-\frac {b n}{6 e^2 (d+e x)^2} \]

[Out]

-1/6*b*n/e^2/(e*x+d)^2+1/6*b*n/d/e^2/(e*x+d)+1/6*b*n*ln(x)/d^2/e^2+1/3*d*(a+b*ln(c*x^n))/e^2/(e*x+d)^3+1/2*(-a
-b*ln(c*x^n))/e^2/(e*x+d)^2-1/6*b*n*ln(e*x+d)/d^2/e^2

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Rubi [A]  time = 0.09, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {43, 2350, 12, 77} \[ -\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}+\frac {b n \log (x)}{6 d^2 e^2}-\frac {b n \log (d+e x)}{6 d^2 e^2}+\frac {b n}{6 d e^2 (d+e x)}-\frac {b n}{6 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*Log[c*x^n]))/(d + e*x)^4,x]

[Out]

-(b*n)/(6*e^2*(d + e*x)^2) + (b*n)/(6*d*e^2*(d + e*x)) + (b*n*Log[x])/(6*d^2*e^2) + (d*(a + b*Log[c*x^n]))/(3*
e^2*(d + e*x)^3) - (a + b*Log[c*x^n])/(2*e^2*(d + e*x)^2) - (b*n*Log[d + e*x])/(6*d^2*e^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*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] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps

\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-(b n) \int \frac {-d-3 e x}{6 e^2 x (d+e x)^3} \, dx\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac {(b n) \int \frac {-d-3 e x}{x (d+e x)^3} \, dx}{6 e^2}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac {(b n) \int \left (-\frac {1}{d^2 x}-\frac {2 e}{(d+e x)^3}+\frac {e}{d (d+e x)^2}+\frac {e}{d^2 (d+e x)}\right ) \, dx}{6 e^2}\\ &=-\frac {b n}{6 e^2 (d+e x)^2}+\frac {b n}{6 d e^2 (d+e x)}+\frac {b n \log (x)}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac {b n \log (d+e x)}{6 d^2 e^2}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 135, normalized size = 1.15 \[ -\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {b n \left (-\frac {2 \log (d+e x)}{d^2}+\frac {2 \log (x)}{d^2}+\frac {2}{d (d+e x)}+\frac {1}{(d+e x)^2}\right )}{6 e^2}+\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x)^4,x]

[Out]

(d*(a + b*Log[c*x^n]))/(3*e^2*(d + e*x)^3) - (a + b*Log[c*x^n])/(2*e^2*(d + e*x)^2) - (b*n*((d + e*x)^(-2) + 2
/(d*(d + e*x)) + (2*Log[x])/d^2 - (2*Log[d + e*x])/d^2))/(6*e^2) + (b*n*(1/(d*(d + e*x)) + Log[x]/d^2 - Log[d
+ e*x]/d^2))/(2*e^2)

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fricas [A]  time = 0.64, size = 162, normalized size = 1.38 \[ \frac {b d e^{2} n x^{2} - a d^{3} + {\left (b d^{2} e n - 3 \, a d^{2} e\right )} x - {\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} + 3 \, b d^{2} e n x + b d^{3} n\right )} \log \left (e x + d\right ) - {\left (3 \, b d^{2} e x + b d^{3}\right )} \log \relax (c) + {\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2}\right )} \log \relax (x)}{6 \, {\left (d^{2} e^{5} x^{3} + 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(b*d*e^2*n*x^2 - a*d^3 + (b*d^2*e*n - 3*a*d^2*e)*x - (b*e^3*n*x^3 + 3*b*d*e^2*n*x^2 + 3*b*d^2*e*n*x + b*d^
3*n)*log(e*x + d) - (3*b*d^2*e*x + b*d^3)*log(c) + (b*e^3*n*x^3 + 3*b*d*e^2*n*x^2)*log(x))/(d^2*e^5*x^3 + 3*d^
3*e^4*x^2 + 3*d^4*e^3*x + d^5*e^2)

