3.1.68 \(\int \frac {x^2 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [68]

Optimal. Leaf size=199 \[ \frac {b d n}{30 e^3 (d+e x)^5}-\frac {7 b n}{120 e^3 (d+e x)^4}+\frac {b n}{180 d e^3 (d+e x)^3}+\frac {b n}{120 d^2 e^3 (d+e x)^2}+\frac {b n}{60 d^3 e^3 (d+e x)}+\frac {b n \log (x)}{60 d^4 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac {b n \log (d+e x)}{60 d^4 e^3} \]

[Out]

1/30*b*d*n/e^3/(e*x+d)^5-7/120*b*n/e^3/(e*x+d)^4+1/180*b*n/d/e^3/(e*x+d)^3+1/120*b*n/d^2/e^3/(e*x+d)^2+1/60*b*
n/d^3/e^3/(e*x+d)+1/60*b*n*ln(x)/d^4/e^3-1/6*d^2*(a+b*ln(c*x^n))/e^3/(e*x+d)^6+2/5*d*(a+b*ln(c*x^n))/e^3/(e*x+
d)^5+1/4*(-a-b*ln(c*x^n))/e^3/(e*x+d)^4-1/60*b*n*ln(e*x+d)/d^4/e^3

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Rubi [A]
time = 0.12, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2382, 12, 907} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}+\frac {b n \log (x)}{60 d^4 e^3}-\frac {b n \log (d+e x)}{60 d^4 e^3}+\frac {b n}{60 d^3 e^3 (d+e x)}+\frac {b n}{120 d^2 e^3 (d+e x)^2}+\frac {b d n}{30 e^3 (d+e x)^5}-\frac {7 b n}{120 e^3 (d+e x)^4}+\frac {b n}{180 d e^3 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(b*d*n)/(30*e^3*(d + e*x)^5) - (7*b*n)/(120*e^3*(d + e*x)^4) + (b*n)/(180*d*e^3*(d + e*x)^3) + (b*n)/(120*d^2*
e^3*(d + e*x)^2) + (b*n)/(60*d^3*e^3*(d + e*x)) + (b*n*Log[x])/(60*d^4*e^3) - (d^2*(a + b*Log[c*x^n]))/(6*e^3*
(d + e*x)^6) + (2*d*(a + b*Log[c*x^n]))/(5*e^3*(d + e*x)^5) - (a + b*Log[c*x^n])/(4*e^3*(d + e*x)^4) - (b*n*Lo
g[d + e*x])/(60*d^4*e^3)

Rule 12

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

Rule 45

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 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2382

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^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}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-(b n) \int \frac {-d^2-6 d e x-15 e^2 x^2}{60 e^3 x (d+e x)^6} \, dx\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac {(b n) \int \frac {-d^2-6 d e x-15 e^2 x^2}{x (d+e x)^6} \, dx}{60 e^3}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac {(b n) \int \left (-\frac {1}{d^4 x}+\frac {10 d e}{(d+e x)^6}-\frac {14 e}{(d+e x)^5}+\frac {e}{d (d+e x)^4}+\frac {e}{d^2 (d+e x)^3}+\frac {e}{d^3 (d+e x)^2}+\frac {e}{d^4 (d+e x)}\right ) \, dx}{60 e^3}\\ &=\frac {b d n}{30 e^3 (d+e x)^5}-\frac {7 b n}{120 e^3 (d+e x)^4}+\frac {b n}{180 d e^3 (d+e x)^3}+\frac {b n}{120 d^2 e^3 (d+e x)^2}+\frac {b n}{60 d^3 e^3 (d+e x)}+\frac {b n \log (x)}{60 d^4 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac {b n \log (d+e x)}{60 d^4 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 192, normalized size = 0.96 \begin {gather*} \frac {-60 a d^6+144 a d^5 (d+e x)+12 b d^5 n (d+e x)-90 a d^4 (d+e x)^2-21 b d^4 n (d+e x)^2+2 b d^3 n (d+e x)^3+3 b d^2 n (d+e x)^4+6 b d n (d+e x)^5+6 b n (d+e x)^6 \log (x)-60 b d^6 \log \left (c x^n\right )+144 b d^5 (d+e x) \log \left (c x^n\right )-90 b d^4 (d+e x)^2 \log \left (c x^n\right )-6 b n (d+e x)^6 \log (d+e x)}{360 d^4 e^3 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(-60*a*d^6 + 144*a*d^5*(d + e*x) + 12*b*d^5*n*(d + e*x) - 90*a*d^4*(d + e*x)^2 - 21*b*d^4*n*(d + e*x)^2 + 2*b*
d^3*n*(d + e*x)^3 + 3*b*d^2*n*(d + e*x)^4 + 6*b*d*n*(d + e*x)^5 + 6*b*n*(d + e*x)^6*Log[x] - 60*b*d^6*Log[c*x^
n] + 144*b*d^5*(d + e*x)*Log[c*x^n] - 90*b*d^4*(d + e*x)^2*Log[c*x^n] - 6*b*n*(d + e*x)^6*Log[d + e*x])/(360*d
^4*e^3*(d + e*x)^6)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 712, normalized size = 3.58

