3.1.70 \(\int \frac {a+b \log (c x^n)}{(d+e x)^7} \, dx\) [70]

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

[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 = 0.05, 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 = {2356, 46} \begin {gather*} -\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} \end {gather*}

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2356

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 = 0.10, size = 99, normalized size = 0.65 \begin {gather*} \frac {-\frac {a+b \log \left (c x^n\right )}{(d+e x)^6}+\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 (x)-60 \log (d+e x)\right )}{60 d^6}}{6 e} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 431, normalized size = 2.84

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right )}{6 e \left (e x +d \right )^{6}}-\frac {60 \ln \left (c \right ) b \,d^{6}+360 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}+900 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}+1200 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}+900 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}+360 \ln \left (e x +d \right ) b \,d^{5} e n x -360 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}-900 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}-1200 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}+30 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-137 b \,d^{6} n +30 i \pi b \,d^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-900 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}-360 \ln \left (-x \right ) b \,d^{5} e n x +60 a \,d^{6}+60 \ln \left (e x +d \right ) b \,d^{6} n -60 \ln \left (-x \right ) b \,d^{6} n -855 b \,d^{4} e^{2} n \,x^{2}-522 b \,d^{5} e n x -60 b d \,e^{5} n \,x^{5}-330 b \,d^{2} e^{4} n \,x^{4}-740 b \,d^{3} e^{3} n \,x^{3}+60 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}-60 \ln \left (-x \right ) b \,e^{6} n \,x^{6}-30 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-30 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{360 d^{6} e \left (e x +d \right )^{6}}\)	\(431\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 259, normalized size = 1.70 \begin {gather*} \frac {1}{360} \, b n {\left (\frac {60 \, x^{4} e^{4} + 270 \, d x^{3} e^{3} + 470 \, d^{2} x^{2} e^{2} + 385 \, d^{3} x e + 137 \, d^{4}}{d^{5} x^{5} e^{6} + 5 \, d^{6} x^{4} e^{5} + 10 \, d^{7} x^{3} e^{4} + 10 \, d^{8} x^{2} e^{3} + 5 \, d^{9} x e^{2} + d^{10} e} - \frac {60 \, e^{\left (-1\right )} \log \left (x e + d\right )}{d^{6}} + \frac {60 \, e^{\left (-1\right )} \log \left (x\right )}{d^{6}}\right )} - \frac {b \log \left (c x^{n}\right )}{6 \, {\left (x^{6} e^{7} + 6 \, d x^{5} e^{6} + 15 \, d^{2} x^{4} e^{5} + 20 \, d^{3} x^{3} e^{4} + 15 \, d^{4} x^{2} e^{3} + 6 \, d^{5} x e^{2} + d^{6} e\right )}} - \frac {a}{6 \, {\left (x^{6} e^{7} + 6 \, d x^{5} e^{6} + 15 \, d^{2} x^{4} e^{5} + 20 \, d^{3} x^{3} e^{4} + 15 \, d^{4} x^{2} e^{3} + 6 \, d^{5} x e^{2} + d^{6} e\right )}} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (135) = 270\).
time = 0.36, size = 295, normalized size = 1.94 \begin {gather*} \frac {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 + 137 \, b d^{6} n - 60 \, b d^{6} \log \left (c\right ) - 60 \, a d^{6} - 60 \, {\left (b n x^{6} e^{6} + 6 \, b d n x^{5} e^{5} + 15 \, b d^{2} n x^{4} e^{4} + 20 \, b d^{3} n x^{3} e^{3} + 15 \, b d^{4} n x^{2} e^{2} + 6 \, b d^{5} n x e + b d^{6} n\right )} \log \left (x e + d\right ) + 60 \, {\left (b n x^{6} e^{6} + 6 \, b d n x^{5} e^{5} + 15 \, b d^{2} n x^{4} e^{4} + 20 \, b d^{3} n x^{3} e^{3} + 15 \, b d^{4} n x^{2} e^{2} + 6 \, b d^{5} n x e\right )} \log \left (x\right )}{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 )}} \end {gather*}

