3.70 \(\int \frac {a+b \log (c x^n)}{(d+e x)^7} \, dx\)

Optimal. Leaf size=152 \[ -\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac {b n \log (x)}{6 d^6 e}-\frac {b n \log (d+e x)}{6 d^6 e}+\frac {b n}{6 d^5 e (d+e x)}+\frac {b n}{12 d^4 e (d+e x)^2}+\frac {b n}{18 d^3 e (d+e x)^3}+\frac {b n}{24 d^2 e (d+e x)^4}+\frac {b n}{30 d e (d+e x)^5} \]

[Out]

1/30*b*n/d/e/(e*x+d)^5+1/24*b*n/d^2/e/(e*x+d)^4+1/18*b*n/d^3/e/(e*x+d)^3+1/12*b*n/d^4/e/(e*x+d)^2+1/6*b*n/d^5/
e/(e*x+d)+1/6*b*n*ln(x)/d^6/e+1/6*(-a-b*ln(c*x^n))/e/(e*x+d)^6-1/6*b*n*ln(e*x+d)/d^6/e

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Rubi [A]  time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2319, 44} \[ -\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac {b n}{6 d^5 e (d+e x)}+\frac {b n}{12 d^4 e (d+e x)^2}+\frac {b n}{18 d^3 e (d+e x)^3}+\frac {b n}{24 d^2 e (d+e x)^4}+\frac {b n \log (x)}{6 d^6 e}-\frac {b n \log (d+e x)}{6 d^6 e}+\frac {b n}{30 d e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x)^7,x]

[Out]

(b*n)/(30*d*e*(d + e*x)^5) + (b*n)/(24*d^2*e*(d + e*x)^4) + (b*n)/(18*d^3*e*(d + e*x)^3) + (b*n)/(12*d^4*e*(d
+ e*x)^2) + (b*n)/(6*d^5*e*(d + e*x)) + (b*n*Log[x])/(6*d^6*e) - (a + b*Log[c*x^n])/(6*e*(d + e*x)^6) - (b*n*L
og[d + e*x])/(6*d^6*e)

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx &=-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac {(b n) \int \frac {1}{x (d+e x)^6} \, dx}{6 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac {(b n) \int \left (\frac {1}{d^6 x}-\frac {e}{d (d+e x)^6}-\frac {e}{d^2 (d+e x)^5}-\frac {e}{d^3 (d+e x)^4}-\frac {e}{d^4 (d+e x)^3}-\frac {e}{d^5 (d+e x)^2}-\frac {e}{d^6 (d+e x)}\right ) \, dx}{6 e}\\ &=\frac {b n}{30 d e (d+e x)^5}+\frac {b n}{24 d^2 e (d+e x)^4}+\frac {b n}{18 d^3 e (d+e x)^3}+\frac {b n}{12 d^4 e (d+e x)^2}+\frac {b n}{6 d^5 e (d+e x)}+\frac {b n \log (x)}{6 d^6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}-\frac {b n \log (d+e x)}{6 d^6 e}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 99, normalized size = 0.65 \[ \frac {\frac {b n \left (\frac {d \left (137 d^4+385 d^3 e x+470 d^2 e^2 x^2+270 d e^3 x^3+60 e^4 x^4\right )}{(d+e x)^5}-60 \log (d+e x)+60 \log (x)\right )}{60 d^6}-\frac {a+b \log \left (c x^n\right )}{(d+e x)^6}}{6 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x)^7,x]

[Out]

(-((a + b*Log[c*x^n])/(d + e*x)^6) + (b*n*((d*(137*d^4 + 385*d^3*e*x + 470*d^2*e^2*x^2 + 270*d*e^3*x^3 + 60*e^
4*x^4))/(d + e*x)^5 + 60*Log[x] - 60*Log[d + e*x]))/(60*d^6))/(6*e)

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fricas [B]  time = 0.85, size = 310, normalized size = 2.04 \[ \frac {60 \, b d e^{5} n x^{5} + 330 \, b d^{2} e^{4} n x^{4} + 740 \, b d^{3} e^{3} n x^{3} + 855 \, b d^{4} e^{2} n x^{2} + 522 \, b d^{5} e n x + 137 \, b d^{6} n - 60 \, b d^{6} \log \relax (c) - 60 \, a d^{6} - 60 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) + 60 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x\right )} \log \relax (x)}{360 \, {\left (d^{6} e^{7} x^{6} + 6 \, d^{7} e^{6} x^{5} + 15 \, d^{8} e^{5} x^{4} + 20 \, d^{9} e^{4} x^{3} + 15 \, d^{10} e^{3} x^{2} + 6 \, d^{11} e^{2} x + d^{12} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/360*(60*b*d*e^5*n*x^5 + 330*b*d^2*e^4*n*x^4 + 740*b*d^3*e^3*n*x^3 + 855*b*d^4*e^2*n*x^2 + 522*b*d^5*e*n*x +
137*b*d^6*n - 60*b*d^6*log(c) - 60*a*d^6 - 60*(b*e^6*n*x^6 + 6*b*d*e^5*n*x^5 + 15*b*d^2*e^4*n*x^4 + 20*b*d^3*e
^3*n*x^3 + 15*b*d^4*e^2*n*x^2 + 6*b*d^5*e*n*x + b*d^6*n)*log(e*x + d) + 60*(b*e^6*n*x^6 + 6*b*d*e^5*n*x^5 + 15
*b*d^2*e^4*n*x^4 + 20*b*d^3*e^3*n*x^3 + 15*b*d^4*e^2*n*x^2 + 6*b*d^5*e*n*x)*log(x))/(d^6*e^7*x^6 + 6*d^7*e^6*x
^5 + 15*d^8*e^5*x^4 + 20*d^9*e^4*x^3 + 15*d^10*e^3*x^2 + 6*d^11*e^2*x + d^12*e)