method	result	size
risch	\(-\frac {b \left (15 e^{2} x^{2}+6 d e x +d^{2}\right ) \ln \left (x^{n}\right )}{60 \left (e x +d \right )^{6} e^{3}}-\frac {6 \ln \left (c \right ) b \,d^{6}+36 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}+90 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}+120 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}+90 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}+36 \ln \left (e x +d \right ) b \,d^{5} e n x -36 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}-90 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}-120 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}+3 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+90 a \,d^{4} e^{2} x^{2}+36 a \,d^{5} e x -2 b \,d^{6} n +45 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+45 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-90 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}-36 \ln \left (-x \right ) b \,d^{5} e n x +6 a \,d^{6}+6 \ln \left (e x +d \right ) b \,d^{6} n -6 \ln \left (-x \right ) b \,d^{6} n -63 b \,d^{4} e^{2} n \,x^{2}-18 b \,d^{5} e n x -6 b d \,e^{5} n \,x^{5}-33 b \,d^{2} e^{4} n \,x^{4}-74 b \,d^{3} e^{3} n \,x^{3}+18 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}-6 \ln \left (-x \right ) b \,e^{6} n \,x^{6}+90 \ln \left (c \right ) b \,d^{4} e^{2} x^{2}+36 \ln \left (c \right ) b \,d^{5} e x -3 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-45 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-18 i \pi b \,d^{5} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-3 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 )-45 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-18 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{360 d^{4} e^{3} \left (e x +d \right )^{6}}\)	\(712\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/60*b*(15*e^2*x^2+6*d*e*x+d^2)/(e*x+d)^6/e^3*ln(x^n)-1/360*(-3*I*Pi*b*d^6*csgn(I*c*x^n)^3+18*I*Pi*b*d^5*e*x*
csgn(I*c)*csgn(I*c*x^n)^2+18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+45*I*Pi*b*d^4*e^2*x^2*csgn(I*c)*csgn(I
*c*x^n)^2-45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3-18*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3-3*I*Pi*b*d^6*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)+6*ln(c)*b*d^6+36*ln(e*x+d)*b*d*e^5*n*x^5+90*ln(e*x+d)*b*d^2*e^4*n*x^4+120*ln(e*x+d)*b*d^3
*e^3*n*x^3+90*ln(e*x+d)*b*d^4*e^2*n*x^2+36*ln(e*x+d)*b*d^5*e*n*x-36*ln(-x)*b*d*e^5*n*x^5-90*ln(-x)*b*d^2*e^4*n
*x^4-120*ln(-x)*b*d^3*e^3*n*x^3+90*a*d^4*e^2*x^2+36*a*d^5*e*x-2*b*d^6*n+45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn
(I*c*x^n)^2+3*I*Pi*b*d^6*csgn(I*c)*csgn(I*c*x^n)^2+3*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-90*ln(-x)*b*d^4*e^
2*n*x^2-36*ln(-x)*b*d^5*e*n*x+6*a*d^6+6*ln(e*x+d)*b*d^6*n-6*ln(-x)*b*d^6*n-63*b*d^4*e^2*n*x^2-18*b*d^5*e*n*x-6
*b*d*e^5*n*x^5-33*b*d^2*e^4*n*x^4-74*b*d^3*e^3*n*x^3+6*ln(e*x+d)*b*e^6*n*x^6-6*ln(-x)*b*e^6*n*x^6+90*ln(c)*b*d
^4*e^2*x^2+36*ln(c)*b*d^5*e*x-45*I*Pi*b*d^4*e^2*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-18*I*Pi*b*d^5*e*x*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n))/d^4/e^3/(e*x+d)^6