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giac [A]  time = 0.29, size = 176, normalized size = 1.50 \[ -\frac {b n x^{3} e^{3} \log \left (x e + d\right ) + 3 \, b d n x^{2} e^{2} \log \left (x e + d\right ) + 3 \, b d^{2} n x e \log \left (x e + d\right ) - b n x^{3} e^{3} \log \relax (x) - 3 \, b d n x^{2} e^{2} \log \relax (x) - b d n x^{2} e^{2} - b d^{2} n x e + b d^{3} n \log \left (x e + d\right ) + 3 \, b d^{2} x e \log \relax (c) + 3 \, a d^{2} x e + b d^{3} \log \relax (c) + a d^{3}}{6 \, {\left (d^{2} x^{3} e^{5} + 3 \, d^{3} x^{2} e^{4} + 3 \, d^{4} x e^{3} + d^{5} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/6*(b*n*x^3*e^3*log(x*e + d) + 3*b*d*n*x^2*e^2*log(x*e + d) + 3*b*d^2*n*x*e*log(x*e + d) - b*n*x^3*e^3*log(x
) - 3*b*d*n*x^2*e^2*log(x) - b*d*n*x^2*e^2 - b*d^2*n*x*e + b*d^3*n*log(x*e + d) + 3*b*d^2*x*e*log(c) + 3*a*d^2
*x*e + b*d^3*log(c) + a*d^3)/(d^2*x^3*e^5 + 3*d^3*x^2*e^4 + 3*d^4*x*e^3 + d^5*e^2)

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maple [C]  time = 0.24, size = 403, normalized size = 3.44 \[ -\frac {\left (3 e x +d \right ) b \ln \left (x^{n}\right )}{6 \left (e x +d \right )^{3} e^{2}}-\frac {-2 b \,e^{3} n \,x^{3} \ln \left (-x \right )+2 b \,e^{3} n \,x^{3} \ln \left (e x +d \right )-3 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-6 b d \,e^{2} n \,x^{2} \ln \left (-x \right )+6 b d \,e^{2} n \,x^{2} \ln \left (e x +d \right )-i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-6 b \,d^{2} e n x \ln \left (-x \right )+6 b \,d^{2} e n x \ln \left (e x +d \right )-2 b d \,e^{2} n \,x^{2}-2 b \,d^{3} n \ln \left (-x \right )+2 b \,d^{3} n \ln \left (e x +d \right )-2 b \,d^{2} e n x +6 b \,d^{2} e x \ln \relax (c )+6 a \,d^{2} e x +2 b \,d^{3} \ln \relax (c )+2 a \,d^{3}}{12 \left (e x +d \right )^{3} d^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*b*(3*e*x+d)/(e*x+d)^3/e^2*ln(x^n)-1/12*(-I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3*I*Pi*b*d^2*e*x*
csgn(I*c*x^n)^3-3*I*Pi*b*d^2*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*Pi*b*d^2*e*x*csgn(I*c)*csgn(I*c*x^n)^
2+I*Pi*b*d^3*csgn(I*c)*csgn(I*c*x^n)^2-I*Pi*b*d^3*csgn(I*c*x^n)^3+3*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2
+I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-2*b*e^3*n*x^3*ln(-x)+2*b*e^3*n*x^3*ln(e*x+d)-6*b*d*e^2*n*x^2*ln(-x)+6*
b*d*e^2*n*x^2*ln(e*x+d)-6*b*d^2*e*n*x*ln(-x)+6*b*d^2*e*n*x*ln(e*x+d)-2*b*d*e^2*n*x^2+6*b*d^2*e*x*ln(c)-2*b*d^3
*n*ln(-x)+2*b*d^3*n*ln(e*x+d)-2*b*d^2*e*n*x+2*b*d^3*ln(c)+6*a*d^2*e*x+2*a*d^3)/d^2/e^2/(e*x+d)^3

