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

Optimal. Leaf size=95 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{b d^2 n}{49 x^7}-\frac{2 b d e n}{25 x^5}-\frac{b e^2 n}{9 x^3} \]

[Out]

-(b*d^2*n)/(49*x^7) - (2*b*d*e*n)/(25*x^5) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(7*x^7) - (2*d*e*(a
+ b*Log[c*x^n]))/(5*x^5) - (e^2*(a + b*Log[c*x^n]))/(3*x^3)

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Rubi [A]  time = 0.0826563, antiderivative size = 74, normalized size of antiderivative = 0.78, 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} \[ -\frac{1}{105} \left (\frac{15 d^2}{x^7}+\frac{42 d e}{x^5}+\frac{35 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{49 x^7}-\frac{2 b d e n}{25 x^5}-\frac{b e^2 n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n)/(49*x^7) - (2*b*d*e*n)/(25*x^5) - (b*e^2*n)/(9*x^3) - (((15*d^2)/x^7 + (42*d*e)/x^5 + (35*e^2)/x^3)
*(a + b*Log[c*x^n]))/105

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 )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\frac{1}{105} \left (\frac{15 d^2}{x^7}+\frac{42 d e}{x^5}+\frac{35 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{105 x^8} \, dx\\ &=-\frac{1}{105} \left (\frac{15 d^2}{x^7}+\frac{42 d e}{x^5}+\frac{35 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{105} (b n) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{x^8} \, dx\\ &=-\frac{1}{105} \left (\frac{15 d^2}{x^7}+\frac{42 d e}{x^5}+\frac{35 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{105} (b n) \int \left (-\frac{15 d^2}{x^8}-\frac{42 d e}{x^6}-\frac{35 e^2}{x^4}\right ) \, dx\\ &=-\frac{b d^2 n}{49 x^7}-\frac{2 b d e n}{25 x^5}-\frac{b e^2 n}{9 x^3}-\frac{1}{105} \left (\frac{15 d^2}{x^7}+\frac{42 d e}{x^5}+\frac{35 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.045484, size = 95, normalized size = 1. \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{b d^2 n}{49 x^7}-\frac{2 b d e n}{25 x^5}-\frac{b e^2 n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n)/(49*x^7) - (2*b*d*e*n)/(25*x^5) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(7*x^7) - (2*d*e*(a
+ b*Log[c*x^n]))/(5*x^5) - (e^2*(a + b*Log[c*x^n]))/(3*x^3)

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Maple [C]  time = 0.109, size = 419, normalized size = 4.4 \begin{align*} -{\frac{b \left ( 35\,{e}^{2}{x}^{4}+42\,de{x}^{2}+15\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{105\,{x}^{7}}}-{\frac{3675\,i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+3675\,i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4410\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +1575\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+7350\,\ln \left ( c \right ) b{e}^{2}{x}^{4}+2450\,b{e}^{2}n{x}^{4}+7350\,a{e}^{2}{x}^{4}-3675\,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 ) -1575\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -1575\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-4410\,i\pi \,bde{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+8820\,\ln \left ( c \right ) bde{x}^{2}+1764\,bden{x}^{2}+8820\,ade{x}^{2}+1575\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4410\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3675\,i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-4410\,i\pi \,bde{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +3150\,\ln \left ( c \right ) b{d}^{2}+450\,b{d}^{2}n+3150\,a{d}^{2}}{22050\,{x}^{7}}} \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^8,x)

[Out]

-1/105*b*(35*e^2*x^4+42*d*e*x^2+15*d^2)/x^7*ln(x^n)-1/22050*(3675*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+3
675*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)+4410*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+1575*I*Pi*b*d^2*csg
n(I*x^n)*csgn(I*c*x^n)^2+7350*ln(c)*b*e^2*x^4+2450*b*e^2*n*x^4+7350*a*e^2*x^4-3675*I*Pi*b*e^2*x^4*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)-1575*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1575*I*Pi*b*d^2*csgn(I*c*x^n)^3-44
10*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^3+8820*ln(c)*b*d*e*x^2+1764*b*d*e*n*x^2+8820*a*d*e*x^2+1575*I*Pi*b*d^2*csgn(I*
c*x^n)^2*csgn(I*c)+4410*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-3675*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^3-4410*I*
Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3150*ln(c)*b*d^2+450*b*d^2*n+3150*a*d^2)/x^7

