3.2.87 \(\int \frac {(d+e x^2)^2 (a+b \log (c x^n))}{x^3} \, dx\) [187]

Optimal. Leaf size=91 \[ -\frac {b d^2 n}{4 x^2}-\frac {1}{4} b e^2 n x^2-b d e n \log ^2(x)-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+2 d e \log (x) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/4*b*d^2*n/x^2-1/4*b*e^2*n*x^2-b*d*e*n*ln(x)^2-1/2*d^2*(a+b*ln(c*x^n))/x^2+1/2*e^2*x^2*(a+b*ln(c*x^n))+2*d*e
*ln(x)*(a+b*ln(c*x^n))

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {272, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+2 d e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{4 x^2}-b d e n \log ^2(x)-\frac {1}{4} b e^2 n x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(b*d^2*n)/x^2 - (b*e^2*n*x^2)/4 - b*d*e*n*Log[x]^2 - (d^2*(a + b*Log[c*x^n]))/(2*x^2) + (e^2*x^2*(a + b*L
og[c*x^n]))/2 + 2*d*e*Log[x]*(a + b*Log[c*x^n])

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac {1}{2} \left (\frac {d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2+e^2 x^4+4 d e x^2 \log (x)}{2 x^3} \, dx\\ &=-\frac {1}{2} \left (\frac {d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \frac {-d^2+e^2 x^4+4 d e x^2 \log (x)}{x^3} \, dx\\ &=-\frac {1}{2} \left (\frac {d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \left (\frac {-d^2+e^2 x^4}{x^3}+\frac {4 d e \log (x)}{x}\right ) \, dx\\ &=-\frac {1}{2} \left (\frac {d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \frac {-d^2+e^2 x^4}{x^3} \, dx-(2 b d e n) \int \frac {\log (x)}{x} \, dx\\ &=-b d e n \log ^2(x)-\frac {1}{2} \left (\frac {d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \left (-\frac {d^2}{x^3}+e^2 x\right ) \, dx\\ &=-\frac {b d^2 n}{4 x^2}-\frac {1}{4} b e^2 n x^2-b d e n \log ^2(x)-\frac {1}{2} \left (\frac {d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 83, normalized size = 0.91 \begin {gather*} \frac {1}{4} \left (-\frac {b d^2 n}{x^2}-b e^2 n x^2-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {4 d e \left (a+b \log \left (c x^n\right )\right )^2}{b n}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*d^2*n)/x^2) - b*e^2*n*x^2 - (2*d^2*(a + b*Log[c*x^n]))/x^2 + 2*e^2*x^2*(a + b*Log[c*x^n]) + (4*d*e*(a +
b*Log[c*x^n])^2)/(b*n))/4

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 433, normalized size = 4.76

method	result	size
risch	\(-\frac {b \left (-e^{2} x^{4}-4 d e \ln \left (x \right ) x^{2}+d^{2}\right ) \ln \left (x^{n}\right )}{2 x^{2}}-\frac {-i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \ln \left (x \right ) \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2}-4 i \ln \left (x \right ) \pi b d e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (x \right ) \pi b d e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{2}+i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+4 i \ln \left (x \right ) \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{2}+i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-2 \ln \left (c \right ) b \,e^{2} x^{4}+4 b d e n \ln \left (x \right )^{2} x^{2}+b \,e^{2} n \,x^{4}-8 \ln \left (x \right ) \ln \left (c \right ) b d e \,x^{2}-2 x^{4} a \,e^{2}-8 \ln \left (x \right ) a d e \,x^{2}+2 d^{2} b \ln \left (c \right )+b \,d^{2} n +2 a \,d^{2}}{4 x^{2}}\)	\(433\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*(-e^2*x^4-4*d*e*ln(x)*x^2+d^2)/x^2*ln(x^n)-1/4*(-I*Pi*b*e^2*x^4*csgn(I*c)*csgn(I*c*x^n)^2-4*I*ln(x)*Pi*
b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^2-4*I*ln(x)*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^2+I*Pi*b*d^2*csgn(I*x^n)*
csgn(I*c*x^n)^2-I*Pi*b*d^2*csgn(I*c*x^n)^3+I*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2+4*I*ln(x)*Pi*b*d*e*csgn(I*c*x^
n)^3*x^2+I*Pi*b*e^2*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+4*I*ln(x)*Pi*b*d*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)*x^2+I*Pi*b*e^2*x^4*csgn(I*c*x^n)^3-I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d^2*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)-2*ln(c)*b*e^2*x^4+4*b*d*e*n*ln(x)^2*x^2+b*e^2*n*x^4-8*ln(x)*ln(c)*b*d*e*x^2-2*x^4*a*e^2-8*
ln(x)*a*d*e*x^2+2*d^2*b*ln(c)+b*d^2*n+2*a*d^2)/x^2

