3.208 \(\int \frac{(d+e x^2)^3 (a+b \log (c x^n))}{x^8} \, dx\)

Optimal. Leaf size=127 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{49 x^7}-\frac{b d e^2 n}{3 x^3}-\frac{b e^3 n}{x} \]

[Out]

-(b*d^3*n)/(49*x^7) - (3*b*d^2*e*n)/(25*x^5) - (b*d*e^2*n)/(3*x^3) - (b*e^3*n)/x - (d^3*(a + b*Log[c*x^n]))/(7
*x^7) - (3*d^2*e*(a + b*Log[c*x^n]))/(5*x^5) - (d*e^2*(a + b*Log[c*x^n]))/x^3 - (e^3*(a + b*Log[c*x^n]))/x

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Rubi [A]  time = 0.100507, antiderivative size = 98, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ -\frac{1}{35} \left (\frac{21 d^2 e}{x^5}+\frac{5 d^3}{x^7}+\frac{35 d e^2}{x^3}+\frac{35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{49 x^7}-\frac{b d e^2 n}{3 x^3}-\frac{b e^3 n}{x} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-(b*d^3*n)/(49*x^7) - (3*b*d^2*e*n)/(25*x^5) - (b*d*e^2*n)/(3*x^3) - (b*e^3*n)/x - (((5*d^3)/x^7 + (21*d^2*e)/
x^5 + (35*d*e^2)/x^3 + (35*e^3)/x)*(a + b*Log[c*x^n]))/35

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]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\frac{1}{35} \left (\frac{5 d^3}{x^7}+\frac{21 d^2 e}{x^5}+\frac{35 d e^2}{x^3}+\frac{35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{35 x^8} \, dx\\ &=-\frac{1}{35} \left (\frac{5 d^3}{x^7}+\frac{21 d^2 e}{x^5}+\frac{35 d e^2}{x^3}+\frac{35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{35} (b n) \int \frac{-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{x^8} \, dx\\ &=-\frac{1}{35} \left (\frac{5 d^3}{x^7}+\frac{21 d^2 e}{x^5}+\frac{35 d e^2}{x^3}+\frac{35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{35} (b n) \int \left (-\frac{5 d^3}{x^8}-\frac{21 d^2 e}{x^6}-\frac{35 d e^2}{x^4}-\frac{35 e^3}{x^2}\right ) \, dx\\ &=-\frac{b d^3 n}{49 x^7}-\frac{3 b d^2 e n}{25 x^5}-\frac{b d e^2 n}{3 x^3}-\frac{b e^3 n}{x}-\frac{1}{35} \left (\frac{5 d^3}{x^7}+\frac{21 d^2 e}{x^5}+\frac{35 d e^2}{x^3}+\frac{35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.057228, size = 127, normalized size = 1. \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{49 x^7}-\frac{b d e^2 n}{3 x^3}-\frac{b e^3 n}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-(b*d^3*n)/(49*x^7) - (3*b*d^2*e*n)/(25*x^5) - (b*d*e^2*n)/(3*x^3) - (b*e^3*n)/x - (d^3*(a + b*Log[c*x^n]))/(7
*x^7) - (3*d^2*e*(a + b*Log[c*x^n]))/(5*x^5) - (d*e^2*(a + b*Log[c*x^n]))/x^3 - (e^3*(a + b*Log[c*x^n]))/x

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Maple [C]  time = 0.132, size = 587, normalized size = 4.6 \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^8,x)

[Out]

-1/35*b*(35*e^3*x^6+35*d*e^2*x^4+21*d^2*e*x^2+5*d^3)/x^7*ln(x^n)-1/7350*(7350*ln(c)*b*d*e^2*x^4+1050*a*d^3+441
0*a*d^2*e*x^2+3675*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+2205*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n
)^2+4410*ln(c)*b*d^2*e*x^2+2205*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+7350*a*d*e^2*x^4+1050*ln(c)*b*d^3+5
25*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+525*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-3675*I*Pi*b*e^3*x^6*csgn(I*
c*x^n)^3+7350*ln(c)*b*e^3*x^6-525*I*Pi*b*d^3*csgn(I*c*x^n)^3+7350*a*e^3*x^6+150*b*d^3*n+3675*I*Pi*b*d*e^2*x^4*
csgn(I*c*x^n)^2*csgn(I*c)-2205*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3-525*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)-3675*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2205*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)-3675*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+7350*b*e^3*n*x^6+2450*b*d*e^2*n*x^4+882*b*d
^2*e*n*x^2+3675*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2+3675*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^2*csgn(I*c)-3675*
I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^3)/x^7

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Maxima [A]  time = 1.01334, size = 193, normalized size = 1.52 \begin{align*} -\frac{b e^{3} n}{x} - \frac{b e^{3} \log \left (c x^{n}\right )}{x} - \frac{a e^{3}}{x} - \frac{b d e^{2} n}{3 \, x^{3}} - \frac{b d e^{2} \log \left (c x^{n}\right )}{x^{3}} - \frac{a d e^{2}}{x^{3}} - \frac{3 \, b d^{2} e n}{25 \, x^{5}} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{3 \, a d^{2} e}{5 \, x^{5}} - \frac{b d^{3} n}{49 \, x^{7}} - \frac{b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac{a d^{3}}{7 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-b*e^3*n/x - b*e^3*log(c*x^n)/x - a*e^3/x - 1/3*b*d*e^2*n/x^3 - b*d*e^2*log(c*x^n)/x^3 - a*d*e^2/x^3 - 3/25*b*
d^2*e*n/x^5 - 3/5*b*d^2*e*log(c*x^n)/x^5 - 3/5*a*d^2*e/x^5 - 1/49*b*d^3*n/x^7 - 1/7*b*d^3*log(c*x^n)/x^7 - 1/7
*a*d^3/x^7

