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

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

[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 = 0.16, 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 = {43, 2350, 12, 893} \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}+\frac {b n}{120 d^2 e^3 (d+e x)^2}+\frac {b n}{60 d^3 e^3 (d+e x)}+\frac {b n \log (x)}{60 d^4 e^3}-\frac {b n \log (d+e x)}{60 d^4 e^3}+\frac {b d n}{30 e^3 (d+e x)^5}-\frac {7 b n}{120 e^3 (d+e x)^4}+\frac {b n}{180 d e^3 (d+e x)^3} \]

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 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 893

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 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^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 = 0.21, size = 192, normalized size = 0.96 \[ \frac {-60 a d^6+144 a d^5 (d+e x)-90 a d^4 (d+e x)^2-60 b d^6 \log \left (c x^n\right )+144 b d^5 (d+e x) \log \left (c x^n\right )-90 b d^4 (d+e x)^2 \log \left (c x^n\right )+12 b d^5 n (d+e x)-21 b d^4 n (d+e x)^2+2 b d^3 n (d+e x)^3+3 b d^2 n (d+e x)^4+6 b d n (d+e x)^5+6 b n \log (x) (d+e x)^6-6 b n (d+e x)^6 \log (d+e x)}{360 d^4 e^3 (d+e x)^6} \]

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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fricas [A]  time = 0.76, size = 333, normalized size = 1.67 \[ \frac {6 \, b d e^{5} n x^{5} + 33 \, b d^{2} e^{4} n x^{4} + 74 \, b d^{3} e^{3} n x^{3} + 2 \, b d^{6} n - 6 \, a d^{6} + 9 \, {\left (7 \, b d^{4} e^{2} n - 10 \, a d^{4} e^{2}\right )} x^{2} + 18 \, {\left (b d^{5} e n - 2 \, a d^{5} e\right )} x - 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 6 \, {\left (15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \relax (c) + 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3}\right )} \log \relax (x)}{360 \, {\left (d^{4} e^{9} x^{6} + 6 \, d^{5} e^{8} x^{5} + 15 \, d^{6} e^{7} x^{4} + 20 \, d^{7} e^{6} x^{3} + 15 \, d^{8} e^{5} x^{2} + 6 \, d^{9} e^{4} x + d^{10} e^{3}\right )}} \]

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

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giac [B]  time = 0.34, size = 362, normalized size = 1.82 \[ -\frac {6 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 36 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 90 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 120 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 90 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 36 \, b d^{5} n x e \log \left (x e + d\right ) - 6 \, b n x^{6} e^{6} \log \relax (x) - 36 \, b d n x^{5} e^{5} \log \relax (x) - 90 \, b d^{2} n x^{4} e^{4} \log \relax (x) - 120 \, b d^{3} n x^{3} e^{3} \log \relax (x) - 6 \, b d n x^{5} e^{5} - 33 \, b d^{2} n x^{4} e^{4} - 74 \, b d^{3} n x^{3} e^{3} - 63 \, b d^{4} n x^{2} e^{2} - 18 \, b d^{5} n x e + 6 \, b d^{6} n \log \left (x e + d\right ) + 90 \, b d^{4} x^{2} e^{2} \log \relax (c) + 36 \, b d^{5} x e \log \relax (c) - 2 \, b d^{6} n + 90 \, a d^{4} x^{2} e^{2} + 36 \, a d^{5} x e + 6 \, b d^{6} \log \relax (c) + 6 \, a d^{6}}{360 \, {\left (d^{4} x^{6} e^{9} + 6 \, d^{5} x^{5} e^{8} + 15 \, d^{6} x^{4} e^{7} + 20 \, d^{7} x^{3} e^{6} + 15 \, d^{8} x^{2} e^{5} + 6 \, d^{9} x e^{4} + d^{10} e^{3}\right )}} \]

