3.209 \(\int \frac{(d+e x^2)^3 (a+b \log (c x^n))}{x^{10}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{3 b d^2 e n}{49 x^7}-\frac{b d^3 n}{81 x^9}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{9 x^3} \]

[Out]

-(b*d^3*n)/(81*x^9) - (3*b*d^2*e*n)/(49*x^7) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*
x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c*x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log
[c*x^n]))/(3*x^3)

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Rubi [A]  time = 0.0977795, antiderivative size = 100, normalized size of antiderivative = 0.75, 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} \[ -\frac{1}{315} \left (\frac{135 d^2 e}{x^7}+\frac{35 d^3}{x^9}+\frac{189 d e^2}{x^5}+\frac{105 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{49 x^7}-\frac{b d^3 n}{81 x^9}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^10,x]

[Out]

-(b*d^3*n)/(81*x^9) - (3*b*d^2*e*n)/(49*x^7) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(9*x^3) - (((35*d^3)/x^9 + (
135*d^2*e)/x^7 + (189*d*e^2)/x^5 + (105*e^3)/x^3)*(a + b*Log[c*x^n]))/315

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^{10}} \, dx &=-\frac{1}{315} \left (\frac{35 d^3}{x^9}+\frac{135 d^2 e}{x^7}+\frac{189 d e^2}{x^5}+\frac{105 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-35 d^3-135 d^2 e x^2-189 d e^2 x^4-105 e^3 x^6}{315 x^{10}} \, dx\\ &=-\frac{1}{315} \left (\frac{35 d^3}{x^9}+\frac{135 d^2 e}{x^7}+\frac{189 d e^2}{x^5}+\frac{105 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{315} (b n) \int \frac{-35 d^3-135 d^2 e x^2-189 d e^2 x^4-105 e^3 x^6}{x^{10}} \, dx\\ &=-\frac{1}{315} \left (\frac{35 d^3}{x^9}+\frac{135 d^2 e}{x^7}+\frac{189 d e^2}{x^5}+\frac{105 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{315} (b n) \int \left (-\frac{35 d^3}{x^{10}}-\frac{135 d^2 e}{x^8}-\frac{189 d e^2}{x^6}-\frac{105 e^3}{x^4}\right ) \, dx\\ &=-\frac{b d^3 n}{81 x^9}-\frac{3 b d^2 e n}{49 x^7}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{9 x^3}-\frac{1}{315} \left (\frac{35 d^3}{x^9}+\frac{135 d^2 e}{x^7}+\frac{189 d e^2}{x^5}+\frac{105 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.059748, size = 133, normalized size = 1. \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{3 b d^2 e n}{49 x^7}-\frac{b d^3 n}{81 x^9}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^10,x]

[Out]

-(b*d^3*n)/(81*x^9) - (3*b*d^2*e*n)/(49*x^7) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*
x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c*x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log
[c*x^n]))/(3*x^3)

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Maple [C]  time = 0.127, size = 587, normalized size = 4.4 \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^10,x)

[Out]

-1/315*b*(105*e^3*x^6+189*d*e^2*x^4+135*d^2*e*x^2+35*d^3)/x^9*ln(x^n)-1/198450*(33075*I*Pi*b*e^3*x^6*csgn(I*c*
x^n)^2*csgn(I*c)+119070*ln(c)*b*d*e^2*x^4+22050*a*d^3+85050*a*d^2*e*x^2+59535*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^2
*csgn(I*c)+85050*ln(c)*b*d^2*e*x^2-33075*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+119070*a*d*e^2*x^4
+42525*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)+22050*ln(c)*b*d^3+11025*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)
^2+66150*ln(c)*b*e^3*x^6+11025*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-33075*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^3-11025
*I*Pi*b*d^3*csgn(I*c*x^n)^3-42525*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+66150*a*e^3*x^6-59535*I
*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2450*b*d^3*n+59535*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x
^n)^2+42525*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-42525*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3-11025*I*Pi*b*d
^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+33075*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2+22050*b*e^3*n*x^6+2381
4*b*d*e^2*n*x^4+12150*b*d^2*e*n*x^2-59535*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^3)/x^9

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Maxima [A]  time = 1.00968, size = 193, normalized size = 1.45 \begin{align*} -\frac{b e^{3} n}{9 \, x^{3}} - \frac{b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a e^{3}}{3 \, x^{3}} - \frac{3 \, b d e^{2} n}{25 \, x^{5}} - \frac{3 \, b d e^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{3 \, a d e^{2}}{5 \, x^{5}} - \frac{3 \, b d^{2} e n}{49 \, x^{7}} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac{3 \, a d^{2} e}{7 \, x^{7}} - \frac{b d^{3} n}{81 \, x^{9}} - \frac{b d^{3} \log \left (c x^{n}\right )}{9 \, x^{9}} - \frac{a d^{3}}{9 \, x^{9}} \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^10,x, algorithm="maxima")

[Out]

-1/9*b*e^3*n/x^3 - 1/3*b*e^3*log(c*x^n)/x^3 - 1/3*a*e^3/x^3 - 3/25*b*d*e^2*n/x^5 - 3/5*b*d*e^2*log(c*x^n)/x^5
- 3/5*a*d*e^2/x^5 - 3/49*b*d^2*e*n/x^7 - 3/7*b*d^2*e*log(c*x^n)/x^7 - 3/7*a*d^2*e/x^7 - 1/81*b*d^3*n/x^9 - 1/9
*b*d^3*log(c*x^n)/x^9 - 1/9*a*d^3/x^9

