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

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

[Out]

-(b*d^2*n)/(4*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])

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Rubi [A]  time = 0.0986328, antiderivative size = 71, normalized size of antiderivative = 0.78, 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 = {266, 43, 2334, 12, 14, 2301} \[ -\frac{1}{2} \left (\frac{d^2}{x^2}-4 d e \log (x)-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{4 x^2}-b d e n \log ^2(x)-\frac{1}{4} b e^2 n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 266

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 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 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 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{\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 = 0.0584254, size = 83, normalized size = 0.91 \[ \frac{1}{4} \left (-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac{4 d e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{x^2}-b e^2 n x^2\right ) \]

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

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Maple [C]  time = 0.237, size = 433, normalized size = 4.8 \begin{align*} -{\frac{b \left ( -{e}^{2}{x}^{4}-4\,de\ln \left ( x \right ){x}^{2}+{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{2\,{x}^{2}}}-{\frac{-4\,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\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}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-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 ) +4\,i\ln \left ( x \right ) \pi \,bde \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}{x}^{2}+i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -2\,\ln \left ( c \right ) b{e}^{2}{x}^{4}+4\,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\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-4\,i\ln \left ( x \right ) \pi \,bde \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ){x}^{2}-i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+4\,bden \left ( \ln \left ( x \right ) \right ) ^{2}{x}^{2}+b{e}^{2}n{x}^{4}-8\,\ln \left ( x \right ) \ln \left ( c \right ) bde{x}^{2}-2\,a{e}^{2}{x}^{4}-8\,\ln \left ( x \right ) ade{x}^{2}+2\,\ln \left ( c \right ) b{d}^{2}+b{d}^{2}n+2\,a{d}^{2}}{4\,{x}^{2}}} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.0803, size = 123, normalized size = 1.35 \begin{align*} -\frac{1}{4} \, b e^{2} n x^{2} + \frac{1}{2} \, b e^{2} x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a e^{2} x^{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{align*}

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

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Fricas [A]  time = 1.234, size = 244, normalized size = 2.68 \begin{align*} \frac{4 \, b d e n x^{2} \log \left (x\right )^{2} -{\left (b e^{2} n - 2 \, a e^{2}\right )} x^{4} - b d^{2} n - 2 \, a d^{2} + 2 \,{\left (b e^{2} x^{4} - b d^{2}\right )} \log \left (c\right ) + 2 \,{\left (b e^{2} n x^{4} + 4 \, b d e x^{2} \log \left (c\right ) + 4 \, a d e x^{2} - b d^{2} n\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}

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

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Sympy [A]  time = 3.54678, size = 136, normalized size = 1.49 \begin{align*} - \frac{a d^{2}}{2 x^{2}} + 2 a d e \log{\left (x \right )} + \frac{a e^{2} x^{2}}{2} - \frac{b d^{2} n \log{\left (x \right )}}{2 x^{2}} - \frac{b d^{2} n}{4 x^{2}} - \frac{b d^{2} \log{\left (c \right )}}{2 x^{2}} + b d e n \log{\left (x \right )}^{2} + 2 b d e \log{\left (c \right )} \log{\left (x \right )} + \frac{b e^{2} n x^{2} \log{\left (x \right )}}{2} - \frac{b e^{2} n x^{2}}{4} + \frac{b e^{2} x^{2} \log{\left (c \right )}}{2} \end{align*}

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]

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

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Giac [A]  time = 1.29697, size = 151, normalized size = 1.66 \begin{align*} \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{align*}

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