\(\int \frac {(d+e x)^2 (a+b \log (c x^n))}{x^6} \, dx\) [18]

Optimal result

Integrand size = 21, antiderivative size = 95 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b d^2 n}{25 x^5}-\frac {b d e n}{8 x^4}-\frac {b e^2 n}{9 x^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]

[Out]

-1/25*b*d^2*n/x^5-1/8*b*d*e*n/x^4-1/9*b*e^2*n/x^3-1/5*d^2*(a+b*ln(c*x^n))/x^5-1/2*d*e*(a+b*ln(c*x^n))/x^4-1/3*
e^2*(a+b*ln(c*x^n))/x^3

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2372, 12, 14} \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {b d^2 n}{25 x^5}-\frac {b d e n}{8 x^4}-\frac {b e^2 n}{9 x^3} \]

[In]

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

[Out]

-1/25*(b*d^2*n)/x^5 - (b*d*e*n)/(8*x^4) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(5*x^5) - (d*e*(a + b*L
og[c*x^n]))/(2*x^4) - (e^2*(a + b*Log[c*x^n]))/(3*x^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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2372

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-(b n) \int \frac {-6 d^2-15 d e x-10 e^2 x^2}{30 x^6} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {1}{30} (b n) \int \frac {-6 d^2-15 d e x-10 e^2 x^2}{x^6} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {1}{30} (b n) \int \left (-\frac {6 d^2}{x^6}-\frac {15 d e}{x^5}-\frac {10 e^2}{x^4}\right ) \, dx \\ & = -\frac {b d^2 n}{25 x^5}-\frac {b d e n}{8 x^4}-\frac {b e^2 n}{9 x^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {60 a \left (6 d^2+15 d e x+10 e^2 x^2\right )+b n \left (72 d^2+225 d e x+200 e^2 x^2\right )+60 b \left (6 d^2+15 d e x+10 e^2 x^2\right ) \log \left (c x^n\right )}{1800 x^5} \]

[In]

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

[Out]

-1/1800*(60*a*(6*d^2 + 15*d*e*x + 10*e^2*x^2) + b*n*(72*d^2 + 225*d*e*x + 200*e^2*x^2) + 60*b*(6*d^2 + 15*d*e*
x + 10*e^2*x^2)*Log[c*x^n])/x^5

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(-\frac {600 b \ln \left (c \,x^{n}\right ) e^{2} x^{2}+200 b \,e^{2} n \,x^{2}+600 a \,e^{2} x^{2}+900 b \ln \left (c \,x^{n}\right ) d e x +225 b d e n x +900 a d e x +360 b \ln \left (c \,x^{n}\right ) d^{2}+72 b \,d^{2} n +360 a \,d^{2}}{1800 x^{5}}\)	\(91\)
risch	\(-\frac {b \left (10 e^{2} x^{2}+15 d e x +6 d^{2}\right ) \ln \left (x^{n}\right )}{30 x^{5}}-\frac {-180 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-450 i \pi b d e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+180 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-180 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+600 \ln \left (c \right ) b \,e^{2} x^{2}+200 b \,e^{2} n \,x^{2}+600 a \,e^{2} x^{2}+300 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+450 i \pi b d e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-300 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+180 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+900 \ln \left (c \right ) b d e x +225 b d e n x +900 a d e x +450 i \pi b d e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+300 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-300 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-450 i \pi b d e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+360 d^{2} b \ln \left (c \right )+72 b \,d^{2} n +360 a \,d^{2}}{1800 x^{5}}\)	\(403\)

[In]

int((e*x+d)^2*(a+b*ln(c*x^n))/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/1800/x^5*(600*b*ln(c*x^n)*e^2*x^2+200*b*e^2*n*x^2+600*a*e^2*x^2+900*b*ln(c*x^n)*d*e*x+225*b*d*e*n*x+900*a*d
*e*x+360*b*ln(c*x^n)*d^2+72*b*d^2*n+360*a*d^2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {72 \, b d^{2} n + 360 \, a d^{2} + 200 \, {\left (b e^{2} n + 3 \, a e^{2}\right )} x^{2} + 225 \, {\left (b d e n + 4 \, a d e\right )} x + 60 \, {\left (10 \, b e^{2} x^{2} + 15 \, b d e x + 6 \, b d^{2}\right )} \log \left (c\right ) + 60 \, {\left (10 \, b e^{2} n x^{2} + 15 \, b d e n x + 6 \, b d^{2} n\right )} \log \left (x\right )}{1800 \, x^{5}} \]