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maxima [A]  time = 0.66, size = 150, normalized size = 1.28 \[ \frac {1}{6} \, b n {\left (\frac {x}{d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e} - \frac {\log \left (e x + d\right )}{d^{2} e^{2}} + \frac {\log \relax (x)}{d^{2} e^{2}}\right )} - \frac {{\left (3 \, e x + d\right )} b \log \left (c x^{n}\right )}{6 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {{\left (3 \, e x + d\right )} a}{6 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*b*n*(x/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - log(e*x + d)/(d^2*e^2) + log(x)/(d^2*e^2)) - 1/6*(3*e*x + d)*b*
log(c*x^n)/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/6*(3*e*x + d)*a/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*
e^3*x + d^3*e^2)

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mupad [B]  time = 3.86, size = 141, normalized size = 1.21 \[ -\frac {a\,d+x\,\left (3\,a\,e-b\,e\,n\right )-\frac {b\,e^2\,n\,x^2}{d}}{6\,d^3\,e^2+18\,d^2\,e^3\,x+18\,d\,e^4\,x^2+6\,e^5\,x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d}{6\,e^2}+\frac {b\,x}{2\,e}\right )}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d^2\,e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (a*d + x*(3*a*e - b*e*n) - (b*e^2*n*x^2)/d)/(6*d^3*e^2 + 6*e^5*x^3 + 18*d^2*e^3*x + 18*d*e^4*x^2) - (log(c*x
^n)*((b*d)/(6*e^2) + (b*x)/(2*e)))/(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x) - (b*n*atanh((2*e*x)/d + 1))/(3*d
^2*e^2)

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sympy [A]  time = 15.69, size = 796, normalized size = 6.80 \[ \begin {cases} \tilde {\infty } \left (- \frac {a}{2 x^{2}} - \frac {b n \log {\relax (x )}}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\relax (c )}}{2 x^{2}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{2 x^{2}} - \frac {b n \log {\relax (x )}}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\relax (c )}}{2 x^{2}}}{e^{4}} & \text {for}\: d = 0 \\\frac {\frac {a x^{2}}{2} + \frac {b n x^{2} \log {\relax (x )}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\relax (c )}}{2}}{d^{4}} & \text {for}\: e = 0 \\- \frac {a d^{3}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {3 a d^{2} e x}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {b d^{3} n \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {3 b d^{2} e n x \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b d^{2} e n x}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {3 b d e^{2} n x^{2} \log {\relax (x )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {3 b d e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b d e^{2} n x^{2}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {3 b d e^{2} x^{2} \log {\relax (c )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b e^{3} n x^{3} \log {\relax (x )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {b e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b e^{3} x^{3} \log {\relax (c )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-a/(2*x**2) - b*n*log(x)/(2*x**2) - b*n/(4*x**2) - b*log(c)/(2*x**2)), Eq(d, 0) & Eq(e, 0)), (
(-a/(2*x**2) - b*n*log(x)/(2*x**2) - b*n/(4*x**2) - b*log(c)/(2*x**2))/e**4, Eq(d, 0)), ((a*x**2/2 + b*n*x**2*
log(x)/2 - b*n*x**2/4 + b*x**2*log(c)/2)/d**4, Eq(e, 0)), (-a*d**3/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**
4*x**2 + 6*d**2*e**5*x**3) - 3*a*d**2*e*x/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3
) - b*d**3*n*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) - 3*b*d**2*e*n
*x*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) + b*d**2*e*n*x/(6*d**5*e
**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) + 3*b*d*e**2*n*x**2*log(x)/(6*d**5*e**2 + 18*d**4
*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) - 3*b*d*e**2*n*x**2*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3*x
+ 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) + b*d*e**2*n*x**2/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 +
6*d**2*e**5*x**3) + 3*b*d*e**2*x**2*log(c)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x*
*3) + b*e**3*n*x**3*log(x)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) - b*e**3*n*x*
*3*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) + b*e**3*x**3*log(c)/(6*
d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3), True))

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