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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maple [C]  time = 0.27, size = 712, normalized size = 3.58 \[ -\frac {\left (15 e^{2} x^{2}+6 d e x +d^{2}\right ) b \ln \left (x^{n}\right )}{60 \left (e x +d \right )^{6} e^{3}}+\frac {3 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+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}+63 b \,d^{4} e^{2} n \,x^{2}+18 b \,d^{5} e n x -6 b \,d^{6} n \ln \left (e x +d \right )+6 b \,d^{6} n \ln \left (-x \right )-90 a \,d^{4} e^{2} x^{2}-36 a \,d^{5} e x -6 a \,d^{6}-6 b \,d^{6} \ln \relax (c )+2 b \,d^{6} n -6 b \,e^{6} n \,x^{6} \ln \left (e x +d \right )+6 b \,e^{6} n \,x^{6} \ln \left (-x \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 )-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}-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}-90 b \,d^{4} e^{2} x^{2} \ln \relax (c )-36 b \,d^{5} e x \ln \relax (c )+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 )-36 b d \,e^{5} n \,x^{5} \ln \left (e x +d \right )-90 b \,d^{2} e^{4} n \,x^{4} \ln \left (e x +d \right )-120 b \,d^{3} e^{3} n \,x^{3} \ln \left (e x +d \right )-90 b \,d^{4} e^{2} n \,x^{2} \ln \left (e x +d \right )-36 b \,d^{5} e n x \ln \left (e x +d \right )+36 b d \,e^{5} n \,x^{5} \ln \left (-x \right )+90 b \,d^{2} e^{4} n \,x^{4} \ln \left (-x \right )+120 b \,d^{3} e^{3} n \,x^{3} \ln \left (-x \right )+90 b \,d^{4} e^{2} n \,x^{2} \ln \left (-x \right )+36 b \,d^{5} e n x \ln \left (-x \right )-3 i \pi b \,d^{6} \mathrm {csgn}\left (i c \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}}{360 \left (e x +d \right )^{6} d^{4} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*(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+63*b*d^4*e^2*n*x^2+18*b*d^5*e*n*x-6*b*d^6*n*ln(e*x+d)+6*b*d^6*n*ln(-x)-90*a*d^4*e^2*x^2-36*a*d^5*e*x-6*a
*d^6-6*b*d^6*ln(c)+2*b*d^6*n-45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)+18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3*I*Pi*b*d^6*csgn(I*c*x^n)^3-6*
b*e^6*n*x^6*ln(e*x+d)+6*b*e^6*n*x^6*ln(-x)-18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*b*d^5*e*x*csg
n(I*c*x^n)^2*csgn(I*c)-45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-90*b*d^4*e^2*x^2*ln(c)-36*b*d^5*e*x*l
n(c)+45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3-3*I*Pi*b*d^6*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b*d^6*csgn(I*x^n)*csg
n(I*c*x^n)^2+18*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3+3*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*b*d*e^5*n*x
^5*ln(e*x+d)-90*b*d^2*e^4*n*x^4*ln(e*x+d)-120*b*d^3*e^3*n*x^3*ln(e*x+d)-90*b*d^4*e^2*n*x^2*ln(e*x+d)-36*b*d^5*
e*n*x*ln(e*x+d)+36*b*d*e^5*n*x^5*ln(-x)+90*b*d^2*e^4*n*x^4*ln(-x)+120*b*d^3*e^3*n*x^3*ln(-x)+90*b*d^4*e^2*n*x^
2*ln(-x)+36*b*d^5*e*n*x*ln(-x))/d^4/e^3/(e*x+d)^6

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maxima [A]  time = 0.67, size = 316, normalized size = 1.59 \[ \frac {1}{360} \, b n {\left (\frac {6 \, e^{4} x^{4} + 27 \, d e^{3} x^{3} + 47 \, d^{2} e^{2} x^{2} + 16 \, d^{3} e x + 2 \, d^{4}}{d^{3} e^{8} x^{5} + 5 \, d^{4} e^{7} x^{4} + 10 \, d^{5} e^{6} x^{3} + 10 \, d^{6} e^{5} x^{2} + 5 \, d^{7} e^{4} x + d^{8} e^{3}} - \frac {6 \, \log \left (e x + d\right )}{d^{4} e^{3}} + \frac {6 \, \log \relax (x)}{d^{4} e^{3}}\right )} - \frac {{\left (15 \, e^{2} x^{2} + 6 \, d e x + d^{2}\right )} b \log \left (c x^{n}\right )}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} - \frac {{\left (15 \, e^{2} x^{2} + 6 \, d e x + d^{2}\right )} a}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]

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

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mupad [B]  time = 3.93, size = 275, normalized size = 1.38 \[ \frac {\frac {b\,d^2\,n}{3}-a\,d^2-x\,\left (6\,a\,d\,e-3\,b\,d\,e\,n\right )-x^2\,\left (15\,a\,e^2-\frac {21\,b\,e^2\,n}{2}\right )+\frac {37\,b\,e^3\,n\,x^3}{3\,d}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^2}+\frac {b\,e^5\,n\,x^5}{d^3}}{60\,d^6\,e^3+360\,d^5\,e^4\,x+900\,d^4\,e^5\,x^2+1200\,d^3\,e^6\,x^3+900\,d^2\,e^7\,x^4+360\,d\,e^8\,x^5+60\,e^9\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{60\,e^3}+\frac {b\,x^2}{4\,e}+\frac {b\,d\,x}{10\,e^2}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{30\,d^4\,e^3} \]

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

[Out]

Timed out

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