\(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^6} \, dx\) [207]

Optimal result

Integrand size = 23, antiderivative size = 118 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b d^3 n}{25 x^5}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{x}-b e^3 n x-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {276, 2372} \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{25 x^5}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{x}-b e^3 n x \]

[In]

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/25*(b*d^3*n)/x^5 - (b*d^2*e*n)/(3*x^3) - (3*b*d*e^2*n)/x - b*e^3*n*x - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (
d^2*e*(a + b*Log[c*x^n]))/x^3 - (3*d*e^2*(a + b*Log[c*x^n]))/x + e^3*x*(a + b*Log[c*x^n])

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^3-\frac {d^3}{5 x^6}-\frac {d^2 e}{x^4}-\frac {3 d e^2}{x^2}\right ) \, dx \\ & = -\frac {b d^3 n}{25 x^5}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{x}-b e^3 n x-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {15 a \left (d^3+5 d^2 e x^2+15 d e^2 x^4-5 e^3 x^6\right )+b n \left (3 d^3+25 d^2 e x^2+225 d e^2 x^4+75 e^3 x^6\right )+15 b \left (d^3+5 d^2 e x^2+15 d e^2 x^4-5 e^3 x^6\right ) \log \left (c x^n\right )}{75 x^5} \]

[In]

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/75*(15*a*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6) + b*n*(3*d^3 + 25*d^2*e*x^2 + 225*d*e^2*x^4 + 75*e^
3*x^6) + 15*b*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6)*Log[c*x^n])/x^5

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.19

method	result	size
parallelrisch	\(-\frac {-75 x^{6} b \ln \left (c \,x^{n}\right ) e^{3}+75 b \,e^{3} n \,x^{6}-75 x^{6} a \,e^{3}+225 x^{4} b \ln \left (c \,x^{n}\right ) d \,e^{2}+225 b d \,e^{2} n \,x^{4}+225 x^{4} a d \,e^{2}+75 b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{2}+25 b \,d^{2} e n \,x^{2}+75 a \,d^{2} e \,x^{2}+15 b \ln \left (c \,x^{n}\right ) d^{3}+3 b \,d^{3} n +15 a \,d^{3}}{75 x^{5}}\)	\(140\)
risch	\(-\frac {b \left (-5 e^{3} x^{6}+15 e^{2} d \,x^{4}+5 d^{2} e \,x^{2}+d^{3}\right ) \ln \left (x^{n}\right )}{5 x^{5}}-\frac {-150 x^{6} a \,e^{3}+450 \ln \left (c \right ) b d \,e^{2} x^{4}-225 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-75 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-15 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+450 x^{4} a d \,e^{2}+150 a \,d^{2} e \,x^{2}+30 a \,d^{3}+30 d^{3} b \ln \left (c \right )-150 \ln \left (c \right ) b \,e^{3} x^{6}+150 e \ln \left (c \right ) b \,d^{2} x^{2}+6 b \,d^{3} n -75 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-15 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+75 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-75 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-75 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+75 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+75 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+225 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+225 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+450 b d \,e^{2} n \,x^{4}+50 b \,d^{2} e n \,x^{2}+150 b \,e^{3} n \,x^{6}+15 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+15 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{150 x^{5}}\)	\(585\)

[In]

int((e*x^2+d)^3*(a+b*ln(c*x^n))/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/75/x^5*(-75*x^6*b*ln(c*x^n)*e^3+75*b*e^3*n*x^6-75*x^6*a*e^3+225*x^4*b*ln(c*x^n)*d*e^2+225*b*d*e^2*n*x^4+225
*x^4*a*d*e^2+75*b*ln(c*x^n)*d^2*e*x^2+25*b*d^2*e*n*x^2+75*a*d^2*e*x^2+15*b*ln(c*x^n)*d^3+3*b*d^3*n+15*a*d^3)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {75 \, {\left (b e^{3} n - a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 225 \, {\left (b d e^{2} n + a d e^{2}\right )} x^{4} + 15 \, a d^{3} + 25 \, {\left (b d^{2} e n + 3 \, a d^{2} e\right )} x^{2} - 15 \, {\left (5 \, b e^{3} x^{6} - 15 \, b d e^{2} x^{4} - 5 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) - 15 \, {\left (5 \, b e^{3} n x^{6} - 15 \, b d e^{2} n x^{4} - 5 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )}{75 \, x^{5}} \]

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^6,x, algorithm="fricas")

[Out]

-1/75*(75*(b*e^3*n - a*e^3)*x^6 + 3*b*d^3*n + 225*(b*d*e^2*n + a*d*e^2)*x^4 + 15*a*d^3 + 25*(b*d^2*e*n + 3*a*d
^2*e)*x^2 - 15*(5*b*e^3*x^6 - 15*b*d*e^2*x^4 - 5*b*d^2*e*x^2 - b*d^3)*log(c) - 15*(5*b*e^3*n*x^6 - 15*b*d*e^2*
n*x^4 - 5*b*d^2*e*n*x^2 - b*d^3*n)*log(x))/x^5