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*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 +
137*b*d^6*n - 60*b*d^6*log(c) - 60*a*d^6 - 60*(b*n*x^6*e^6 + 6*b*d*n*x^5*e^5 + 15*b*d^2*n*x^4*e^4 + 20*b*d^3*n
*x^3*e^3 + 15*b*d^4*n*x^2*e^2 + 6*b*d^5*n*x*e + b*d^6*n)*log(x*e + d) + 60*(b*n*x^6*e^6 + 6*b*d*n*x^5*e^5 + 15
*b*d^2*n*x^4*e^4 + 20*b*d^3*n*x^3*e^3 + 15*b*d^4*n*x^2*e^2 + 6*b*d^5*n*x*e)*log(x))/(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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1955 vs. \(2 (134) = 268\).
time = 84.98, size = 1955, normalized size = 12.86 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{6 x^{6}} - \frac {b n}{36 x^{6}} - \frac {b \log {\left (c x^{n} \right )}}{6 x^{6}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{6 x^{6}} - \frac {b n}{36 x^{6}} - \frac {b \log {\left (c x^{n} \right )}}{6 x^{6}}}{e^{7}} & \text {for}\: d = 0 \\\frac {a x - b n x + b x \log {\left (c x^{n} \right )}}{d^{7}} & \text {for}\: e = 0 \\- \frac {60 a d^{6}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {60 b d^{6} n \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {137 b d^{6} n}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {360 b d^{5} e n x \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {522 b d^{5} e n x}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {360 b d^{5} e x \log {\left (c x^{n} \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {900 b d^{4} e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {855 b d^{4} e^{2} n x^{2}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {900 b d^{4} e^{2} x^{2} \log {\left (c x^{n} \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {1200 b d^{3} e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {740 b d^{3} e^{3} n x^{3}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {1200 b d^{3} e^{3} x^{3} \log {\left (c x^{n} \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {900 b d^{2} e^{4} n x^{4} \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {330 b d^{2} e^{4} n x^{4}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {900 b d^{2} e^{4} x^{4} \log {\left (c x^{n} \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {360 b d e^{5} n x^{5} \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {60 b d e^{5} n x^{5}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {360 b d e^{5} x^{5} \log {\left (c x^{n} \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} - \frac {60 b e^{6} n x^{6} \log {\left (\frac {d}{e} + x \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} + \frac {60 b e^{6} x^{6} \log {\left (c x^{n} \right )}}{360 d^{12} e + 2160 d^{11} e^{2} x + 5400 d^{10} e^{3} x^{2} + 7200 d^{9} e^{4} x^{3} + 5400 d^{8} e^{5} x^{4} + 2160 d^{7} e^{6} x^{5} + 360 d^{6} e^{7} x^{6}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-a/(6*x**6) - b*n/(36*x**6) - b*log(c*x**n)/(6*x**6)), Eq(d, 0) & Eq(e, 0)), ((-a/(6*x**6) - b
*n/(36*x**6) - b*log(c*x**n)/(6*x**6))/e**7, Eq(d, 0)), ((a*x - b*n*x + b*x*log(c*x**n))/d**7, Eq(e, 0)), (-60
*a*d**6/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 +
2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 60*b*d**6*n*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d
**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 137*b
*d**6*n/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 +
2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 360*b*d**5*e*n*x*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5
400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) +
522*b*d**5*e*n*x/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**
5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 360*b*d**5*e*x*log(c*x**n)/(360*d**12*e + 2160*d**11*e**2
*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x*
*6) - 900*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*
e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 855*b*d**4*e**2*n*x**2/(360*d**1
2*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x*
*5 + 360*d**6*e**7*x**6) + 900*b*d**4*e**2*x**2*log(c*x**n)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3
*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 1200*b*d**3*e*
*3*n*x**3*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d*
*8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 740*b*d**3*e**3*n*x**3/(360*d**12*e + 2160*d**11*e*
*2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*
x**6) + 1200*b*d**3*e**3*x**3*log(c*x**n)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*
e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 900*b*d**2*e**4*n*x**4*log(d/e +
x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160
*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 330*b*d**2*e**4*n*x**4/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e
**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 900*b*d**2*
e**4*x**4*log(c*x**n)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**
8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 360*b*d*e**5*n*x**5*log(d/e + x)/(360*d**12*e + 2160
*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d
**6*e**7*x**6) + 60*b*d*e**5*n*x**5/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x
**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 360*b*d*e**5*x**5*log(c*x**n)/(360*d**
12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x
**5 + 360*d**6*e**7*x**6) - 60*b*e**6*n*x**6*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x
**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 60*b*e**6*x**6*l
og(c*x**n)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4
+ 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (135) = 270\).
time = 5.03, size = 344, normalized size = 2.26 \begin {gather*} -\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 \left (x\right ) - 360 \, b d n x^{5} e^{5} \log \left (x\right ) - 900 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 1200 \, b d^{3} n x^{3} e^{3} \log \left (x\right ) - 900 \, b d^{4} n x^{2} e^{2} \log \left (x\right ) - 360 \, b d^{5} n x e \log \left (x\right ) - 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 \left (c\right ) + 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 )}} \end {gather*}

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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Mupad [B]
time = 3.93, size = 232, normalized size = 1.53 \begin {gather*} \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} \end {gather*}

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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