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Maxima [A]
time = 0.30, size = 293, normalized size = 1.47 \begin {gather*} \frac {1}{360} \, b n {\left (\frac {6 \, x^{4} e^{4} + 27 \, d x^{3} e^{3} + 47 \, d^{2} x^{2} e^{2} + 16 \, d^{3} x e + 2 \, d^{4}}{d^{3} x^{5} e^{8} + 5 \, d^{4} x^{4} e^{7} + 10 \, d^{5} x^{3} e^{6} + 10 \, d^{6} x^{2} e^{5} + 5 \, d^{7} x e^{4} + d^{8} e^{3}} - \frac {6 \, e^{\left (-3\right )} \log \left (x e + d\right )}{d^{4}} + \frac {6 \, e^{\left (-3\right )} \log \left (x\right )}{d^{4}}\right )} - \frac {{\left (15 \, x^{2} e^{2} + 6 \, d x e + d^{2}\right )} b \log \left (c x^{n}\right )}{60 \, {\left (x^{6} e^{9} + 6 \, d x^{5} e^{8} + 15 \, d^{2} x^{4} e^{7} + 20 \, d^{3} x^{3} e^{6} + 15 \, d^{4} x^{2} e^{5} + 6 \, d^{5} x e^{4} + d^{6} e^{3}\right )}} - \frac {{\left (15 \, x^{2} e^{2} + 6 \, d x e + d^{2}\right )} a}{60 \, {\left (x^{6} e^{9} + 6 \, d x^{5} e^{8} + 15 \, d^{2} x^{4} e^{7} + 20 \, d^{3} x^{3} e^{6} + 15 \, d^{4} x^{2} e^{5} + 6 \, d^{5} x e^{4} + d^{6} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/360*b*n*((6*x^4*e^4 + 27*d*x^3*e^3 + 47*d^2*x^2*e^2 + 16*d^3*x*e + 2*d^4)/(d^3*x^5*e^8 + 5*d^4*x^4*e^7 + 10*
d^5*x^3*e^6 + 10*d^6*x^2*e^5 + 5*d^7*x*e^4 + d^8*e^3) - 6*e^(-3)*log(x*e + d)/d^4 + 6*e^(-3)*log(x)/d^4) - 1/6
0*(15*x^2*e^2 + 6*d*x*e + d^2)*b*log(c*x^n)/(x^6*e^9 + 6*d*x^5*e^8 + 15*d^2*x^4*e^7 + 20*d^3*x^3*e^6 + 15*d^4*
x^2*e^5 + 6*d^5*x*e^4 + d^6*e^3) - 1/60*(15*x^2*e^2 + 6*d*x*e + d^2)*a/(x^6*e^9 + 6*d*x^5*e^8 + 15*d^2*x^4*e^7
+ 20*d^3*x^3*e^6 + 15*d^4*x^2*e^5 + 6*d^5*x*e^4 + d^6*e^3)