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giac [B]  time = 0.31, size = 344, normalized size = 2.26 \[ -\frac {60 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 360 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 900 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 1200 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 900 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 360 \, b d^{5} n x e \log \left (x e + d\right ) - 60 \, b n x^{6} e^{6} \log \relax (x) - 360 \, b d n x^{5} e^{5} \log \relax (x) - 900 \, b d^{2} n x^{4} e^{4} \log \relax (x) - 1200 \, b d^{3} n x^{3} e^{3} \log \relax (x) - 900 \, b d^{4} n x^{2} e^{2} \log \relax (x) - 360 \, b d^{5} n x e \log \relax (x) - 60 \, b d n x^{5} e^{5} - 330 \, b d^{2} n x^{4} e^{4} - 740 \, b d^{3} n x^{3} e^{3} - 855 \, b d^{4} n x^{2} e^{2} - 522 \, b d^{5} n x e + 60 \, b d^{6} n \log \left (x e + d\right ) - 137 \, b d^{6} n + 60 \, b d^{6} \log \relax (c) + 60 \, a d^{6}}{360 \, {\left (d^{6} x^{6} e^{7} + 6 \, d^{7} x^{5} e^{6} + 15 \, d^{8} x^{4} e^{5} + 20 \, d^{9} x^{3} e^{4} + 15 \, d^{10} x^{2} e^{3} + 6 \, d^{11} x e^{2} + d^{12} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/360*(60*b*n*x^6*e^6*log(x*e + d) + 360*b*d*n*x^5*e^5*log(x*e + d) + 900*b*d^2*n*x^4*e^4*log(x*e + d) + 1200
*b*d^3*n*x^3*e^3*log(x*e + d) + 900*b*d^4*n*x^2*e^2*log(x*e + d) + 360*b*d^5*n*x*e*log(x*e + d) - 60*b*n*x^6*e
^6*log(x) - 360*b*d*n*x^5*e^5*log(x) - 900*b*d^2*n*x^4*e^4*log(x) - 1200*b*d^3*n*x^3*e^3*log(x) - 900*b*d^4*n*
x^2*e^2*log(x) - 360*b*d^5*n*x*e*log(x) - 60*b*d*n*x^5*e^5 - 330*b*d^2*n*x^4*e^4 - 740*b*d^3*n*x^3*e^3 - 855*b
*d^4*n*x^2*e^2 - 522*b*d^5*n*x*e + 60*b*d^6*n*log(x*e + d) - 137*b*d^6*n + 60*b*d^6*log(c) + 60*a*d^6)/(d^6*x^
6*e^7 + 6*d^7*x^5*e^6 + 15*d^8*x^4*e^5 + 20*d^9*x^3*e^4 + 15*d^10*x^2*e^3 + 6*d^11*x*e^2 + d^12*e)

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maple [C]  time = 0.21, size = 431, normalized size = 2.84 \[ -\frac {b \ln \left (x^{n}\right )}{6 \left (e x +d \right )^{6} e}-\frac {-30 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-60 b d \,e^{5} n \,x^{5}-330 b \,d^{2} e^{4} n \,x^{4}-740 b \,d^{3} e^{3} n \,x^{3}-855 b \,d^{4} e^{2} n \,x^{2}-522 b \,d^{5} e n x +60 b \,d^{6} n \ln \left (e x +d \right )-60 b \,d^{6} n \ln \left (-x \right )+60 a \,d^{6}+60 b \,d^{6} \ln \relax (c )-137 b \,d^{6} n +60 b \,e^{6} n \,x^{6} \ln \left (e x +d \right )-60 b \,e^{6} n \,x^{6} \ln \left (-x \right )-30 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+360 b d \,e^{5} n \,x^{5} \ln \left (e x +d \right )+900 b \,d^{2} e^{4} n \,x^{4} \ln \left (e x +d \right )+1200 b \,d^{3} e^{3} n \,x^{3} \ln \left (e x +d \right )+900 b \,d^{4} e^{2} n \,x^{2} \ln \left (e x +d \right )+360 b \,d^{5} e n x \ln \left (e x +d \right )-360 b d \,e^{5} n \,x^{5} \ln \left (-x \right )-900 b \,d^{2} e^{4} n \,x^{4} \ln \left (-x \right )-1200 b \,d^{3} e^{3} n \,x^{3} \ln \left (-x \right )-900 b \,d^{4} e^{2} n \,x^{2} \ln \left (-x \right )-360 b \,d^{5} e n x \ln \left (-x \right )+30 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+30 i \pi b \,d^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{360 \left (e x +d \right )^{6} d^{6} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)/(e*x+d)^7,x)

