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

Optimal. Leaf size=89 \[ d^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+d e x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} b d^2 n \log ^2(x)-\frac{1}{2} b d e n x^2-\frac{1}{16} b e^2 n x^4 \]

[Out]

-(b*d*e*n*x^2)/2 - (b*e^2*n*x^4)/16 - (b*d^2*n*Log[x]^2)/2 + d*e*x^2*(a + b*Log[c*x^n]) + (e^2*x^4*(a + b*Log[
c*x^n]))/4 + d^2*Log[x]*(a + b*Log[c*x^n])

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Rubi [A]  time = 0.0818464, antiderivative size = 73, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {266, 43, 2334, 2301} \[ \frac{1}{4} \left (4 d^2 \log (x)+4 d e x^2+e^2 x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} b d^2 n \log ^2(x)-\frac{1}{2} b d e n x^2-\frac{1}{16} b e^2 n x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*e*n*x^2)/2 - (b*e^2*n*x^4)/16 - (b*d^2*n*Log[x]^2)/2 + ((4*d*e*x^2 + e^2*x^4 + 4*d^2*Log[x])*(a + b*Log[
c*x^n]))/4

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 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} \, dx &=\frac{1}{4} \left (4 d e x^2+e^2 x^4+4 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d e x+\frac{e^2 x^3}{4}+\frac{d^2 \log (x)}{x}\right ) \, dx\\ &=-\frac{1}{2} b d e n x^2-\frac{1}{16} b e^2 n x^4+\frac{1}{4} \left (4 d e x^2+e^2 x^4+4 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^2 n\right ) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{1}{2} b d e n x^2-\frac{1}{16} b e^2 n x^4-\frac{1}{2} b d^2 n \log ^2(x)+\frac{1}{4} \left (4 d e x^2+e^2 x^4+4 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0495569, size = 82, normalized size = 0.92 \[ \frac{1}{16} \left (\frac{8 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+16 d e x^2 \left (a+b \log \left (c x^n\right )\right )+4 e^2 x^4 \left (a+b \log \left (c x^n\right )\right )-8 b d e n x^2-b e^2 n x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*d*e*n*x^2 - b*e^2*n*x^4 + 16*d*e*x^2*(a + b*Log[c*x^n]) + 4*e^2*x^4*(a + b*Log[c*x^n]) + (8*d^2*(a + b*L
og[c*x^n])^2)/(b*n))/16

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Maple [C]  time = 0.215, size = 423, normalized size = 4.8 \begin{align*} \left ({\frac{b{e}^{2}{x}^{4}}{4}}+bde{x}^{2}+b{d}^{2}\ln \left ( x \right ) \right ) \ln \left ({x}^{n} \right ) -{\frac{b{d}^{2}n \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}+{\frac{i}{8}}\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{2}}\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{2}}\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{8}}\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 ) +{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{4}}{4}}-{\frac{b{e}^{2}n{x}^{4}}{16}}+{\frac{a{e}^{2}{x}^{4}}{4}}+\ln \left ( c \right ) bde{x}^{2}-{\frac{bden{x}^{2}}{2}}+ade{x}^{2}-{\frac{i}{2}}\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{8}}\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{2}}\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+\ln \left ( x \right ) \ln \left ( c \right ) b{d}^{2}+\ln \left ( x \right ) a{d}^{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,x)

[Out]

(1/4*b*e^2*x^4+b*d*e*x^2+b*d^2*ln(x))*ln(x^n)-1/2*b*d^2*n*ln(x)^2+1/8*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)
+1/2*I*ln(x)*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*ln(x)*Pi*b*d^2*csgn(I*c*x^n)^3-1/2*I*ln(x)*Pi*b*d^2*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+1/2*I*Pi*b*d*e*x^2*csgn(I*x^n)*
csgn(I*c*x^n)^2-1/8*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*ln(x)*Pi*b*d^2*csgn(I*c*x^n)^2*cs
gn(I*c)+1/4*ln(c)*b*e^2*x^4-1/16*b*e^2*n*x^4+1/4*a*e^2*x^4+ln(c)*b*d*e*x^2-1/2*b*d*e*n*x^2+a*d*e*x^2-1/2*I*Pi*
b*d*e*x^2*csgn(I*c*x^n)^3+1/8*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)-1/8*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^3+ln(x)*ln(c)*b*d^2+ln(x)*a*d^2