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Fricas [A]
time = 0.35, size = 312, normalized size = 1.57 \begin {gather*} \frac {6 \, b d n x^{5} e^{5} + 33 \, b d^{2} n x^{4} e^{4} + 74 \, b d^{3} n x^{3} e^{3} + 2 \, b d^{6} n - 6 \, a d^{6} + 9 \, {\left (7 \, b d^{4} n - 10 \, a d^{4}\right )} x^{2} e^{2} + 18 \, {\left (b d^{5} n - 2 \, a d^{5}\right )} x e - 6 \, {\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 ) - 6 \, {\left (15 \, b d^{4} x^{2} e^{2} + 6 \, b d^{5} x e + b d^{6}\right )} \log \left (c\right ) + 6 \, {\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}\right )} \log \left (x\right )}{360 \, {\left (d^{4} x^{6} e^{9} + 6 \, d^{5} x^{5} e^{8} + 15 \, d^{6} x^{4} e^{7} + 20 \, d^{7} x^{3} e^{6} + 15 \, d^{8} x^{2} e^{5} + 6 \, d^{9} x e^{4} + d^{10} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/360*(6*b*d*n*x^5*e^5 + 33*b*d^2*n*x^4*e^4 + 74*b*d^3*n*x^3*e^3 + 2*b*d^6*n - 6*a*d^6 + 9*(7*b*d^4*n - 10*a*d
^4)*x^2*e^2 + 18*(b*d^5*n - 2*a*d^5)*x*e - 6*(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) - 6*(15*b*d^4*x^2*e^2 + 6*b*d^5*x*e + b*d
^6)*log(c) + 6*(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)*log(x))/(d^4*x^6*e^9
+ 6*d^5*x^5*e^8 + 15*d^6*x^4*e^7 + 20*d^7*x^3*e^6 + 15*d^8*x^2*e^5 + 6*d^9*x*e^4 + d^10*e^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1986 vs. \(2 (196) = 392\).
time = 85.27, size = 1986, normalized size = 9.98 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{4 x^{4}} - \frac {b n}{16 x^{4}} - \frac {b \log {\left (c x^{n} \right )}}{4 x^{4}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{4 x^{4}} - \frac {b n}{16 x^{4}} - \frac {b \log {\left (c x^{n} \right )}}{4 x^{4}}}{e^{7}} & \text {for}\: d = 0 \\\frac {\frac {a x^{3}}{3} - \frac {b n x^{3}}{9} + \frac {b x^{3} \log {\left (c x^{n} \right )}}{3}}{d^{7}} & \text {for}\: e = 0 \\- \frac {6 a d^{6}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {36 a d^{5} e x}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {90 a d^{4} e^{2} x^{2}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {6 b d^{6} n \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {2 b d^{6} n}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {36 b d^{5} e n x \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {18 b d^{5} e n x}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {90 b d^{4} e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {63 b d^{4} e^{2} n x^{2}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {120 b d^{3} e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {74 b d^{3} e^{3} n x^{3}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {120 b d^{3} e^{3} x^{3} \log {\left (c x^{n} \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {90 b d^{2} e^{4} n x^{4} \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {33 b d^{2} e^{4} n x^{4}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {90 b d^{2} e^{4} x^{4} \log {\left (c x^{n} \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {36 b d e^{5} n x^{5} \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {6 b d e^{5} n x^{5}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {36 b d e^{5} x^{5} \log {\left (c x^{n} \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} - \frac {6 b e^{6} n x^{6} \log {\left (\frac {d}{e} + x \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} + \frac {6 b e^{6} x^{6} \log {\left (c x^{n} \right )}}{360 d^{10} e^{3} + 2160 d^{9} e^{4} x + 5400 d^{8} e^{5} x^{2} + 7200 d^{7} e^{6} x^{3} + 5400 d^{6} e^{7} x^{4} + 2160 d^{5} e^{8} x^{5} + 360 d^{4} e^{9} x^{6}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

