3.207 \(\int \frac{(d+e x^2)^3 (a+b \log (c x^n))}{x^6} \, dx\)

Optimal. Leaf size=118 \[ -\frac{d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 e n}{3 x^3}-\frac{b d^3 n}{25 x^5}-\frac{3 b d e^2 n}{x}-b e^3 n x \]

[Out]

-(b*d^3*n)/(25*x^5) - (b*d^2*e*n)/(3*x^3) - (3*b*d*e^2*n)/x - b*e^3*n*x - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (
d^2*e*(a + b*Log[c*x^n]))/x^3 - (3*d*e^2*(a + b*Log[c*x^n]))/x + e^3*x*(a + b*Log[c*x^n])

________________________________________________________________________________________

Rubi [A]  time = 0.0860084, antiderivative size = 91, normalized size of antiderivative = 0.77, 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} \[ -\frac{1}{5} \left (\frac{5 d^2 e}{x^3}+\frac{d^3}{x^5}+\frac{15 d e^2}{x}-5 e^3 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 e n}{3 x^3}-\frac{b d^3 n}{25 x^5}-\frac{3 b d e^2 n}{x}-b e^3 n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^3*n)/(25*x^5) - (b*d^2*e*n)/(3*x^3) - (3*b*d*e^2*n)/x - b*e^3*n*x - ((d^3/x^5 + (5*d^2*e)/x^3 + (15*d*e^
2)/x - 5*e^3*x)*(a + b*Log[c*x^n]))/5

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 )^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac{1}{5} \left (\frac{d^3}{x^5}+\frac{5 d^2 e}{x^3}+\frac{15 d e^2}{x}-5 e^3 x\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^3-\frac{d^3}{5 x^6}-\frac{d^2 e}{x^4}-\frac{3 d e^2}{x^2}\right ) \, dx\\ &=-\frac{b d^3 n}{25 x^5}-\frac{b d^2 e n}{3 x^3}-\frac{3 b d e^2 n}{x}-b e^3 n x-\frac{1}{5} \left (\frac{d^3}{x^5}+\frac{5 d^2 e}{x^3}+\frac{15 d e^2}{x}-5 e^3 x\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0540893, size = 115, normalized size = 0.97 \[ -\frac{15 a \left (5 d^2 e x^2+d^3+15 d e^2 x^4-5 e^3 x^6\right )+15 b \left (5 d^2 e x^2+d^3+15 d e^2 x^4-5 e^3 x^6\right ) \log \left (c x^n\right )+b n \left (25 d^2 e x^2+3 d^3+225 d e^2 x^4+75 e^3 x^6\right )}{75 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(15*a*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6) + b*n*(3*d^3 + 25*d^2*e*x^2 + 225*d*e^2*x^4 + 75*e^3*x^6
) + 15*b*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6)*Log[c*x^n])/(75*x^5)

________________________________________________________________________________________

Maple [C]  time = 0.242, size = 585, normalized size = 5. \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^6,x)

[Out]

-1/5*b*(-5*e^3*x^6+15*d*e^2*x^4+5*d^2*e*x^2+d^3)/x^5*ln(x^n)-1/150*(450*ln(c)*b*d*e^2*x^4+30*a*d^3-75*I*Pi*b*d
^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+150*a*d^2*e*x^2+150*ln(c)*b*d^2*e*x^2+450*a*d*e^2*x^4+75*I*Pi*b*e
^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+225*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+30*ln(c)*b*d^3-150
*ln(c)*b*e^3*x^6-15*I*Pi*b*d^3*csgn(I*c*x^n)^3-150*a*e^3*x^6+6*b*d^3*n-225*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)+225*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)+75*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)
^2+75*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+15*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+15*I*Pi*b*d^3*csgn(
I*c*x^n)^2*csgn(I*c)+75*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^3+150*b*e^3*n*x^6+450*b*d*e^2*n*x^4+50*b*d^2*e*n*x^2-75*I
*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3-15*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-75*I*Pi*b*e^3*x^6*csgn(I*x^n
)*csgn(I*c*x^n)^2-75*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^2*csgn(I*c)-225*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^3)/x^5

________________________________________________________________________________________

Maxima [A]  time = 1.0347, size = 182, normalized size = 1.54 \begin{align*} -b e^{3} n x + b e^{3} x \log \left (c x^{n}\right ) + a e^{3} x - \frac{3 \, b d e^{2} n}{x} - \frac{3 \, b d e^{2} \log \left (c x^{n}\right )}{x} - \frac{3 \, a d e^{2}}{x} - \frac{b d^{2} e n}{3 \, x^{3}} - \frac{b d^{2} e \log \left (c x^{n}\right )}{x^{3}} - \frac{a d^{2} e}{x^{3}} - \frac{b d^{3} n}{25 \, x^{5}} - \frac{b d^{3} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{a d^{3}}{5 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