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=- \frac {a d^{3}}{5 x^{5}} - \frac {a d^{2} e}{x^{3}} - \frac {3 a d e^{2}}{x} + a e^{3} x - \frac {b d^{3} n}{25 x^{5}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b d^{2} e n}{3 x^{3}} - \frac {b d^{2} e \log {\left (c x^{n} \right )}}{x^{3}} - \frac {3 b d e^{2} n}{x} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{x} - b e^{3} n x + b e^{3} x \log {\left (c x^{n} \right )} \]

[In]

integrate((e*x**2+d)**3*(a+b*ln(c*x**n))/x**6,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-b e^{3} n x + b e^{3} x \log \left (c x^{n}\right ) + a e^{3} x - \frac {3 \, b d e^{2} n}{x} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{x} - \frac {3 \, a d e^{2}}{x} - \frac {b d^{2} e n}{3 \, x^{3}} - \frac {b d^{2} e \log \left (c x^{n}\right )}{x^{3}} - \frac {a d^{2} e}{x^{3}} - \frac {b d^{3} n}{25 \, x^{5}} - \frac {b d^{3} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d^{3}}{5 \, x^{5}} \]

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^6,x, algorithm="maxima")

[Out]

-b*e^3*n*x + b*e^3*x*log(c*x^n) + a*e^3*x - 3*b*d*e^2*n/x - 3*b*d*e^2*log(c*x^n)/x - 3*a*d*e^2/x - 1/3*b*d^2*e
*n/x^3 - b*d^2*e*log(c*x^n)/x^3 - a*d^2*e/x^3 - 1/25*b*d^3*n/x^5 - 1/5*b*d^3*log(c*x^n)/x^5 - 1/5*a*d^3/x^5

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-{\left (b e^{3} n - b e^{3} \log \left (c\right ) - a e^{3}\right )} x + \frac {1}{5} \, {\left (5 \, b e^{3} n x - \frac {15 \, b d e^{2} n x^{4} + 5 \, b d^{2} e n x^{2} + b d^{3} n}{x^{5}}\right )} \log \left (x\right ) - \frac {225 \, b d e^{2} n x^{4} + 225 \, b d e^{2} x^{4} \log \left (c\right ) + 225 \, a d e^{2} x^{4} + 25 \, b d^{2} e n x^{2} + 75 \, b d^{2} e x^{2} \log \left (c\right ) + 75 \, a d^{2} e x^{2} + 3 \, b d^{3} n + 15 \, b d^{3} \log \left (c\right ) + 15 \, a d^{3}}{75 \, x^{5}} \]

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^6,x, algorithm="giac")

[Out]

-(b*e^3*n - b*e^3*log(c) - a*e^3)*x + 1/5*(5*b*e^3*n*x - (15*b*d*e^2*n*x^4 + 5*b*d^2*e*n*x^2 + b*d^3*n)/x^5)*l
og(x) - 1/75*(225*b*d*e^2*n*x^4 + 225*b*d*e^2*x^4*log(c) + 225*a*d*e^2*x^4 + 25*b*d^2*e*n*x^2 + 75*b*d^2*e*x^2
*log(c) + 75*a*d^2*e*x^2 + 3*b*d^3*n + 15*b*d^3*log(c) + 15*a*d^3)/x^5

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=e^3\,x\,\left (a-b\,n\right )-\frac {a\,d^3+x^2\,\left (5\,a\,d^2\,e+\frac {5\,b\,d^2\,e\,n}{3}\right )+x^4\,\left (15\,a\,d\,e^2+15\,b\,d\,e^2\,n\right )+\frac {b\,d^3\,n}{5}}{5\,x^5}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^3}{5}+b\,d^2\,e\,x^2+3\,b\,d\,e^2\,x^4+\frac {11\,b\,e^3\,x^6}{5}}{x^5}-\frac {16\,b\,e^3\,x}{5}\right ) \]

[In]

int(((d + e*x^2)^3*(a + b*log(c*x^n)))/x^6,x)

[Out]

e^3*x*(a - b*n) - (a*d^3 + x^2*(5*a*d^2*e + (5*b*d^2*e*n)/3) + x^4*(15*a*d*e^2 + 15*b*d*e^2*n) + (b*d^3*n)/5)/
(5*x^5) - log(c*x^n)*(((b*d^3)/5 + (11*b*e^3*x^6)/5 + b*d^2*e*x^2 + 3*b*d*e^2*x^4)/x^5 - (16*b*e^3*x)/5)