3.1.14 \(\int \frac {(d+e x)^2 (a+b \log (c x^n))}{x^2} \, dx\) [14]

Optimal. Leaf size=78 \[ -\frac {b d^2 n}{x}-b e^2 n x-b d e n \log ^2(x)-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )+2 d e \log (x) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {45, 2372, 2338} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{x}+2 d e \log (x) \left (a+b \log \left (c x^n\right )\right )+e^2 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{x}-b d e n \log ^2(x)-b e^2 n x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.04, size = 76, normalized size = 0.97 \begin {gather*} -\frac {b d^2 n}{x}+a e^2 x-b e^2 n x+b e^2 x \log \left (c x^n\right )-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 419, normalized size = 5.37

method	result	size
risch	\(-\frac {b \left (-2 d e x \ln \left (x \right )-e^{2} x^{2}+d^{2}\right ) \ln \left (x^{n}\right )}{x}-\frac {-i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \ln \left (x \right ) \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x -i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 i \ln \left (x \right ) \pi b d e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x -2 i \ln \left (x \right ) \pi b d e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x +2 i \ln \left (x \right ) \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x +2 b d e n \ln \left (x \right )^{2} x -4 \ln \left (x \right ) \ln \left (c \right ) b d e x -2 \ln \left (c \right ) b \,e^{2} x^{2}+2 b \,e^{2} n \,x^{2}-4 \ln \left (x \right ) a d e x -2 a \,e^{2} x^{2}+2 d^{2} b \ln \left (c \right )+2 b \,d^{2} n +2 a \,d^{2}}{2 x}\)	\(419\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 82, normalized size = 1.05 \begin {gather*} -b n x e^{2} + b x e^{2} \log \left (c x^{n}\right ) + \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}{x} + a x e^{2} - \frac {b d^{2} \log \left (c x^{n}\right )}{x} - \frac {a d^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 95, normalized size = 1.22 \begin {gather*} \frac {b d n x e \log \left (x\right )^{2} - b d^{2} n - {\left (b n - a\right )} x^{2} e^{2} - a d^{2} + {\left (b x^{2} e^{2} - b d^{2}\right )} \log \left (c\right ) + {\left (b n x^{2} e^{2} + 2 \, b d x e \log \left (c\right ) - b d^{2} n + 2 \, a d x e\right )} \log \left (x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.38, size = 112, normalized size = 1.44 \begin {gather*} \begin {cases} - \frac {a d^{2}}{x} + \frac {2 a d e \log {\left (c x^{n} \right )}}{n} + a e^{2} x - \frac {b d^{2} n}{x} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{x} + \frac {b d e \log {\left (c x^{n} \right )}^{2}}{n} - b e^{2} n x + b e^{2} x \log {\left (c x^{n} \right )} & \text {for}\: n \neq 0 \\\left (a + b \log {\left (c \right )}\right ) \left (- \frac {d^{2}}{x} + 2 d e \log {\left (x \right )} + e^{2} x\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.00, size = 101, normalized size = 1.29 \begin {gather*} \frac {b d n x e \log \left (x\right )^{2} + b n x^{2} e^{2} \log \left (x\right ) + 2 \, b d x e \log \left (c\right ) \log \left (x\right ) - b n x^{2} e^{2} + b x^{2} e^{2} \log \left (c\right ) - b d^{2} n \log \left (x\right ) + 2 \, a d x e \log \left (x\right ) - b d^{2} n + a x^{2} e^{2} - b d^{2} \log \left (c\right ) - a d^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.66, size = 99, normalized size = 1.27 \begin {gather*} \ln \left (x\right )\,\left (2\,a\,d\,e+2\,b\,d\,e\,n\right )-\frac {a\,d^2+b\,d^2\,n}{x}-\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2+2\,b\,d\,e\,x+b\,e^2\,x^2}{x}-2\,b\,e^2\,x\right )+e^2\,x\,\left (a-b\,n\right )+\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(((a + b*log(c*x^n))*(d + e*x)^2)/x^2,x)

[Out]

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

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