[Out]

-1/6*b/e/(e*x+d)^6*ln(x^n)-1/360*(-60*b*d*e^5*n*x^5-330*b*d^2*e^4*n*x^4-740*b*d^3*e^3*n*x^3-855*b*d^4*e^2*n*x^
2-522*b*d^5*e*n*x+60*b*d^6*n*ln(e*x+d)-60*b*d^6*n*ln(-x)+60*a*d^6+60*b*d^6*ln(c)-137*b*d^6*n-30*I*Pi*b*d^6*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-30*I*Pi*b*d^6*csgn(I*c*x^n)^3+60*b*e^6*n*x^6*ln(e*x+d)-60*b*e^6*n*x^6*ln(-x)+
30*I*Pi*b*d^6*csgn(I*c*x^n)^2*csgn(I*c)+30*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2+360*b*d*e^5*n*x^5*ln(e*x+d)+
900*b*d^2*e^4*n*x^4*ln(e*x+d)+1200*b*d^3*e^3*n*x^3*ln(e*x+d)+900*b*d^4*e^2*n*x^2*ln(e*x+d)+360*b*d^5*e*n*x*ln(
e*x+d)-360*b*d*e^5*n*x^5*ln(-x)-900*b*d^2*e^4*n*x^4*ln(-x)-1200*b*d^3*e^3*n*x^3*ln(-x)-900*b*d^4*e^2*n*x^2*ln(
-x)-360*b*d^5*e*n*x*ln(-x))/d^6/e/(e*x+d)^6

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maxima [B]  time = 0.89, size = 276, normalized size = 1.82 \[ \frac {1}{360} \, b n {\left (\frac {60 \, e^{4} x^{4} + 270 \, d e^{3} x^{3} + 470 \, d^{2} e^{2} x^{2} + 385 \, d^{3} e x + 137 \, d^{4}}{d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e} - \frac {60 \, \log \left (e x + d\right )}{d^{6} e} + \frac {60 \, \log \relax (x)}{d^{6} e}\right )} - \frac {b \log \left (c x^{n}\right )}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {a}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/360*b*n*((60*e^4*x^4 + 270*d*e^3*x^3 + 470*d^2*e^2*x^2 + 385*d^3*e*x + 137*d^4)/(d^5*e^6*x^5 + 5*d^6*e^5*x^4
+ 10*d^7*e^4*x^3 + 10*d^8*e^3*x^2 + 5*d^9*e^2*x + d^10*e) - 60*log(e*x + d)/(d^6*e) + 60*log(x)/(d^6*e)) - 1/
6*b*log(c*x^n)/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e
) - 1/6*a/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e)

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mupad [B]  time = 3.93, size = 232, normalized size = 1.53 \[ \frac {\frac {137\,b\,n}{60}-a+\frac {57\,b\,e^2\,n\,x^2}{4\,d^2}+\frac {37\,b\,e^3\,n\,x^3}{3\,d^3}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^4}+\frac {b\,e^5\,n\,x^5}{d^5}+\frac {87\,b\,e\,n\,x}{10\,d}}{6\,d^6\,e+36\,d^5\,e^2\,x+90\,d^4\,e^3\,x^2+120\,d^3\,e^4\,x^3+90\,d^2\,e^5\,x^4+36\,d\,e^6\,x^5+6\,e^7\,x^6}-\frac {b\,\ln \left (c\,x^n\right )}{6\,e\,\left (d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6\right )}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d^6\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*x)^7,x)

[Out]

((137*b*n)/60 - a + (57*b*e^2*n*x^2)/(4*d^2) + (37*b*e^3*n*x^3)/(3*d^3) + (11*b*e^4*n*x^4)/(2*d^4) + (b*e^5*n*
x^5)/d^5 + (87*b*e*n*x)/(10*d))/(6*d^6*e + 6*e^7*x^6 + 36*d^5*e^2*x + 36*d*e^6*x^5 + 90*d^4*e^3*x^2 + 120*d^3*
e^4*x^3 + 90*d^2*e^5*x^4) - (b*log(c*x^n))/(6*e*(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3
+ 15*d^2*e^4*x^4 + 6*d^5*e*x)) - (b*n*atanh((2*e*x)/d + 1))/(3*d^6*e)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

Timed out

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