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 90, normalized size = 0.99 \begin {gather*} -\frac {1}{4} \, b n x^{2} e^{2} + \frac {1}{2} \, b x^{2} e^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a x^{2} e^{2} + \frac {b d e \log \left (c x^{n}\right )^{2}}{n} + 2 \, a d e \log \left (x\right ) - \frac {b d^{2} n}{4 \, x^{2}} - \frac {b d^{2} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a d^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*n*x^2*e^2 + 1/2*b*x^2*e^2*log(c*x^n) + 1/2*a*x^2*e^2 + b*d*e*log(c*x^n)^2/n + 2*a*d*e*log(x) - 1/4*b*d^
2*n/x^2 - 1/2*b*d^2*log(c*x^n)/x^2 - 1/2*a*d^2/x^2

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 105, normalized size = 1.15 \begin {gather*} \frac {4 \, b d n x^{2} e \log \left (x\right )^{2} - {\left (b n - 2 \, a\right )} x^{4} e^{2} - b d^{2} n - 2 \, a d^{2} + 2 \, {\left (b x^{4} e^{2} - b d^{2}\right )} \log \left (c\right ) + 2 \, {\left (b n x^{4} e^{2} + 4 \, b d x^{2} e \log \left (c\right ) + 4 \, a d x^{2} e - b d^{2} n\right )} \log \left (x\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*b*d*n*x^2*e*log(x)^2 - (b*n - 2*a)*x^4*e^2 - b*d^2*n - 2*a*d^2 + 2*(b*x^4*e^2 - b*d^2)*log(c) + 2*(b*n*
x^4*e^2 + 4*b*d*x^2*e*log(c) + 4*a*d*x^2*e - b*d^2*n)*log(x))/x^2

________________________________________________________________________________________

Sympy [A]
time = 0.77, size = 139, normalized size = 1.53 \begin {gather*} \begin {cases} - \frac {a d^{2}}{2 x^{2}} + \frac {2 a d e \log {\left (c x^{n} \right )}}{n} + \frac {a e^{2} x^{2}}{2} - \frac {b d^{2} n}{4 x^{2}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{2 x^{2}} + \frac {b d e \log {\left (c x^{n} \right )}^{2}}{n} - \frac {b e^{2} n x^{2}}{4} + \frac {b e^{2} x^{2} \log {\left (c x^{n} \right )}}{2} & \text {for}\: n \neq 0 \\\left (a + b \log {\left (c \right )}\right ) \left (- \frac {d^{2}}{2 x^{2}} + 2 d e \log {\left (x \right )} + \frac {e^{2} x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*d**2/(2*x**2) + 2*a*d*e*log(c*x**n)/n + a*e**2*x**2/2 - b*d**2*n/(4*x**2) - b*d**2*log(c*x**n)/(
2*x**2) + b*d*e*log(c*x**n)**2/n - b*e**2*n*x**2/4 + b*e**2*x**2*log(c*x**n)/2, Ne(n, 0)), ((a + b*log(c))*(-d
**2/(2*x**2) + 2*d*e*log(x) + e**2*x**2/2), True))

________________________________________________________________________________________

Giac [A]
time = 3.46, size = 112, normalized size = 1.23 \begin {gather*} \frac {2 \, b n x^{4} e^{2} \log \left (x\right ) + 4 \, b d n x^{2} e \log \left (x\right )^{2} - b n x^{4} e^{2} + 2 \, b x^{4} e^{2} \log \left (c\right ) + 8 \, b d x^{2} e \log \left (c\right ) \log \left (x\right ) + 2 \, a x^{4} e^{2} + 8 \, a d x^{2} e \log \left (x\right ) - 2 \, b d^{2} n \log \left (x\right ) - b d^{2} n - 2 \, b d^{2} \log \left (c\right ) - 2 \, a d^{2}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 3.74, size = 110, normalized size = 1.21 \begin {gather*} \ln \left (x\right )\,\left (2\,a\,d\,e+b\,d\,e\,n\right )-\frac {\frac {a\,d^2}{2}+\frac {b\,d^2\,n}{4}}{x^2}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^2}{2}+b\,d\,e\,x^2+\frac {b\,e^2\,x^4}{2}}{x^2}-b\,e^2\,x^2\right )+\frac {e^2\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {b\,d\,e\,{\ln \left (c\,x^n\right )}^2}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*(2*a*d*e + b*d*e*n) - ((a*d^2)/2 + (b*d^2*n)/4)/x^2 - log(c*x^n)*(((b*d^2)/2 + (b*e^2*x^4)/2 + b*d*e*x^
2)/x^2 - b*e^2*x^2) + (e^2*x^2*(2*a - b*n))/4 + (b*d*e*log(c*x^n)^2)/n

________________________________________________________________________________________