3.3.5 \(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^2} \, dx\) [205]

Optimal. Leaf size=118 \[ -\frac {b d^3 n}{x}-3 b d^2 e n x-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-b*d^3*n/x-3*b*d^2*e*n*x-1/3*b*d*e^2*n*x^3-1/25*b*e^3*n*x^5-d^3*(a+b*ln(c*x^n))/x+3*d^2*e*x*(a+b*ln(c*x^n))+d*
e^2*x^3*(a+b*ln(c*x^n))+1/5*e^3*x^5*(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 )}{x}+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{x}-3 b d^2 e n x-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.04, size = 123, normalized size = 1.04 \begin {gather*} -\frac {b d^3 n}{x}+3 a d^2 e x-3 b d^2 e n x-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5+3 b d^2 e x \log \left (c x^n\right )-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 587, normalized size = 4.97

method	result	size
risch	\(-\frac {b \left (-e^{3} x^{6}-5 d \,e^{2} x^{4}-15 d^{2} e \,x^{2}+5 d^{3}\right ) \ln \left (x^{n}\right )}{5 x}-\frac {-30 \ln \left (c \right ) b \,e^{3} x^{6}-30 x^{6} a \,e^{3}-15 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e -75 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-75 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-150 x^{4} a d \,e^{2}-450 a \,d^{2} x^{2} e +150 a \,d^{3}+75 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 )+15 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+15 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 )+150 b \,d^{3} n +150 d^{3} b \ln \left (c \right )-450 \ln \left (c \right ) b \,d^{2} x^{2} e -150 \ln \left (c \right ) b d \,e^{2} x^{4}+225 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 )+6 b \,e^{3} n \,x^{6}+225 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-15 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+50 b d \,e^{2} n \,x^{4}+450 b \,d^{2} e n \,x^{2}+75 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-75 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 )-75 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{150 x}\)	\(587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 132, normalized size = 1.12 \begin {gather*} -\frac {1}{25} \, b n x^{5} e^{3} + \frac {1}{5} \, b x^{5} e^{3} \log \left (c x^{n}\right ) + \frac {1}{5} \, a x^{5} e^{3} - \frac {1}{3} \, b d n x^{3} e^{2} + b d x^{3} e^{2} \log \left (c x^{n}\right ) + a d x^{3} e^{2} - 3 \, b d^{2} n x e + 3 \, b d^{2} x e \log \left (c x^{n}\right ) + 3 \, a d^{2} x e - \frac {b d^{3} n}{x} - \frac {b d^{3} \log \left (c x^{n}\right )}{x} - \frac {a d^{3}}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 149, normalized size = 1.26 \begin {gather*} -\frac {3 \, {\left (b n - 5 \, a\right )} x^{6} e^{3} + 25 \, {\left (b d n - 3 \, a d\right )} x^{4} e^{2} + 75 \, b d^{3} n + 75 \, a d^{3} + 225 \, {\left (b d^{2} n - a d^{2}\right )} x^{2} e - 15 \, {\left (b x^{6} e^{3} + 5 \, b d x^{4} e^{2} + 15 \, b d^{2} x^{2} e - 5 \, b d^{3}\right )} \log \left (c\right ) - 15 \, {\left (b n x^{6} e^{3} + 5 \, b d n x^{4} e^{2} + 15 \, b d^{2} n x^{2} e - 5 \, b d^{3} n\right )} \log \left (x\right )}{75 \, x} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.97, size = 146, normalized size = 1.24 \begin {gather*} - \frac {a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac {a e^{3} x^{5}}{5} - \frac {b d^{3} n}{x} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{x} - 3 b d^{2} e n x + 3 b d^{2} e x \log {\left (c x^{n} \right )} - \frac {b d e^{2} n x^{3}}{3} + b d e^{2} x^{3} \log {\left (c x^{n} \right )} - \frac {b e^{3} n x^{5}}{25} + \frac {b e^{3} x^{5} \log {\left (c x^{n} \right )}}{5} \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**2,x)

[Out]

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

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Giac [A]
time = 3.84, size = 166, normalized size = 1.41 \begin {gather*} \frac {15 \, b n x^{6} e^{3} \log \left (x\right ) - 3 \, b n x^{6} e^{3} + 15 \, b x^{6} e^{3} \log \left (c\right ) + 75 \, b d n x^{4} e^{2} \log \left (x\right ) + 15 \, a x^{6} e^{3} - 25 \, b d n x^{4} e^{2} + 75 \, b d x^{4} e^{2} \log \left (c\right ) + 225 \, b d^{2} n x^{2} e \log \left (x\right ) + 75 \, a d x^{4} e^{2} - 225 \, b d^{2} n x^{2} e + 225 \, b d^{2} x^{2} e \log \left (c\right ) + 225 \, a d^{2} x^{2} e - 75 \, b d^{3} n \log \left (x\right ) - 75 \, b d^{3} n - 75 \, b d^{3} \log \left (c\right ) - 75 \, a d^{3}}{75 \, x} \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^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.52, size = 145, normalized size = 1.23 \begin {gather*} \ln \left (c\,x^n\right )\,\left (\frac {6\,b\,d^2\,e\,x^2+4\,b\,d\,e^2\,x^4+\frac {6\,b\,e^3\,x^6}{5}}{x}-\frac {b\,d^3+3\,b\,d^2\,e\,x^2+3\,b\,d\,e^2\,x^4+b\,e^3\,x^6}{x}\right )-\frac {a\,d^3+b\,d^3\,n}{x}+\frac {e^3\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {d\,e^2\,x^3\,\left (3\,a-b\,n\right )}{3}+3\,d^2\,e\,x\,\left (a-b\,n\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^2,x)

[Out]

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

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