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Fricas [A]  time = 1.32141, size = 413, normalized size = 3.11 \begin{align*} -\frac{11025 \,{\left (b e^{3} n + 3 \, a e^{3}\right )} x^{6} + 1225 \, b d^{3} n + 11907 \,{\left (b d e^{2} n + 5 \, a d e^{2}\right )} x^{4} + 11025 \, a d^{3} + 6075 \,{\left (b d^{2} e n + 7 \, a d^{2} e\right )} x^{2} + 315 \,{\left (105 \, b e^{3} x^{6} + 189 \, b d e^{2} x^{4} + 135 \, b d^{2} e x^{2} + 35 \, b d^{3}\right )} \log \left (c\right ) + 315 \,{\left (105 \, b e^{3} n x^{6} + 189 \, b d e^{2} n x^{4} + 135 \, b d^{2} e n x^{2} + 35 \, b d^{3} n\right )} \log \left (x\right )}{99225 \, x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/99225*(11025*(b*e^3*n + 3*a*e^3)*x^6 + 1225*b*d^3*n + 11907*(b*d*e^2*n + 5*a*d*e^2)*x^4 + 11025*a*d^3 + 607
5*(b*d^2*e*n + 7*a*d^2*e)*x^2 + 315*(105*b*e^3*x^6 + 189*b*d*e^2*x^4 + 135*b*d^2*e*x^2 + 35*b*d^3)*log(c) + 31
5*(105*b*e^3*n*x^6 + 189*b*d*e^2*n*x^4 + 135*b*d^2*e*n*x^2 + 35*b*d^3*n)*log(x))/x^9

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Sympy [A]  time = 31.7602, size = 231, normalized size = 1.74 \begin{align*} - \frac{a d^{3}}{9 x^{9}} - \frac{3 a d^{2} e}{7 x^{7}} - \frac{3 a d e^{2}}{5 x^{5}} - \frac{a e^{3}}{3 x^{3}} - \frac{b d^{3} n \log{\left (x \right )}}{9 x^{9}} - \frac{b d^{3} n}{81 x^{9}} - \frac{b d^{3} \log{\left (c \right )}}{9 x^{9}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{7 x^{7}} - \frac{3 b d^{2} e n}{49 x^{7}} - \frac{3 b d^{2} e \log{\left (c \right )}}{7 x^{7}} - \frac{3 b d e^{2} n \log{\left (x \right )}}{5 x^{5}} - \frac{3 b d e^{2} n}{25 x^{5}} - \frac{3 b d e^{2} \log{\left (c \right )}}{5 x^{5}} - \frac{b e^{3} n \log{\left (x \right )}}{3 x^{3}} - \frac{b e^{3} n}{9 x^{3}} - \frac{b e^{3} \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)**3*(a+b*ln(c*x**n))/x**10,x)

[Out]

-a*d**3/(9*x**9) - 3*a*d**2*e/(7*x**7) - 3*a*d*e**2/(5*x**5) - a*e**3/(3*x**3) - b*d**3*n*log(x)/(9*x**9) - b*
d**3*n/(81*x**9) - b*d**3*log(c)/(9*x**9) - 3*b*d**2*e*n*log(x)/(7*x**7) - 3*b*d**2*e*n/(49*x**7) - 3*b*d**2*e
*log(c)/(7*x**7) - 3*b*d*e**2*n*log(x)/(5*x**5) - 3*b*d*e**2*n/(25*x**5) - 3*b*d*e**2*log(c)/(5*x**5) - b*e**3
*n*log(x)/(3*x**3) - b*e**3*n/(9*x**3) - b*e**3*log(c)/(3*x**3)

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Giac [A]  time = 1.41871, size = 224, normalized size = 1.68 \begin{align*} -\frac{33075 \, b n x^{6} e^{3} \log \left (x\right ) + 11025 \, b n x^{6} e^{3} + 33075 \, b x^{6} e^{3} \log \left (c\right ) + 59535 \, b d n x^{4} e^{2} \log \left (x\right ) + 33075 \, a x^{6} e^{3} + 11907 \, b d n x^{4} e^{2} + 59535 \, b d x^{4} e^{2} \log \left (c\right ) + 42525 \, b d^{2} n x^{2} e \log \left (x\right ) + 59535 \, a d x^{4} e^{2} + 6075 \, b d^{2} n x^{2} e + 42525 \, b d^{2} x^{2} e \log \left (c\right ) + 42525 \, a d^{2} x^{2} e + 11025 \, b d^{3} n \log \left (x\right ) + 1225 \, b d^{3} n + 11025 \, b d^{3} \log \left (c\right ) + 11025 \, a d^{3}}{99225 \, x^{9}} \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^10,x, algorithm="giac")

[Out]

-1/99225*(33075*b*n*x^6*e^3*log(x) + 11025*b*n*x^6*e^3 + 33075*b*x^6*e^3*log(c) + 59535*b*d*n*x^4*e^2*log(x) +
33075*a*x^6*e^3 + 11907*b*d*n*x^4*e^2 + 59535*b*d*x^4*e^2*log(c) + 42525*b*d^2*n*x^2*e*log(x) + 59535*a*d*x^4
*e^2 + 6075*b*d^2*n*x^2*e + 42525*b*d^2*x^2*e*log(c) + 42525*a*d^2*x^2*e + 11025*b*d^3*n*log(x) + 1225*b*d^3*n
+ 11025*b*d^3*log(c) + 11025*a*d^3)/x^9