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

Optimal. Leaf size=83 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{x}+2 d e x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} e^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{x}-2 b d e n x-\frac{1}{9} b e^2 n x^3 \]

[Out]

-((b*d^2*n)/x) - 2*b*d*e*n*x - (b*e^2*n*x^3)/9 - (d^2*(a + b*Log[c*x^n]))/x + 2*d*e*x*(a + b*Log[c*x^n]) + (e^
2*x^3*(a + b*Log[c*x^n]))/3

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Rubi [A]  time = 0.0706047, antiderivative size = 66, normalized size of antiderivative = 0.8, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {270, 2334} \[ -\frac{1}{3} \left (\frac{3 d^2}{x}-6 d e x-e^2 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{x}-2 b d e n x-\frac{1}{9} b e^2 n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d^2*n)/x) - 2*b*d*e*n*x - (b*e^2*n*x^3)/9 - (((3*d^2)/x - 6*d*e*x - e^2*x^3)*(a + b*Log[c*x^n]))/3

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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])

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{1}{3} \left (\frac{3 d^2}{x}-6 d e x-e^2 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (2 d e-\frac{d^2}{x^2}+\frac{e^2 x^2}{3}\right ) \, dx\\ &=-\frac{b d^2 n}{x}-2 b d e n x-\frac{1}{9} b e^2 n x^3-\frac{1}{3} \left (\frac{3 d^2}{x}-6 d e x-e^2 x^3\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0346713, size = 86, normalized size = 1.04 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{3} e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+2 a d e x+2 b d e x \log \left (c x^n\right )-\frac{b d^2 n}{x}-2 b d e n x-\frac{1}{9} b e^2 n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d^2*n)/x) + 2*a*d*e*x - 2*b*d*e*n*x - (b*e^2*n*x^3)/9 + 2*b*d*e*x*Log[c*x^n] - (d^2*(a + b*Log[c*x^n]))/x
+ (e^2*x^3*(a + b*Log[c*x^n]))/3

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Maple [C]  time = 0.207, size = 419, normalized size = 5.1 \begin{align*} -{\frac{b \left ( -{e}^{2}{x}^{4}-6\,de{x}^{2}+3\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{3\,x}}-{\frac{-9\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -3\,i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -3\,i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+18\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-18\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+3\,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 ) -9\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+9\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+3\,i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+18\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +9\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -18\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -6\,\ln \left ( c \right ) b{e}^{2}{x}^{4}+2\,b{e}^{2}n{x}^{4}-6\,a{e}^{2}{x}^{4}-36\,\ln \left ( c \right ) bde{x}^{2}+36\,bden{x}^{2}-36\,ade{x}^{2}+18\,\ln \left ( c \right ) b{d}^{2}+18\,b{d}^{2}n+18\,a{d}^{2}}{18\,x}} \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^2,x)

[Out]

-1/3*b*(-e^2*x^4-6*d*e*x^2+3*d^2)/x*ln(x^n)-1/18*(-9*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-3*I*Pi*b*e
^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+18*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^
3-18*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-9*I*Pi*b*
d^2*csgn(I*c*x^n)^3+9*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^3+18*I*Pi*b*d*e*x^
2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+9*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^2
*csgn(I*c)-6*ln(c)*b*e^2*x^4+2*b*e^2*n*x^4-6*a*e^2*x^4-36*ln(c)*b*d*e*x^2+36*b*d*e*n*x^2-36*a*d*e*x^2+18*ln(c)
*b*d^2+18*b*d^2*n+18*a*d^2)/x

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Maxima [A]  time = 1.13785, size = 127, normalized size = 1.53 \begin{align*} -\frac{1}{9} \, b e^{2} n x^{3} + \frac{1}{3} \, b e^{2} x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a e^{2} x^{3} - 2 \, b d e n x + 2 \, b d e x \log \left (c x^{n}\right ) + 2 \, a d e x - \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^2+d)^2*(a+b*log(c*x^n))/x^2,x, algorithm="maxima")

[Out]

-1/9*b*e^2*n*x^3 + 1/3*b*e^2*x^3*log(c*x^n) + 1/3*a*e^2*x^3 - 2*b*d*e*n*x + 2*b*d*e*x*log(c*x^n) + 2*a*d*e*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 = 1.31644, size = 247, normalized size = 2.98 \begin{align*} -\frac{{\left (b e^{2} n - 3 \, a e^{2}\right )} x^{4} + 9 \, b d^{2} n + 9 \, a d^{2} + 18 \,{\left (b d e n - a d e\right )} x^{2} - 3 \,{\left (b e^{2} x^{4} + 6 \, b d e x^{2} - 3 \, b d^{2}\right )} \log \left (c\right ) - 3 \,{\left (b e^{2} n x^{4} + 6 \, b d e n x^{2} - 3 \, b d^{2} n\right )} \log \left (x\right )}{9 \, x} \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^2,x, algorithm="fricas")

[Out]

-1/9*((b*e^2*n - 3*a*e^2)*x^4 + 9*b*d^2*n + 9*a*d^2 + 18*(b*d*e*n - a*d*e)*x^2 - 3*(b*e^2*x^4 + 6*b*d*e*x^2 -
3*b*d^2)*log(c) - 3*(b*e^2*n*x^4 + 6*b*d*e*n*x^2 - 3*b*d^2*n)*log(x))/x

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Sympy [A]  time = 4.8064, size = 131, normalized size = 1.58 \begin{align*} - \frac{a d^{2}}{x} + 2 a d e x + \frac{a e^{2} x^{3}}{3} - \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} + 2 b d e n x \log{\left (x \right )} - 2 b d e n x + 2 b d e x \log{\left (c \right )} + \frac{b e^{2} n x^{3} \log{\left (x \right )}}{3} - \frac{b e^{2} n x^{3}}{9} + \frac{b e^{2} x^{3} \log{\left (c \right )}}{3} \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**2,x)

[Out]

-a*d**2/x + 2*a*d*e*x + a*e**2*x**3/3 - b*d**2*n*log(x)/x - b*d**2*n/x - b*d**2*log(c)/x + 2*b*d*e*n*x*log(x)
- 2*b*d*e*n*x + 2*b*d*e*x*log(c) + b*e**2*n*x**3*log(x)/3 - b*e**2*n*x**3/9 + b*e**2*x**3*log(c)/3

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Giac [A]  time = 1.28094, size = 157, normalized size = 1.89 \begin{align*} \frac{3 \, b n x^{4} e^{2} \log \left (x\right ) - b n x^{4} e^{2} + 3 \, b x^{4} e^{2} \log \left (c\right ) + 18 \, b d n x^{2} e \log \left (x\right ) + 3 \, a x^{4} e^{2} - 18 \, b d n x^{2} e + 18 \, b d x^{2} e \log \left (c\right ) + 18 \, a d x^{2} e - 9 \, b d^{2} n \log \left (x\right ) - 9 \, b d^{2} n - 9 \, b d^{2} \log \left (c\right ) - 9 \, a d^{2}}{9 \, x} \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^2,x, algorithm="giac")

[Out]

1/9*(3*b*n*x^4*e^2*log(x) - b*n*x^4*e^2 + 3*b*x^4*e^2*log(c) + 18*b*d*n*x^2*e*log(x) + 3*a*x^4*e^2 - 18*b*d*n*
x^2*e + 18*b*d*x^2*e*log(c) + 18*a*d*x^2*e - 9*b*d^2*n*log(x) - 9*b*d^2*n - 9*b*d^2*log(c) - 9*a*d^2)/x