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Maxima [A]  time = 1.207, size = 119, normalized size = 1.34 \begin{align*} -\frac{1}{16} \, b e^{2} n x^{4} + \frac{1}{4} \, b e^{2} x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a e^{2} x^{4} - \frac{1}{2} \, b d e n x^{2} + b d e x^{2} \log \left (c x^{n}\right ) + a d e x^{2} + \frac{b d^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{2} \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

-1/16*b*e^2*n*x^4 + 1/4*b*e^2*x^4*log(c*x^n) + 1/4*a*e^2*x^4 - 1/2*b*d*e*n*x^2 + b*d*e*x^2*log(c*x^n) + a*d*e*
x^2 + 1/2*b*d^2*log(c*x^n)^2/n + a*d^2*log(x)

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Fricas [A]  time = 1.32966, size = 259, normalized size = 2.91 \begin{align*} \frac{1}{2} \, b d^{2} n \log \left (x\right )^{2} - \frac{1}{16} \,{\left (b e^{2} n - 4 \, a e^{2}\right )} x^{4} - \frac{1}{2} \,{\left (b d e n - 2 \, a d e\right )} x^{2} + \frac{1}{4} \,{\left (b e^{2} x^{4} + 4 \, b d e x^{2}\right )} \log \left (c\right ) + \frac{1}{4} \,{\left (b e^{2} n x^{4} + 4 \, b d e n x^{2} + 4 \, b d^{2} \log \left (c\right ) + 4 \, a d^{2}\right )} \log \left (x\right ) \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,x, algorithm="fricas")

[Out]

1/2*b*d^2*n*log(x)^2 - 1/16*(b*e^2*n - 4*a*e^2)*x^4 - 1/2*(b*d*e*n - 2*a*d*e)*x^2 + 1/4*(b*e^2*x^4 + 4*b*d*e*x
^2)*log(c) + 1/4*(b*e^2*n*x^4 + 4*b*d*e*n*x^2 + 4*b*d^2*log(c) + 4*a*d^2)*log(x)

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Sympy [A]  time = 3.57275, size = 129, normalized size = 1.45 \begin{align*} a d^{2} \log{\left (x \right )} + a d e x^{2} + \frac{a e^{2} x^{4}}{4} + \frac{b d^{2} n \log{\left (x \right )}^{2}}{2} + b d^{2} \log{\left (c \right )} \log{\left (x \right )} + b d e n x^{2} \log{\left (x \right )} - \frac{b d e n x^{2}}{2} + b d e x^{2} \log{\left (c \right )} + \frac{b e^{2} n x^{4} \log{\left (x \right )}}{4} - \frac{b e^{2} n x^{4}}{16} + \frac{b e^{2} x^{4} \log{\left (c \right )}}{4} \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,x)

[Out]

a*d**2*log(x) + a*d*e*x**2 + a*e**2*x**4/4 + b*d**2*n*log(x)**2/2 + b*d**2*log(c)*log(x) + b*d*e*n*x**2*log(x)
- b*d*e*n*x**2/2 + b*d*e*x**2*log(c) + b*e**2*n*x**4*log(x)/4 - b*e**2*n*x**4/16 + b*e**2*x**4*log(c)/4

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Giac [A]  time = 1.36055, size = 142, normalized size = 1.6 \begin{align*} \frac{1}{4} \, b n x^{4} e^{2} \log \left (x\right ) - \frac{1}{16} \, b n x^{4} e^{2} + \frac{1}{4} \, b x^{4} e^{2} \log \left (c\right ) + b d n x^{2} e \log \left (x\right ) + \frac{1}{4} \, a x^{4} e^{2} - \frac{1}{2} \, b d n x^{2} e + b d x^{2} e \log \left (c\right ) + \frac{1}{2} \, b d^{2} n \log \left (x\right )^{2} + a d x^{2} e + b d^{2} \log \left (c\right ) \log \left (x\right ) + a d^{2} \log \left (x\right ) \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,x, algorithm="giac")

[Out]

1/4*b*n*x^4*e^2*log(x) - 1/16*b*n*x^4*e^2 + 1/4*b*x^4*e^2*log(c) + b*d*n*x^2*e*log(x) + 1/4*a*x^4*e^2 - 1/2*b*
d*n*x^2*e + b*d*x^2*e*log(c) + 1/2*b*d^2*n*log(x)^2 + a*d*x^2*e + b*d^2*log(c)*log(x) + a*d^2*log(x)