3.193 \(\int \frac{(d+e x^2)^2 (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=82 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{9 x^3}-\frac{2 b d e n}{x}-b e^2 n x \]

[Out]

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

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Rubi [A]  time = 0.0725627, antiderivative size = 65, normalized size of antiderivative = 0.79, 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} \[ -\frac{1}{3} \left (\frac{d^2}{x^3}+\frac{6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{9 x^3}-\frac{2 b d e n}{x}-b e^2 n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n)/(9*x^3) - (2*b*d*e*n)/x - b*e^2*n*x - ((d^2/x^3 + (6*d*e)/x - 3*e^2*x)*(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^4} \, dx &=-\frac{1}{3} \left (\frac{d^2}{x^3}+\frac{6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^2-\frac{d^2}{3 x^4}-\frac{2 d e}{x^2}\right ) \, dx\\ &=-\frac{b d^2 n}{9 x^3}-\frac{2 b d e n}{x}-b e^2 n x-\frac{1}{3} \left (\frac{d^2}{x^3}+\frac{6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0397359, size = 80, normalized size = 0.98 \[ -\frac{3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )+3 b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \log \left (c x^n\right )+b n \left (d^2+18 d e x^2+9 e^2 x^4\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4) + b*n*(d^2 + 18*d*e*x^2 + 9*e^2*x^4) + 3*b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*L
og[c*x^n])/(9*x^3)

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Maple [C]  time = 0.213, size = 417, normalized size = 5.1 \begin{align*} -{\frac{b \left ( -3\,{e}^{2}{x}^{4}+6\,de{x}^{2}+{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{3\,{x}^{3}}}-{\frac{3\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+18\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+9\,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 ) +18\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -9\,i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-9\,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{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -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 ) -18\,\ln \left ( c \right ) b{e}^{2}{x}^{4}-18\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-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 ) -3\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+9\,i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+18\,b{e}^{2}n{x}^{4}-18\,a{e}^{2}{x}^{4}+36\,\ln \left ( c \right ) bde{x}^{2}+36\,bden{x}^{2}+36\,ade{x}^{2}+6\,\ln \left ( c \right ) b{d}^{2}+2\,b{d}^{2}n+6\,a{d}^{2}}{18\,{x}^{3}}} \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^4,x)

[Out]

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

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Maxima [A]  time = 1.07545, size = 124, normalized size = 1.51 \begin{align*} -b e^{2} n x + b e^{2} x \log \left (c x^{n}\right ) + a e^{2} x - \frac{2 \, b d e n}{x} - \frac{2 \, b d e \log \left (c x^{n}\right )}{x} - \frac{2 \, a d e}{x} - \frac{b d^{2} n}{9 \, x^{3}} - \frac{b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a d^{2}}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.31971, size = 247, normalized size = 3.01 \begin{align*} -\frac{9 \,{\left (b e^{2} n - a e^{2}\right )} x^{4} + b d^{2} n + 3 \, a d^{2} + 18 \,{\left (b d e n + a d e\right )} x^{2} - 3 \,{\left (3 \, b e^{2} x^{4} - 6 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) - 3 \,{\left (3 \, b e^{2} n x^{4} - 6 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )}{9 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.25941, size = 131, normalized size = 1.6 \begin{align*} - \frac{a d^{2}}{3 x^{3}} - \frac{2 a d e}{x} + a e^{2} x - \frac{b d^{2} n \log{\left (x \right )}}{3 x^{3}} - \frac{b d^{2} n}{9 x^{3}} - \frac{b d^{2} \log{\left (c \right )}}{3 x^{3}} - \frac{2 b d e n \log{\left (x \right )}}{x} - \frac{2 b d e n}{x} - \frac{2 b d e \log{\left (c \right )}}{x} + 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**2+d)**2*(a+b*ln(c*x**n))/x**4,x)

[Out]

-a*d**2/(3*x**3) - 2*a*d*e/x + a*e**2*x - b*d**2*n*log(x)/(3*x**3) - b*d**2*n/(9*x**3) - b*d**2*log(c)/(3*x**3
) - 2*b*d*e*n*log(x)/x - 2*b*d*e*n/x - 2*b*d*e*log(c)/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 = 1.51212, size = 157, normalized size = 1.91 \begin{align*} \frac{9 \, b n x^{4} e^{2} \log \left (x\right ) - 9 \, b n x^{4} e^{2} + 9 \, b x^{4} e^{2} \log \left (c\right ) - 18 \, b d n x^{2} e \log \left (x\right ) + 9 \, 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 - 3 \, b d^{2} n \log \left (x\right ) - b d^{2} n - 3 \, b d^{2} \log \left (c\right ) - 3 \, a d^{2}}{9 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

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