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Maxima [A]  time = 1.15342, size = 135, normalized size = 1.42 \begin{align*} -\frac{b e^{2} n}{9 \, x^{3}} - \frac{b e^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a e^{2}}{3 \, x^{3}} - \frac{2 \, b d e n}{25 \, x^{5}} - \frac{2 \, b d e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{2 \, a d e}{5 \, x^{5}} - \frac{b d^{2} n}{49 \, x^{7}} - \frac{b d^{2} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac{a d^{2}}{7 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-1/9*b*e^2*n/x^3 - 1/3*b*e^2*log(c*x^n)/x^3 - 1/3*a*e^2/x^3 - 2/25*b*d*e*n/x^5 - 2/5*b*d*e*log(c*x^n)/x^5 - 2/
5*a*d*e/x^5 - 1/49*b*d^2*n/x^7 - 1/7*b*d^2*log(c*x^n)/x^7 - 1/7*a*d^2/x^7

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Fricas [A]  time = 1.32808, size = 292, normalized size = 3.07 \begin{align*} -\frac{1225 \,{\left (b e^{2} n + 3 \, a e^{2}\right )} x^{4} + 225 \, b d^{2} n + 1575 \, a d^{2} + 882 \,{\left (b d e n + 5 \, a d e\right )} x^{2} + 105 \,{\left (35 \, b e^{2} x^{4} + 42 \, b d e x^{2} + 15 \, b d^{2}\right )} \log \left (c\right ) + 105 \,{\left (35 \, b e^{2} n x^{4} + 42 \, b d e n x^{2} + 15 \, b d^{2} n\right )} \log \left (x\right )}{11025 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/11025*(1225*(b*e^2*n + 3*a*e^2)*x^4 + 225*b*d^2*n + 1575*a*d^2 + 882*(b*d*e*n + 5*a*d*e)*x^2 + 105*(35*b*e^
2*x^4 + 42*b*d*e*x^2 + 15*b*d^2)*log(c) + 105*(35*b*e^2*n*x^4 + 42*b*d*e*n*x^2 + 15*b*d^2*n)*log(x))/x^7

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Sympy [A]  time = 13.3084, size = 160, normalized size = 1.68 \begin{align*} - \frac{a d^{2}}{7 x^{7}} - \frac{2 a d e}{5 x^{5}} - \frac{a e^{2}}{3 x^{3}} - \frac{b d^{2} n \log{\left (x \right )}}{7 x^{7}} - \frac{b d^{2} n}{49 x^{7}} - \frac{b d^{2} \log{\left (c \right )}}{7 x^{7}} - \frac{2 b d e n \log{\left (x \right )}}{5 x^{5}} - \frac{2 b d e n}{25 x^{5}} - \frac{2 b d e \log{\left (c \right )}}{5 x^{5}} - \frac{b e^{2} n \log{\left (x \right )}}{3 x^{3}} - \frac{b e^{2} n}{9 x^{3}} - \frac{b e^{2} \log{\left (c \right )}}{3 x^{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**8,x)

[Out]

-a*d**2/(7*x**7) - 2*a*d*e/(5*x**5) - a*e**2/(3*x**3) - b*d**2*n*log(x)/(7*x**7) - b*d**2*n/(49*x**7) - b*d**2
*log(c)/(7*x**7) - 2*b*d*e*n*log(x)/(5*x**5) - 2*b*d*e*n/(25*x**5) - 2*b*d*e*log(c)/(5*x**5) - b*e**2*n*log(x)
/(3*x**3) - b*e**2*n/(9*x**3) - b*e**2*log(c)/(3*x**3)

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Giac [A]  time = 1.34968, size = 157, normalized size = 1.65 \begin{align*} -\frac{3675 \, b n x^{4} e^{2} \log \left (x\right ) + 1225 \, b n x^{4} e^{2} + 3675 \, b x^{4} e^{2} \log \left (c\right ) + 4410 \, b d n x^{2} e \log \left (x\right ) + 3675 \, a x^{4} e^{2} + 882 \, b d n x^{2} e + 4410 \, b d x^{2} e \log \left (c\right ) + 4410 \, a d x^{2} e + 1575 \, b d^{2} n \log \left (x\right ) + 225 \, b d^{2} n + 1575 \, b d^{2} \log \left (c\right ) + 1575 \, a d^{2}}{11025 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

-1/11025*(3675*b*n*x^4*e^2*log(x) + 1225*b*n*x^4*e^2 + 3675*b*x^4*e^2*log(c) + 4410*b*d*n*x^2*e*log(x) + 3675*
a*x^4*e^2 + 882*b*d*n*x^2*e + 4410*b*d*x^2*e*log(c) + 4410*a*d*x^2*e + 1575*b*d^2*n*log(x) + 225*b*d^2*n + 157
5*b*d^2*log(c) + 1575*a*d^2)/x^7