3.206 \(\int \frac{(d+e x^2)^3 (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{9 x^3}-3 b d e^2 n x-\frac{1}{9} b e^3 n x^3 \]

[Out]

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

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Rubi [A]  time = 0.0910002, antiderivative size = 91, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {270, 2334, 12} \[ -\frac{1}{3} \left (\frac{9 d^2 e}{x}+\frac{d^3}{x^3}-9 d e^2 x-e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{9 x^3}-3 b d e^2 n x-\frac{1}{9} b e^3 n x^3 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

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

Mathematica [A]  time = 0.0539169, size = 112, normalized size = 0.93 \[ -\frac{3 a \left (9 d^2 e x^2+d^3-9 d e^2 x^4-e^3 x^6\right )+3 b \left (9 d^2 e x^2+d^3-9 d e^2 x^4-e^3 x^6\right ) \log \left (c x^n\right )+b n \left (27 d^2 e x^2+d^3+27 d e^2 x^4+e^3 x^6\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-(3*a*(d^3 + 9*d^2*e*x^2 - 9*d*e^2*x^4 - e^3*x^6) + b*n*(d^3 + 27*d^2*e*x^2 + 27*d*e^2*x^4 + e^3*x^6) + 3*b*(d
^3 + 9*d^2*e*x^2 - 9*d*e^2*x^4 - e^3*x^6)*Log[c*x^n])/(9*x^3)

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Maple [C]  time = 0.23, size = 585, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(a+b*ln(c*x^n))/x^4,x)

[Out]

-1/3*b*(-e^3*x^6-9*d*e^2*x^4+9*d^2*e*x^2+d^3)/x^3*ln(x^n)-1/18*(-54*ln(c)*b*d*e^2*x^4+6*a*d^3+54*a*d^2*e*x^2-3
*I*Pi*b*d^3*csgn(I*c*x^n)^3+54*ln(c)*b*d^2*e*x^2-54*a*d*e^2*x^4+6*ln(c)*b*d^3+27*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n
)^2*csgn(I*c)-27*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)-6*ln(c)*b*e^3*x^6+3*I*Pi*b*e^3*x^6*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)-27*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+27*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*
x^n)^2+3*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+27*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3*I*Pi
*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+3*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^3-6*a*e^3*x^6-27*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)+2*b*d^3*n+2*b*e^3*n*x^6-27*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3-3*I*Pi*b*d^3*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)+27*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^3-3*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi
*b*e^3*x^6*csgn(I*c*x^n)^2*csgn(I*c)+54*b*d*e^2*n*x^4+54*b*d^2*e*n*x^2)/x^3

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Maxima [A]  time = 1.026, size = 185, normalized size = 1.53 \begin{align*} -\frac{1}{9} \, b e^{3} n x^{3} + \frac{1}{3} \, b e^{3} x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a e^{3} x^{3} - 3 \, b d e^{2} n x + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x - \frac{3 \, b d^{2} e n}{x} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{x} - \frac{3 \, a d^{2} e}{x} - \frac{b d^{3} n}{9 \, x^{3}} - \frac{b d^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a d^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.31218, size = 340, normalized size = 2.81 \begin{align*} -\frac{{\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} + b d^{3} n + 27 \,{\left (b d e^{2} n - a d e^{2}\right )} x^{4} + 3 \, a d^{3} + 27 \,{\left (b d^{2} e n + a d^{2} e\right )} x^{2} - 3 \,{\left (b e^{3} x^{6} + 9 \, b d e^{2} x^{4} - 9 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) - 3 \,{\left (b e^{3} n x^{6} + 9 \, b d e^{2} n x^{4} - 9 \, b d^{2} e n x^{2} - b d^{3} 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)^3*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")

[Out]

-1/9*((b*e^3*n - 3*a*e^3)*x^6 + b*d^3*n + 27*(b*d*e^2*n - a*d*e^2)*x^4 + 3*a*d^3 + 27*(b*d^2*e*n + a*d^2*e)*x^
2 - 3*(b*e^3*x^6 + 9*b*d*e^2*x^4 - 9*b*d^2*e*x^2 - b*d^3)*log(c) - 3*(b*e^3*n*x^6 + 9*b*d*e^2*n*x^4 - 9*b*d^2*
e*n*x^2 - b*d^3*n)*log(x))/x^3

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Sympy [A]  time = 9.09706, size = 202, normalized size = 1.67 \begin{align*} - \frac{a d^{3}}{3 x^{3}} - \frac{3 a d^{2} e}{x} + 3 a d e^{2} x + \frac{a e^{3} x^{3}}{3} - \frac{b d^{3} n \log{\left (x \right )}}{3 x^{3}} - \frac{b d^{3} n}{9 x^{3}} - \frac{b d^{3} \log{\left (c \right )}}{3 x^{3}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{x} - \frac{3 b d^{2} e n}{x} - \frac{3 b d^{2} e \log{\left (c \right )}}{x} + 3 b d e^{2} n x \log{\left (x \right )} - 3 b d e^{2} n x + 3 b d e^{2} x \log{\left (c \right )} + \frac{b e^{3} n x^{3} \log{\left (x \right )}}{3} - \frac{b e^{3} n x^{3}}{9} + \frac{b e^{3} 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)**3*(a+b*ln(c*x**n))/x**4,x)

[Out]

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

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Giac [A]  time = 1.27344, size = 224, normalized size = 1.85 \begin{align*} \frac{3 \, b n x^{6} e^{3} \log \left (x\right ) - b n x^{6} e^{3} + 3 \, b x^{6} e^{3} \log \left (c\right ) + 27 \, b d n x^{4} e^{2} \log \left (x\right ) + 3 \, a x^{6} e^{3} - 27 \, b d n x^{4} e^{2} + 27 \, b d x^{4} e^{2} \log \left (c\right ) - 27 \, b d^{2} n x^{2} e \log \left (x\right ) + 27 \, a d x^{4} e^{2} - 27 \, b d^{2} n x^{2} e - 27 \, b d^{2} x^{2} e \log \left (c\right ) - 27 \, a d^{2} x^{2} e - 3 \, b d^{3} n \log \left (x\right ) - b d^{3} n - 3 \, b d^{3} \log \left (c\right ) - 3 \, a d^{3}}{9 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="giac")

[Out]

1/9*(3*b*n*x^6*e^3*log(x) - b*n*x^6*e^3 + 3*b*x^6*e^3*log(c) + 27*b*d*n*x^4*e^2*log(x) + 3*a*x^6*e^3 - 27*b*d*
n*x^4*e^2 + 27*b*d*x^4*e^2*log(c) - 27*b*d^2*n*x^2*e*log(x) + 27*a*d*x^4*e^2 - 27*b*d^2*n*x^2*e - 27*b*d^2*x^2
*e*log(c) - 27*a*d^2*x^2*e - 3*b*d^3*n*log(x) - b*d^3*n - 3*b*d^3*log(c) - 3*a*d^3)/x^3