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

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

[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 = 0.08, antiderivative size = 61, normalized size of antiderivative = 0.78, 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 = {43, 2334, 2301} \[ -\left (\frac {d^2}{x}-2 d e \log (x)-e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{x}-b d e n \log ^2(x)-b e^2 n x \]

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

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 2301

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 = 0.06, size = 76, normalized size = 0.97 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+a e^2 x+b e^2 x \log \left (c x^n\right )-\frac {b d^2 n}{x}-b e^2 n x \]

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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fricas [A]  time = 0.64, size = 98, normalized size = 1.26 \[ \frac {b d e n x \log \relax (x)^{2} - b d^{2} n - a d^{2} - {\left (b e^{2} n - a e^{2}\right )} x^{2} + {\left (b e^{2} x^{2} - b d^{2}\right )} \log \relax (c) + {\left (b e^{2} n x^{2} + 2 \, b d e x \log \relax (c) - b d^{2} n + 2 \, a d e x\right )} \log \relax (x)}{x} \]

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

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giac [A]  time = 0.30, size = 101, normalized size = 1.29 \[ \frac {b d n x e \log \relax (x)^{2} + b n x^{2} e^{2} \log \relax (x) + 2 \, b d x e \log \relax (c) \log \relax (x) - b n x^{2} e^{2} + b x^{2} e^{2} \log \relax (c) - b d^{2} n \log \relax (x) + 2 \, a d x e \log \relax (x) - b d^{2} n + a x^{2} e^{2} - b d^{2} \log \relax (c) - a d^{2}}{x} \]

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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maple [C]  time = 0.32, size = 419, normalized size = 5.37 \[ -\frac {\left (-2 d e x \ln \relax (x )-e^{2} x^{2}+d^{2}\right ) b \ln \left (x^{n}\right )}{x}-\frac {2 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )-2 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )-2 i \pi b d e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )+2 i \pi b d e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )+i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b d e n x \ln \relax (x )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b d e x \ln \relax (c ) \ln \relax (x )+2 b \,e^{2} n \,x^{2}-2 b \,e^{2} x^{2} \ln \relax (c )-4 a d e x \ln \relax (x )-2 a \,e^{2} x^{2}+2 b \,d^{2} n +2 b \,d^{2} \ln \relax (c )+2 a \,d^{2}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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maxima [A]  time = 0.53, size = 83, normalized size = 1.06 \[ -b e^{2} n x + b e^{2} x \log \left (c x^{n}\right ) + a e^{2} x + \frac {b d e \log \left (c x^{n}\right )^{2}}{n} + 2 \, a d e \log \relax (x) - \frac {b d^{2} n}{x} - \frac {b d^{2} \log \left (c x^{n}\right )}{x} - \frac {a d^{2}}{x} \]

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

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mupad [B]  time = 3.66, size = 99, normalized size = 1.27 \[ \ln \relax (x)\,\left (2\,a\,d\,e+2\,b\,d\,e\,n\right )-\frac {a\,d^2+b\,d^2\,n}{x}-\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2+2\,b\,d\,e\,x+b\,e^2\,x^2}{x}-2\,b\,e^2\,x\right )+e^2\,x\,\left (a-b\,n\right )+\frac {b\,d\,e\,{\ln \left (c\,x^n\right )}^2}{n} \]

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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sympy [A]  time = 1.01, size = 109, normalized size = 1.40 \[ - \frac {a d^{2}}{x} + 2 a d e \log {\relax (x )} + a e^{2} x - \frac {b d^{2} n \log {\relax (x )}}{x} - \frac {b d^{2} n}{x} - \frac {b d^{2} \log {\relax (c )}}{x} + b d e n \log {\relax (x )}^{2} + 2 b d e \log {\relax (c )} \log {\relax (x )} + b e^{2} n x \log {\relax (x )} - b e^{2} n x + b e^{2} x \log {\relax (c )} \]

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]

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

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