3.3.7 \(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^6} \, dx\) [207]

Optimal. Leaf size=118 \[ -\frac {b d^3 n}{25 x^5}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{x}-b e^3 n x-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {276, 2372} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+e^3 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{25 x^5}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{x}-b e^3 n x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/25*(b*d^3*n)/x^5 - (b*d^2*e*n)/(3*x^3) - (3*b*d*e^2*n)/x - b*e^3*n*x - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (
d^2*e*(a + b*Log[c*x^n]))/x^3 - (3*d*e^2*(a + b*Log[c*x^n]))/x + e^3*x*(a + b*Log[c*x^n])

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

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Mathematica [A]
time = 0.04, size = 115, normalized size = 0.97 \begin {gather*} -\frac {15 a \left (d^3+5 d^2 e x^2+15 d e^2 x^4-5 e^3 x^6\right )+b n \left (3 d^3+25 d^2 e x^2+225 d e^2 x^4+75 e^3 x^6\right )+15 b \left (d^3+5 d^2 e x^2+15 d e^2 x^4-5 e^3 x^6\right ) \log \left (c x^n\right )}{75 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/75*(15*a*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6) + b*n*(3*d^3 + 25*d^2*e*x^2 + 225*d*e^2*x^4 + 75*e^
3*x^6) + 15*b*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6)*Log[c*x^n])/x^5

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 585, normalized size = 4.96

method	result	size
risch	\(-\frac {b \left (-5 e^{3} x^{6}+15 d \,e^{2} x^{4}+5 d^{2} e \,x^{2}+d^{3}\right ) \ln \left (x^{n}\right )}{5 x^{5}}-\frac {-150 \ln \left (c \right ) b \,e^{3} x^{6}-150 x^{6} a \,e^{3}-75 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e +225 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+225 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+450 x^{4} a d \,e^{2}+150 a \,d^{2} x^{2} e +30 a \,d^{3}-225 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 )+75 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+75 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 )+6 b \,d^{3} n +30 d^{3} b \ln \left (c \right )+150 \ln \left (c \right ) b \,d^{2} x^{2} e +450 \ln \left (c \right ) b d \,e^{2} x^{4}-75 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 )+150 b \,e^{3} n \,x^{6}-75 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-75 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+450 b d \,e^{2} n \,x^{4}+50 b \,d^{2} e n \,x^{2}+15 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+15 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-15 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 )-15 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{150 x^{5}}\)	\(585\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(a+b*ln(c*x^n))/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*b*(-5*e^3*x^6+15*d*e^2*x^4+5*d^2*e*x^2+d^3)/x^5*ln(x^n)-1/150*(-150*ln(c)*b*e^3*x^6-150*x^6*a*e^3-75*I*Pi
*b*e^3*x^6*csgn(I*c)*csgn(I*c*x^n)^2-75*I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2+450*x^4*a*d*e^2+150*a*d^2*x
^2*e+30*a*d^3-15*I*Pi*b*d^3*csgn(I*c*x^n)^3+225*I*Pi*b*d*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+75*I*Pi*b*d^2*e*x
^2*csgn(I*c)*csgn(I*c*x^n)^2+75*I*Pi*b*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2*e+6*b*d^3*n+75*I*Pi*b*e^3*x^6*csgn(
I*c)*csgn(I*x^n)*csgn(I*c*x^n)+30*d^3*b*ln(c)+150*ln(c)*b*d^2*x^2*e+450*ln(c)*b*d*e^2*x^4+225*I*Pi*b*d*e^2*x^4
*csgn(I*c)*csgn(I*c*x^n)^2-225*I*Pi*b*d*e^2*x^4*csgn(I*c*x^n)^3-75*I*Pi*b*d^2*e*x^2*csgn(I*c*x^n)^3+150*b*e^3*
n*x^6+450*b*d*e^2*n*x^4+50*b*d^2*e*n*x^2-15*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-225*I*Pi*b*d*e^2*x^
4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-75*I*Pi*b*d^2*e*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+15*I*Pi*b*d^3*cs
gn(I*c)*csgn(I*c*x^n)^2+15*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+75*I*Pi*b*e^3*x^6*csgn(I*c*x^n)^3)/x^5