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Fricas [A]  time = 1.32045, size = 389, normalized size = 3.06 \begin{align*} -\frac{3675 \,{\left (b e^{3} n + a e^{3}\right )} x^{6} + 75 \, b d^{3} n + 1225 \,{\left (b d e^{2} n + 3 \, a d e^{2}\right )} x^{4} + 525 \, a d^{3} + 441 \,{\left (b d^{2} e n + 5 \, a d^{2} e\right )} x^{2} + 105 \,{\left (35 \, b e^{3} x^{6} + 35 \, b d e^{2} x^{4} + 21 \, b d^{2} e x^{2} + 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (35 \, b e^{3} n x^{6} + 35 \, b d e^{2} n x^{4} + 21 \, b d^{2} e n x^{2} + 5 \, b d^{3} n\right )} \log \left (x\right )}{3675 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/3675*(3675*(b*e^3*n + a*e^3)*x^6 + 75*b*d^3*n + 1225*(b*d*e^2*n + 3*a*d*e^2)*x^4 + 525*a*d^3 + 441*(b*d^2*e
*n + 5*a*d^2*e)*x^2 + 105*(35*b*e^3*x^6 + 35*b*d*e^2*x^4 + 21*b*d^2*e*x^2 + 5*b*d^3)*log(c) + 105*(35*b*e^3*n*
x^6 + 35*b*d*e^2*n*x^4 + 21*b*d^2*e*n*x^2 + 5*b*d^3*n)*log(x))/x^7

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Sympy [A]  time = 13.4882, size = 206, normalized size = 1.62 \begin{align*} - \frac{a d^{3}}{7 x^{7}} - \frac{3 a d^{2} e}{5 x^{5}} - \frac{a d e^{2}}{x^{3}} - \frac{a e^{3}}{x} - \frac{b d^{3} n \log{\left (x \right )}}{7 x^{7}} - \frac{b d^{3} n}{49 x^{7}} - \frac{b d^{3} \log{\left (c \right )}}{7 x^{7}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{5 x^{5}} - \frac{3 b d^{2} e n}{25 x^{5}} - \frac{3 b d^{2} e \log{\left (c \right )}}{5 x^{5}} - \frac{b d e^{2} n \log{\left (x \right )}}{x^{3}} - \frac{b d e^{2} n}{3 x^{3}} - \frac{b d e^{2} \log{\left (c \right )}}{x^{3}} - \frac{b e^{3} n \log{\left (x \right )}}{x} - \frac{b e^{3} n}{x} - \frac{b e^{3} \log{\left (c \right )}}{x} \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**8,x)

[Out]

-a*d**3/(7*x**7) - 3*a*d**2*e/(5*x**5) - a*d*e**2/x**3 - a*e**3/x - b*d**3*n*log(x)/(7*x**7) - b*d**3*n/(49*x*
*7) - b*d**3*log(c)/(7*x**7) - 3*b*d**2*e*n*log(x)/(5*x**5) - 3*b*d**2*e*n/(25*x**5) - 3*b*d**2*e*log(c)/(5*x*
*5) - b*d*e**2*n*log(x)/x**3 - b*d*e**2*n/(3*x**3) - b*d*e**2*log(c)/x**3 - b*e**3*n*log(x)/x - b*e**3*n/x - b
*e**3*log(c)/x

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Giac [A]  time = 1.27793, size = 224, normalized size = 1.76 \begin{align*} -\frac{3675 \, b n x^{6} e^{3} \log \left (x\right ) + 3675 \, b n x^{6} e^{3} + 3675 \, b x^{6} e^{3} \log \left (c\right ) + 3675 \, b d n x^{4} e^{2} \log \left (x\right ) + 3675 \, a x^{6} e^{3} + 1225 \, b d n x^{4} e^{2} + 3675 \, b d x^{4} e^{2} \log \left (c\right ) + 2205 \, b d^{2} n x^{2} e \log \left (x\right ) + 3675 \, a d x^{4} e^{2} + 441 \, b d^{2} n x^{2} e + 2205 \, b d^{2} x^{2} e \log \left (c\right ) + 2205 \, a d^{2} x^{2} e + 525 \, b d^{3} n \log \left (x\right ) + 75 \, b d^{3} n + 525 \, b d^{3} \log \left (c\right ) + 525 \, a d^{3}}{3675 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

-1/3675*(3675*b*n*x^6*e^3*log(x) + 3675*b*n*x^6*e^3 + 3675*b*x^6*e^3*log(c) + 3675*b*d*n*x^4*e^2*log(x) + 3675
*a*x^6*e^3 + 1225*b*d*n*x^4*e^2 + 3675*b*d*x^4*e^2*log(c) + 2205*b*d^2*n*x^2*e*log(x) + 3675*a*d*x^4*e^2 + 441
*b*d^2*n*x^2*e + 2205*b*d^2*x^2*e*log(c) + 2205*a*d^2*x^2*e + 525*b*d^3*n*log(x) + 75*b*d^3*n + 525*b*d^3*log(
c) + 525*a*d^3)/x^7