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))/x^6,x, algorithm="fricas")

[Out]

-1/1800*(72*b*d^2*n + 360*a*d^2 + 200*(b*e^2*n + 3*a*e^2)*x^2 + 225*(b*d*e*n + 4*a*d*e)*x + 60*(10*b*e^2*x^2 +
15*b*d*e*x + 6*b*d^2)*log(c) + 60*(10*b*e^2*n*x^2 + 15*b*d*e*n*x + 6*b*d^2*n)*log(x))/x^5

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=- \frac {a d^{2}}{5 x^{5}} - \frac {a d e}{2 x^{4}} - \frac {a e^{2}}{3 x^{3}} - \frac {b d^{2} n}{25 x^{5}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b d e n}{8 x^{4}} - \frac {b d e \log {\left (c x^{n} \right )}}{2 x^{4}} - \frac {b e^{2} n}{9 x^{3}} - \frac {b e^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} \]

[In]

integrate((e*x+d)**2*(a+b*ln(c*x**n))/x**6,x)

[Out]

-a*d**2/(5*x**5) - a*d*e/(2*x**4) - a*e**2/(3*x**3) - b*d**2*n/(25*x**5) - b*d**2*log(c*x**n)/(5*x**5) - b*d*e
*n/(8*x**4) - b*d*e*log(c*x**n)/(2*x**4) - b*e**2*n/(9*x**3) - b*e**2*log(c*x**n)/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b e^{2} n}{9 \, x^{3}} - \frac {b e^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {b d e n}{8 \, x^{4}} - \frac {a e^{2}}{3 \, x^{3}} - \frac {b d e \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac {b d^{2} n}{25 \, x^{5}} - \frac {a d e}{2 \, x^{4}} - \frac {b d^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d^{2}}{5 \, x^{5}} \]

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))/x^6,x, algorithm="maxima")

[Out]

-1/9*b*e^2*n/x^3 - 1/3*b*e^2*log(c*x^n)/x^3 - 1/8*b*d*e*n/x^4 - 1/3*a*e^2/x^3 - 1/2*b*d*e*log(c*x^n)/x^4 - 1/2
5*b*d^2*n/x^5 - 1/2*a*d*e/x^4 - 1/5*b*d^2*log(c*x^n)/x^5 - 1/5*a*d^2/x^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {{\left (10 \, b e^{2} n x^{2} + 15 \, b d e n x + 6 \, b d^{2} n\right )} \log \left (x\right )}{30 \, x^{5}} - \frac {200 \, b e^{2} n x^{2} + 600 \, b e^{2} x^{2} \log \left (c\right ) + 225 \, b d e n x + 600 \, a e^{2} x^{2} + 900 \, b d e x \log \left (c\right ) + 72 \, b d^{2} n + 900 \, a d e x + 360 \, b d^{2} \log \left (c\right ) + 360 \, a d^{2}}{1800 \, x^{5}} \]

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))/x^6,x, algorithm="giac")

[Out]

-1/30*(10*b*e^2*n*x^2 + 15*b*d*e*n*x + 6*b*d^2*n)*log(x)/x^5 - 1/1800*(200*b*e^2*n*x^2 + 600*b*e^2*x^2*log(c)
+ 225*b*d*e*n*x + 600*a*e^2*x^2 + 900*b*d*e*x*log(c) + 72*b*d^2*n + 900*a*d*e*x + 360*b*d^2*log(c) + 360*a*d^2
)/x^5

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {x^2\,\left (10\,a\,e^2+\frac {10\,b\,e^2\,n}{3}\right )+6\,a\,d^2+x\,\left (15\,a\,d\,e+\frac {15\,b\,d\,e\,n}{4}\right )+\frac {6\,b\,d^2\,n}{5}}{30\,x^5}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{5}+\frac {b\,d\,e\,x}{2}+\frac {b\,e^2\,x^2}{3}\right )}{x^5} \]

[In]

int(((a + b*log(c*x^n))*(d + e*x)^2)/x^6,x)

[Out]

- (x^2*(10*a*e^2 + (10*b*e^2*n)/3) + 6*a*d^2 + x*(15*a*d*e + (15*b*d*e*n)/4) + (6*b*d^2*n)/5)/(30*x^5) - (log(
c*x^n)*((b*d^2)/5 + (b*e^2*x^2)/3 + (b*d*e*x)/2))/x^5