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

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

[Out]

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

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Rubi [A]  time = 0.10, 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 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 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^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*}

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Mathematica [A]  time = 0.06, size = 127, normalized size = 1.00 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/49*(b*d^3*n)/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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fricas [A]  time = 0.62, size = 160, normalized size = 1.26 \[ -\frac {3675 \, {\left (b e^{3} n + a e^{3}\right )} x^{6} + 75 \, b d^{3} n + 1225 \, {\left (b d e^{2} n + 3 \, a d e^{2}\right )} x^{4} + 525 \, a d^{3} + 441 \, {\left (b d^{2} e n + 5 \, a d^{2} e\right )} x^{2} + 105 \, {\left (35 \, b e^{3} x^{6} + 35 \, b d e^{2} x^{4} + 21 \, b d^{2} e x^{2} + 5 \, b d^{3}\right )} \log \relax (c) + 105 \, {\left (35 \, b e^{3} n x^{6} + 35 \, b d e^{2} n x^{4} + 21 \, b d^{2} e n x^{2} + 5 \, b d^{3} n\right )} \log \relax (x)}{3675 \, x^{7}} \]

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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giac [A]  time = 0.27, size = 166, normalized size = 1.31 \[ -\frac {3675 \, b n x^{6} e^{3} \log \relax (x) + 3675 \, b n x^{6} e^{3} + 3675 \, b x^{6} e^{3} \log \relax (c) + 3675 \, b d n x^{4} e^{2} \log \relax (x) + 3675 \, a x^{6} e^{3} + 1225 \, b d n x^{4} e^{2} + 3675 \, b d x^{4} e^{2} \log \relax (c) + 2205 \, b d^{2} n x^{2} e \log \relax (x) + 3675 \, a d x^{4} e^{2} + 441 \, b d^{2} n x^{2} e + 2205 \, b d^{2} x^{2} e \log \relax (c) + 2205 \, a d^{2} x^{2} e + 525 \, b d^{3} n \log \relax (x) + 75 \, b d^{3} n + 525 \, b d^{3} \log \relax (c) + 525 \, a d^{3}}{3675 \, x^{7}} \]

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

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maple [C]  time = 0.18, size = 587, normalized size = 4.62 \[ -\frac {\left (35 e^{3} x^{6}+35 d \,e^{2} x^{4}+21 d^{2} e \,x^{2}+5 d^{3}\right ) b \ln \left (x^{n}\right )}{35 x^{7}}-\frac {7350 a \,e^{3} x^{6}+7350 b d \,e^{2} x^{4} \ln \relax (c )+7350 a d \,e^{2} x^{4}+1050 a \,d^{3}+7350 b \,e^{3} x^{6} \ln \relax (c )+150 b \,d^{3} n +1050 b \,d^{3} \ln \relax (c )+4410 a \,d^{2} e \,x^{2}-525 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 )+4410 b \,d^{2} e \,x^{2} \ln \relax (c )-2205 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 )+7350 b \,e^{3} n \,x^{6}-3675 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 )+3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-525 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2205 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2205 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+525 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+525 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2450 b d \,e^{2} n \,x^{4}+882 b \,d^{2} e n \,x^{2}-3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2205 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-3675 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 )}{7350 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(b*ln(c*x^n)+a)/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*a*e^3*x^6+7350*b*d*e^2*x^4*ln(c)
+7350*a*d*e^2*x^4+1050*a*d^3-3675*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2205*I*Pi*b*d^2*e*x^2*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+7350*b*e^3*x^6*ln(c)+150*b*d^3*n+1050*b*d^3*ln(c)+4410*a*d^2*e*x^2+4410*b*d
^2*e*x^2*ln(c)+7350*b*e^3*n*x^6-525*I*Pi*b*d^3*csgn(I*c*x^n)^3+2205*I*Pi*b*d^2*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)
^2+2205*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)-3675*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+367
5*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)+3675*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-3675*I*Pi*b*e^3
*x^6*csgn(I*c*x^n)^3+525*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)-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)^2+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*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 = 0.68, size = 143, normalized size = 1.13 \[ -\frac {b e^{3} n}{x} - \frac {b e^{3} \log \left (c x^{n}\right )}{x} - \frac {a e^{3}}{x} - \frac {b d e^{2} n}{3 \, x^{3}} - \frac {b d e^{2} \log \left (c x^{n}\right )}{x^{3}} - \frac {a d e^{2}}{x^{3}} - \frac {3 \, b d^{2} e n}{25 \, x^{5}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {b d^{3} n}{49 \, x^{7}} - \frac {b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {a d^{3}}{7 \, x^{7}} \]

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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mupad [B]  time = 3.80, size = 123, normalized size = 0.97 \[ -\frac {x^6\,\left (35\,a\,e^3+35\,b\,e^3\,n\right )+5\,a\,d^3+x^2\,\left (21\,a\,d^2\,e+\frac {21\,b\,d^2\,e\,n}{5}\right )+x^4\,\left (35\,a\,d\,e^2+\frac {35\,b\,d\,e^2\,n}{3}\right )+\frac {5\,b\,d^3\,n}{7}}{35\,x^7}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{7}+\frac {3\,b\,d^2\,e\,x^2}{5}+b\,d\,e^2\,x^4+b\,e^3\,x^6\right )}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^3*(a + b*log(c*x^n)))/x^8,x)

[Out]

- (x^6*(35*a*e^3 + 35*b*e^3*n) + 5*a*d^3 + x^2*(21*a*d^2*e + (21*b*d^2*e*n)/5) + x^4*(35*a*d*e^2 + (35*b*d*e^2
*n)/3) + (5*b*d^3*n)/7)/(35*x^7) - (log(c*x^n)*((b*d^3)/7 + b*e^3*x^6 + (3*b*d^2*e*x^2)/5 + b*d*e^2*x^4))/x^7

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sympy [A]  time = 10.39, size = 206, normalized size = 1.62 \[ - \frac {a d^{3}}{7 x^{7}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {a d e^{2}}{x^{3}} - \frac {a e^{3}}{x} - \frac {b d^{3} n \log {\relax (x )}}{7 x^{7}} - \frac {b d^{3} n}{49 x^{7}} - \frac {b d^{3} \log {\relax (c )}}{7 x^{7}} - \frac {3 b d^{2} e n \log {\relax (x )}}{5 x^{5}} - \frac {3 b d^{2} e n}{25 x^{5}} - \frac {3 b d^{2} e \log {\relax (c )}}{5 x^{5}} - \frac {b d e^{2} n \log {\relax (x )}}{x^{3}} - \frac {b d e^{2} n}{3 x^{3}} - \frac {b d e^{2} \log {\relax (c )}}{x^{3}} - \frac {b e^{3} n \log {\relax (x )}}{x} - \frac {b e^{3} n}{x} - \frac {b e^{3} \log {\relax (c )}}{x} \]

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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