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Maxima [A]
time = 0.28, size = 132, normalized size = 1.12 \begin {gather*} -b n x e^{3} + b x e^{3} \log \left (c x^{n}\right ) + a x e^{3} - \frac {3 \, b d n e^{2}}{x} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{x} - \frac {3 \, a d e^{2}}{x} - \frac {b d^{2} n e}{3 \, x^{3}} - \frac {b d^{2} e \log \left (c x^{n}\right )}{x^{3}} - \frac {a d^{2} e}{x^{3}} - \frac {b d^{3} n}{25 \, x^{5}} - \frac {b d^{3} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d^{3}}{5 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 150, normalized size = 1.27 \begin {gather*} -\frac {75 \, {\left (b n - a\right )} x^{6} e^{3} + 225 \, {\left (b d n + a d\right )} x^{4} e^{2} + 3 \, b d^{3} n + 15 \, a d^{3} + 25 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} x^{2} e - 15 \, {\left (5 \, b x^{6} e^{3} - 15 \, b d x^{4} e^{2} - 5 \, b d^{2} x^{2} e - b d^{3}\right )} \log \left (c\right ) - 15 \, {\left (5 \, b n x^{6} e^{3} - 15 \, b d n x^{4} e^{2} - 5 \, b d^{2} n x^{2} e - b d^{3} n\right )} \log \left (x\right )}{75 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/75*(75*(b*n - a)*x^6*e^3 + 225*(b*d*n + a*d)*x^4*e^2 + 3*b*d^3*n + 15*a*d^3 + 25*(b*d^2*n + 3*a*d^2)*x^2*e
- 15*(5*b*x^6*e^3 - 15*b*d*x^4*e^2 - 5*b*d^2*x^2*e - b*d^3)*log(c) - 15*(5*b*n*x^6*e^3 - 15*b*d*n*x^4*e^2 - 5*
b*d^2*n*x^2*e - b*d^3*n)*log(x))/x^5

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Sympy [A]
time = 1.11, size = 146, normalized size = 1.24 \begin {gather*} - \frac {a d^{3}}{5 x^{5}} - \frac {a d^{2} e}{x^{3}} - \frac {3 a d e^{2}}{x} + a e^{3} x - \frac {b d^{3} n}{25 x^{5}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b d^{2} e n}{3 x^{3}} - \frac {b d^{2} e \log {\left (c x^{n} \right )}}{x^{3}} - \frac {3 b d e^{2} n}{x} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{x} - b e^{3} n x + b e^{3} x \log {\left (c x^{n} \right )} \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**6,x)

[Out]

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

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Giac [A]
time = 5.65, size = 166, normalized size = 1.41 \begin {gather*} \frac {75 \, b n x^{6} e^{3} \log \left (x\right ) - 75 \, b n x^{6} e^{3} + 75 \, b x^{6} e^{3} \log \left (c\right ) - 225 \, b d n x^{4} e^{2} \log \left (x\right ) + 75 \, a x^{6} e^{3} - 225 \, b d n x^{4} e^{2} - 225 \, b d x^{4} e^{2} \log \left (c\right ) - 75 \, b d^{2} n x^{2} e \log \left (x\right ) - 225 \, a d x^{4} e^{2} - 25 \, b d^{2} n x^{2} e - 75 \, b d^{2} x^{2} e \log \left (c\right ) - 75 \, a d^{2} x^{2} e - 15 \, b d^{3} n \log \left (x\right ) - 3 \, b d^{3} n - 15 \, b d^{3} \log \left (c\right ) - 15 \, a d^{3}}{75 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

1/75*(75*b*n*x^6*e^3*log(x) - 75*b*n*x^6*e^3 + 75*b*x^6*e^3*log(c) - 225*b*d*n*x^4*e^2*log(x) + 75*a*x^6*e^3 -
225*b*d*n*x^4*e^2 - 225*b*d*x^4*e^2*log(c) - 75*b*d^2*n*x^2*e*log(x) - 225*a*d*x^4*e^2 - 25*b*d^2*n*x^2*e - 7
5*b*d^2*x^2*e*log(c) - 75*a*d^2*x^2*e - 15*b*d^3*n*log(x) - 3*b*d^3*n - 15*b*d^3*log(c) - 15*a*d^3)/x^5

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Mupad [B]
time = 3.58, size = 125, normalized size = 1.06 \begin {gather*} e^3\,x\,\left (a-b\,n\right )-\frac {a\,d^3+x^2\,\left (5\,a\,d^2\,e+\frac {5\,b\,d^2\,e\,n}{3}\right )+x^4\,\left (15\,a\,d\,e^2+15\,b\,d\,e^2\,n\right )+\frac {b\,d^3\,n}{5}}{5\,x^5}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^3}{5}+b\,d^2\,e\,x^2+3\,b\,d\,e^2\,x^4+\frac {11\,b\,e^3\,x^6}{5}}{x^5}-\frac {16\,b\,e^3\,x}{5}\right ) \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^6,x)

[Out]

e^3*x*(a - b*n) - (a*d^3 + x^2*(5*a*d^2*e + (5*b*d^2*e*n)/3) + x^4*(15*a*d*e^2 + 15*b*d*e^2*n) + (b*d^3*n)/5)/
(5*x^5) - log(c*x^n)*(((b*d^3)/5 + (11*b*e^3*x^6)/5 + b*d^2*e*x^2 + 3*b*d*e^2*x^4)/x^5 - (16*b*e^3*x)/5)

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