Piecewise((zoo*(-a/(4*x**4) - b*n/(16*x**4) - b*log(c*x**n)/(4*x**4)), Eq(d, 0) & Eq(e, 0)), ((-a/(4*x**4) - b
*n/(16*x**4) - b*log(c*x**n)/(4*x**4))/e**7, Eq(d, 0)), ((a*x**3/3 - b*n*x**3/9 + b*x**3*log(c*x**n)/3)/d**7,
Eq(e, 0)), (-6*a*d**6/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d*
*6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 36*a*d**5*e*x/(360*d**10*e**3 + 2160*d**9*e**4*x +
5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) -
90*a*d**4*e**2*x**2/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6
*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 6*b*d**6*n*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e
**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*
x**6) + 2*b*d**6*n/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*
e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 36*b*d**5*e*n*x*log(d/e + x)/(360*d**10*e**3 + 2160*d*
*9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e
**9*x**6) + 18*b*d**5*e*n*x/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5
400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 90*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**10
*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x*
*5 + 360*d**4*e**9*x**6) + 63*b*d**4*e**2*n*x**2/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 72
00*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 120*b*d**3*e**3*n*x**3*l
og(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x*
*4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 74*b*d**3*e**3*n*x**3/(360*d**10*e**3 + 2160*d**9*e**4*x + 54
00*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 12
0*b*d**3*e**3*x**3*log(c*x**n)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3
+ 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 90*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d*
*10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8
*x**5 + 360*d**4*e**9*x**6) + 33*b*d**2*e**4*n*x**4/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 +
7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 90*b*d**2*e**4*x**4*l
og(c*x**n)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**
4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 36*b*d*e**5*n*x**5*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e*
*4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x
**6) + 6*b*d*e**5*n*x**5/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400
*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 36*b*d*e**5*x**5*log(c*x**n)/(360*d**10*e**3 + 2
160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*
d**4*e**9*x**6) - 6*b*e**6*n*x**6*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200
*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 6*b*e**6*x**6*log(c*x**n)/
(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d*
*5*e**8*x**5 + 360*d**4*e**9*x**6), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (178) = 356\).
time = 3.80, size = 362, normalized size = 1.82 \begin {gather*} -\frac {6 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 36 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 90 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 120 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 90 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 36 \, b d^{5} n x e \log \left (x e + d\right ) - 6 \, b n x^{6} e^{6} \log \left (x\right ) - 36 \, b d n x^{5} e^{5} \log \left (x\right ) - 90 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 120 \, b d^{3} n x^{3} e^{3} \log \left (x\right ) - 6 \, b d n x^{5} e^{5} - 33 \, b d^{2} n x^{4} e^{4} - 74 \, b d^{3} n x^{3} e^{3} - 63 \, b d^{4} n x^{2} e^{2} - 18 \, b d^{5} n x e + 6 \, b d^{6} n \log \left (x e + d\right ) + 90 \, b d^{4} x^{2} e^{2} \log \left (c\right ) + 36 \, b d^{5} x e \log \left (c\right ) - 2 \, b d^{6} n + 90 \, a d^{4} x^{2} e^{2} + 36 \, a d^{5} x e + 6 \, b d^{6} \log \left (c\right ) + 6 \, a d^{6}}{360 \, {\left (d^{4} x^{6} e^{9} + 6 \, d^{5} x^{5} e^{8} + 15 \, d^{6} x^{4} e^{7} + 20 \, d^{7} x^{3} e^{6} + 15 \, d^{8} x^{2} e^{5} + 6 \, d^{9} x e^{4} + d^{10} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/360*(6*b*n*x^6*e^6*log(x*e + d) + 36*b*d*n*x^5*e^5*log(x*e + d) + 90*b*d^2*n*x^4*e^4*log(x*e + d) + 120*b*d
^3*n*x^3*e^3*log(x*e + d) + 90*b*d^4*n*x^2*e^2*log(x*e + d) + 36*b*d^5*n*x*e*log(x*e + d) - 6*b*n*x^6*e^6*log(
x) - 36*b*d*n*x^5*e^5*log(x) - 90*b*d^2*n*x^4*e^4*log(x) - 120*b*d^3*n*x^3*e^3*log(x) - 6*b*d*n*x^5*e^5 - 33*b
*d^2*n*x^4*e^4 - 74*b*d^3*n*x^3*e^3 - 63*b*d^4*n*x^2*e^2 - 18*b*d^5*n*x*e + 6*b*d^6*n*log(x*e + d) + 90*b*d^4*
x^2*e^2*log(c) + 36*b*d^5*x*e*log(c) - 2*b*d^6*n + 90*a*d^4*x^2*e^2 + 36*a*d^5*x*e + 6*b*d^6*log(c) + 6*a*d^6)
/(d^4*x^6*e^9 + 6*d^5*x^5*e^8 + 15*d^6*x^4*e^7 + 20*d^7*x^3*e^6 + 15*d^8*x^2*e^5 + 6*d^9*x*e^4 + d^10*e^3)

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Mupad [B]
time = 3.93, size = 275, normalized size = 1.38 \begin {gather*} \frac {\frac {b\,d^2\,n}{3}-a\,d^2-x\,\left (6\,a\,d\,e-3\,b\,d\,e\,n\right )-x^2\,\left (15\,a\,e^2-\frac {21\,b\,e^2\,n}{2}\right )+\frac {37\,b\,e^3\,n\,x^3}{3\,d}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^2}+\frac {b\,e^5\,n\,x^5}{d^3}}{60\,d^6\,e^3+360\,d^5\,e^4\,x+900\,d^4\,e^5\,x^2+1200\,d^3\,e^6\,x^3+900\,d^2\,e^7\,x^4+360\,d\,e^8\,x^5+60\,e^9\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{60\,e^3}+\frac {b\,x^2}{4\,e}+\frac {b\,d\,x}{10\,e^2}\right )}{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}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{30\,d^4\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*log(c*x^n)))/(d + e*x)^7,x)

[Out]

((b*d^2*n)/3 - a*d^2 - x*(6*a*d*e - 3*b*d*e*n) - x^2*(15*a*e^2 - (21*b*e^2*n)/2) + (37*b*e^3*n*x^3)/(3*d) + (1
1*b*e^4*n*x^4)/(2*d^2) + (b*e^5*n*x^5)/d^3)/(60*d^6*e^3 + 60*e^9*x^6 + 360*d^5*e^4*x + 360*d*e^8*x^5 + 900*d^4
*e^5*x^2 + 1200*d^3*e^6*x^3 + 900*d^2*e^7*x^4) - (log(c*x^n)*((b*d^2)/(60*e^3) + (b*x^2)/(4*e) + (b*d*x)/(10*e
^2)))/(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*atan
h((2*e*x)/d + 1))/(30*d^4*e^3)

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