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*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])

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Rubi [A]  time = 0.0759305, 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 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])

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]

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*}

Mathematica [A]  time = 0.0537332, 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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Maple [C]  time = 0.253, size = 419, normalized size = 5.4 \begin{align*} -{\frac{b \left ( -2\,dex\ln \left ( x \right ) -{e}^{2}{x}^{2}+{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{x}}-{\frac{2\,i\ln \left ( x \right ) \pi \,bde{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) x-2\,i\ln \left ( x \right ) \pi \,bde \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x-i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-2\,i\ln \left ( x \right ) \pi \,bde{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}x+2\,i\ln \left ( x \right ) \pi \,bde \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}x+2\,bden \left ( \ln \left ( x \right ) \right ) ^{2}x-4\,\ln \left ( x \right ) \ln \left ( c \right ) bdex-2\,\ln \left ( c \right ) b{e}^{2}{x}^{2}+2\,b{e}^{2}n{x}^{2}-4\,\ln \left ( x \right ) adex-2\,a{e}^{2}{x}^{2}+2\,\ln \left ( c \right ) b{d}^{2}+2\,b{d}^{2}n+2\,a{d}^{2}}{2\,x}} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.15123, size = 112, normalized size = 1.44 \begin{align*} -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 \left (x\right ) - \frac{b d^{2} n}{x} - \frac{b d^{2} \log \left (c x^{n}\right )}{x} - \frac{a d^{2}}{x} \end{align*}

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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Fricas [A]  time = 1.02175, size = 215, normalized size = 2.76 \begin{align*} \frac{b d e n x \log \left (x\right )^{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 \left (c\right ) +{\left (b e^{2} n x^{2} + 2 \, b d e x \log \left (c\right ) - b d^{2} n + 2 \, a d e x\right )} \log \left (x\right )}{x} \end{align*}

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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Sympy [A]  time = 5.91223, size = 109, normalized size = 1.4 \begin{align*} - \frac{a d^{2}}{x} + 2 a d e \log{\left (x \right )} + a e^{2} x - \frac{b d^{2} n \log{\left (x \right )}}{x} - \frac{b d^{2} n}{x} - \frac{b d^{2} \log{\left (c \right )}}{x} + b d e n \log{\left (x \right )}^{2} + 2 b d e \log{\left (c \right )} \log{\left (x \right )} + b e^{2} n x \log{\left (x \right )} - b e^{2} n x + b e^{2} x \log{\left (c \right )} \end{align*}

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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Giac [A]  time = 1.44243, size = 136, normalized size = 1.74 \begin{align*} \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{align*}

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