3.3.9 \(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^{10}} \, dx\) [209]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, 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 = {276, 2372, 12, 14} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\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 {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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/81*(b*d^3*n)/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)

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]]

Rule 276

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 2372

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]}, Dist[a + b*Log[c*x^n], u, 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^{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*}

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Mathematica [A]
time = 0.04, size = 133, normalized size = 1.00 \begin {gather*} -\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 {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/81*(b*d^3*n)/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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 587, normalized size = 4.41

method	result	size
risch	\(-\frac {b \left (105 e^{3} x^{6}+189 d \,e^{2} x^{4}+135 d^{2} e \,x^{2}+35 d^{3}\right ) \ln \left (x^{n}\right )}{315 x^{9}}-\frac {66150 \ln \left (c \right ) b \,e^{3} x^{6}+66150 x^{6} a \,e^{3}+33075 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+42525 i \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e +59535 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+59535 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+119070 x^{4} a d \,e^{2}+85050 a \,d^{2} x^{2} e +22050 a \,d^{3}-59535 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-33075 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-33075 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2450 b \,d^{3} n +22050 d^{3} b \ln \left (c \right )+85050 \ln \left (c \right ) b \,d^{2} x^{2} e +119070 \ln \left (c \right ) b d \,e^{2} x^{4}-42525 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+22050 b \,e^{3} n \,x^{6}-42525 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+33075 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-59535 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+23814 b d \,e^{2} n \,x^{4}+12150 b \,d^{2} e n \,x^{2}+11025 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+11025 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+42525 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-11025 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-11025 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{198450 x^{9}}\)	\(587\)

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,method=_RETURNVERBOSE)

[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*(66150*ln(c)*b*e^3*x^6+66150*x^
6*a*e^3-33075*I*Pi*b*e^3*x^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+119070*x^4*a*d*e^2+33075*I*Pi*b*e^3*x^6*csgn(
I*x^n)*csgn(I*c*x^n)^2+85050*a*d^2*x^2*e+22050*a*d^3+59535*I*Pi*b*d*e^2*x^4*csgn(I*c)*csgn(I*c*x^n)^2+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*c)*csgn(I*c*x^n)^2+42525*I*Pi*b*d^2*x
^2*csgn(I*x^n)*csgn(I*c*x^n)^2*e+2450*b*d^3*n+22050*d^3*b*ln(c)+85050*ln(c)*b*d^2*x^2*e+119070*ln(c)*b*d*e^2*x
^4-11025*I*Pi*b*d^3*csgn(I*c*x^n)^3+33075*I*Pi*b*e^3*x^6*csgn(I*c)*csgn(I*c*x^n)^2+22050*b*e^3*n*x^6-59535*I*P
i*b*d*e^2*x^4*csgn(I*c*x^n)^3-42525*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3-42525*I*Pi*b*d^2*e*x^2*csgn(I*c)*csgn(I*x
^n)*csgn(I*c*x^n)-11025*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+23814*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)*csgn(I*x^n)*csgn(I*c*x^n)-33075*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^3+11025*I*Pi*b*
d^3*csgn(I*c)*csgn(I*c*x^n)^2+11025*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2)/x^9

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Maxima [A]
time = 0.26, size = 140, normalized size = 1.05 \begin {gather*} -\frac {b n e^{3}}{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 n e^{2}}{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} n e}{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 {gather*}

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*n*e^3/x^3 - 1/3*b*e^3*log(c*x^n)/x^3 - 1/3*a*e^3/x^3 - 3/25*b*d*n*e^2/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*n*e/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 = 0.35, size = 151, normalized size = 1.14 \begin {gather*} -\frac {11025 \, {\left (b n + 3 \, a\right )} x^{6} e^{3} + 11907 \, {\left (b d n + 5 \, a d\right )} x^{4} e^{2} + 1225 \, b d^{3} n + 11025 \, a d^{3} + 6075 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} x^{2} e + 315 \, {\left (105 \, b x^{6} e^{3} + 189 \, b d x^{4} e^{2} + 135 \, b d^{2} x^{2} e + 35 \, b d^{3}\right )} \log \left (c\right ) + 315 \, {\left (105 \, b n x^{6} e^{3} + 189 \, b d n x^{4} e^{2} + 135 \, b d^{2} n x^{2} e + 35 \, b d^{3} n\right )} \log \left (x\right )}{99225 \, x^{9}} \end {gather*}

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*n + 3*a)*x^6*e^3 + 11907*(b*d*n + 5*a*d)*x^4*e^2 + 1225*b*d^3*n + 11025*a*d^3 + 6075*(b*d^2
*n + 7*a*d^2)*x^2*e + 315*(105*b*x^6*e^3 + 189*b*d*x^4*e^2 + 135*b*d^2*x^2*e + 35*b*d^3)*log(c) + 315*(105*b*n
*x^6*e^3 + 189*b*d*n*x^4*e^2 + 135*b*d^2*n*x^2*e + 35*b*d^3*n)*log(x))/x^9

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Sympy [A]
time = 2.67, size = 177, normalized size = 1.33 \begin {gather*} - \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}{81 x^{9}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{9 x^{9}} - \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 b d e^{2} n}{25 x^{5}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b e^{3} n}{9 x^{3}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} \end {gather*}

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

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Giac [A]
time = 4.72, size = 166, normalized size = 1.25 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 3.70, size = 125, normalized size = 0.94 \begin {gather*} -\frac {x^6\,\left (105\,a\,e^3+35\,b\,e^3\,n\right )+35\,a\,d^3+x^2\,\left (135\,a\,d^2\,e+\frac {135\,b\,d^2\,e\,n}{7}\right )+x^4\,\left (189\,a\,d\,e^2+\frac {189\,b\,d\,e^2\,n}{5}\right )+\frac {35\,b\,d^3\,n}{9}}{315\,x^9}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{9}+\frac {3\,b\,d^2\,e\,x^2}{7}+\frac {3\,b\,d\,e^2\,x^4}{5}+\frac {b\,e^3\,x^6}{3}\right )}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^3*(a + b*log(c*x^n)))/x^10,x)

[Out]

- (x^6*(105*a*e^3 + 35*b*e^3*n) + 35*a*d^3 + x^2*(135*a*d^2*e + (135*b*d^2*e*n)/7) + x^4*(189*a*d*e^2 + (189*b
*d*e^2*n)/5) + (35*b*d^3*n)/9)/(315*x^9) - (log(c*x^n)*((b*d^3)/9 + (b*e^3*x^6)/3 + (3*b*d^2*e*x^2)/7 + (3*b*d
*e^2*x^4)/5))/x^9

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