-b*e^3*n*x + b*e^3*x*log(c*x^n) + a*e^3*x - 3*b*d*e^2*n/x - 3*b*d*e^2*log(c*x^n)/x - 3*a*d*e^2/x - 1/3*b*d^2*e
*n/x^3 - b*d^2*e*log(c*x^n)/x^3 - a*d^2*e/x^3 - 1/25*b*d^3*n/x^5 - 1/5*b*d^3*log(c*x^n)/x^5 - 1/5*a*d^3/x^5

________________________________________________________________________________________

Fricas [A]  time = 1.34191, size = 362, normalized size = 3.07 \begin{align*} -\frac{75 \,{\left (b e^{3} n - a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 225 \,{\left (b d e^{2} n + a d e^{2}\right )} x^{4} + 15 \, a d^{3} + 25 \,{\left (b d^{2} e n + 3 \, a d^{2} e\right )} x^{2} - 15 \,{\left (5 \, b e^{3} x^{6} - 15 \, b d e^{2} x^{4} - 5 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) - 15 \,{\left (5 \, b e^{3} n x^{6} - 15 \, b d e^{2} n x^{4} - 5 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )}{75 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/75*(75*(b*e^3*n - a*e^3)*x^6 + 3*b*d^3*n + 225*(b*d*e^2*n + a*d*e^2)*x^4 + 15*a*d^3 + 25*(b*d^2*e*n + 3*a*d
^2*e)*x^2 - 15*(5*b*e^3*x^6 - 15*b*d*e^2*x^4 - 5*b*d^2*e*x^2 - b*d^3)*log(c) - 15*(5*b*e^3*n*x^6 - 15*b*d*e^2*
n*x^4 - 5*b*d^2*e*n*x^2 - b*d^3*n)*log(x))/x^5

________________________________________________________________________________________

Sympy [A]  time = 9.12959, size = 190, normalized size = 1.61 \begin{align*} - \frac{a d^{3}}{5 x^{5}} - \frac{a d^{2} e}{x^{3}} - \frac{3 a d e^{2}}{x} + a e^{3} x - \frac{b d^{3} n \log{\left (x \right )}}{5 x^{5}} - \frac{b d^{3} n}{25 x^{5}} - \frac{b d^{3} \log{\left (c \right )}}{5 x^{5}} - \frac{b d^{2} e n \log{\left (x \right )}}{x^{3}} - \frac{b d^{2} e n}{3 x^{3}} - \frac{b d^{2} e \log{\left (c \right )}}{x^{3}} - \frac{3 b d e^{2} n \log{\left (x \right )}}{x} - \frac{3 b d e^{2} n}{x} - \frac{3 b d e^{2} \log{\left (c \right )}}{x} + b e^{3} n x \log{\left (x \right )} - b e^{3} n x + b e^{3} x \log{\left (c \right )} \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**6,x)

[Out]

-a*d**3/(5*x**5) - a*d**2*e/x**3 - 3*a*d*e**2/x + a*e**3*x - b*d**3*n*log(x)/(5*x**5) - b*d**3*n/(25*x**5) - b
*d**3*log(c)/(5*x**5) - b*d**2*e*n*log(x)/x**3 - b*d**2*e*n/(3*x**3) - b*d**2*e*log(c)/x**3 - 3*b*d*e**2*n*log
(x)/x - 3*b*d*e**2*n/x - 3*b*d*e**2*log(c)/x + b*e**3*n*x*log(x) - b*e**3*n*x + b*e**3*x*log(c)

________________________________________________________________________________________

Giac [A]  time = 1.29483, size = 224, normalized size = 1.9 \begin{align*} \frac{75 \, b n x^{6} e^{3} \log \left (x\right ) - 75 \, b n x^{6} e^{3} + 75 \, b x^{6} e^{3} \log \left (c\right ) - 225 \, b d n x^{4} e^{2} \log \left (x\right ) + 75 \, a x^{6} e^{3} - 225 \, b d n x^{4} e^{2} - 225 \, b d x^{4} e^{2} \log \left (c\right ) - 75 \, b d^{2} n x^{2} e \log \left (x\right ) - 225 \, a d x^{4} e^{2} - 25 \, b d^{2} n x^{2} e - 75 \, b d^{2} x^{2} e \log \left (c\right ) - 75 \, a d^{2} x^{2} e - 15 \, b d^{3} n \log \left (x\right ) - 3 \, b d^{3} n - 15 \, b d^{3} \log \left (c\right ) - 15 \, a d^{3}}{75 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

1/75*(75*b*n*x^6*e^3*log(x) - 75*b*n*x^6*e^3 + 75*b*x^6*e^3*log(c) - 225*b*d*n*x^4*e^2*log(x) + 75*a*x^6*e^3 -
225*b*d*n*x^4*e^2 - 225*b*d*x^4*e^2*log(c) - 75*b*d^2*n*x^2*e*log(x) - 225*a*d*x^4*e^2 - 25*b*d^2*n*x^2*e - 7
5*b*d^2*x^2*e*log(c) - 75*a*d^2*x^2*e - 15*b*d^3*n*log(x) - 3*b*d^3*n - 15*b*d^3*log(c) - 15*a*d^3)/x^5