3.1.20 \(\int x^2 (d+e x)^3 (a+b \log (c x^n)) \, dx\) [20]

Optimal. Leaf size=100 \[ -\frac {1}{9} b d^3 n x^3-\frac {3}{16} b d^2 e n x^4-\frac {3}{25} b d e^2 n x^5-\frac {1}{36} b e^3 n x^6+\frac {1}{60} \left (20 d^3 x^3+45 d^2 e x^4+36 d e^2 x^5+10 e^3 x^6\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/9*b*d^3*n*x^3-3/16*b*d^2*e*n*x^4-3/25*b*d*e^2*n*x^5-1/36*b*e^3*n*x^6+1/60*(10*e^3*x^6+36*d*e^2*x^5+45*d^2*e
*x^4+20*d^3*x^3)*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2371, 12, 14} \begin {gather*} \frac {1}{60} \left (20 d^3 x^3+45 d^2 e x^4+36 d e^2 x^5+10 e^3 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^3 n x^3-\frac {3}{16} b d^2 e n x^4-\frac {3}{25} b d e^2 n x^5-\frac {1}{36} b e^3 n x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(b*d^3*n*x^3) - (3*b*d^2*e*n*x^4)/16 - (3*b*d*e^2*n*x^5)/25 - (b*e^3*n*x^6)/36 + ((20*d^3*x^3 + 45*d^2*e*
x^4 + 36*d*e^2*x^5 + 10*e^3*x^6)*(a + b*Log[c*x^n]))/60

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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2371

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] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int x^2 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{60} \left (20 d^3 x^3+45 d^2 e x^4+36 d e^2 x^5+10 e^3 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{60} x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right ) \, dx\\ &=\frac {1}{60} \left (20 d^3 x^3+45 d^2 e x^4+36 d e^2 x^5+10 e^3 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right ) \, dx\\ &=\frac {1}{60} \left (20 d^3 x^3+45 d^2 e x^4+36 d e^2 x^5+10 e^3 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int \left (20 d^3 x^2+45 d^2 e x^3+36 d e^2 x^4+10 e^3 x^5\right ) \, dx\\ &=-\frac {1}{9} b d^3 n x^3-\frac {3}{16} b d^2 e n x^4-\frac {3}{25} b d e^2 n x^5-\frac {1}{36} b e^3 n x^6+\frac {1}{60} \left (20 d^3 x^3+45 d^2 e x^4+36 d e^2 x^5+10 e^3 x^6\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 = 1.33 \begin {gather*} -\frac {1}{9} b d^3 n x^3-\frac {3}{16} b d^2 e n x^4-\frac {3}{25} b d e^2 n x^5-\frac {1}{36} b e^3 n x^6+\frac {1}{3} d^3 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} d^2 e x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{6} e^3 x^6 \left (a+b \log \left (c x^n\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(b*d^3*n*x^3) - (3*b*d^2*e*n*x^4)/16 - (3*b*d*e^2*n*x^5)/25 - (b*e^3*n*x^6)/36 + (d^3*x^3*(a + b*Log[c*x^
n]))/3 + (3*d^2*e*x^4*(a + b*Log[c*x^n]))/4 + (3*d*e^2*x^5*(a + b*Log[c*x^n]))/5 + (e^3*x^6*(a + b*Log[c*x^n])
)/6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 91.93, size = 600, normalized size = 6.00

method	result	size
risch	\(\frac {\ln \left (c \right ) b \,d^{3} x^{3}}{3}+\frac {\ln \left (c \right ) b \,e^{3} x^{6}}{6}+\frac {x^{6} a \,e^{3}}{6}+\frac {x^{3} a \,d^{3}}{3}+\frac {3 \ln \left (c \right ) b d \,e^{2} x^{5}}{5}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{4}}{4}+\frac {b \,x^{3} \left (10 e^{3} x^{3}+36 d \,e^{2} x^{2}+45 d^{2} e x +20 d^{3}\right ) \ln \left (x^{n}\right )}{60}+\frac {3 x^{4} a \,d^{2} e}{4}+\frac {3 x^{5} a d \,e^{2}}{5}-\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{10}-\frac {3 i \pi b \,d^{2} e \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}-\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}-\frac {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 )}{12}-\frac {b \,e^{3} n \,x^{6}}{36}+\frac {i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {3 i \pi b \,d^{2} e \,x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {b \,d^{3} n \,x^{3}}{9}-\frac {3 b \,d^{2} e n \,x^{4}}{16}-\frac {3 b d \,e^{2} n \,x^{5}}{25}+\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {3 i \pi b \,d^{2} e \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {3 i \pi b \,d^{2} e \,x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{12}-\frac {i \pi b \,d^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}\)	\(600\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(c)*b*d^3*x^3+1/6*ln(c)*b*e^3*x^6+1/6*x^6*a*e^3+1/3*x^3*a*d^3+3/5*ln(c)*b*d*e^2*x^5+3/4*ln(c)*b*d^2*e*x^
4+1/60*b*x^3*(10*e^3*x^3+36*d*e^2*x^2+45*d^2*e*x+20*d^3)*ln(x^n)+3/10*I*Pi*b*d*e^2*x^5*csgn(I*c)*csgn(I*c*x^n)
^2+3/10*I*Pi*b*d*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2+3/4*x^4*a*d^2*e+3/5*x^5*a*d*e^2+3/8*I*Pi*b*d^2*e*x^4*csgn
(I*c)*csgn(I*c*x^n)^2+3/8*I*Pi*b*d^2*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I*Pi*b*d^3*x^3*csgn(I*c)*csgn(I*x^n
)*csgn(I*c*x^n)-1/12*I*Pi*b*e^3*x^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/36*b*e^3*n*x^6-1/9*b*d^3*n*x^3+1/12*
I*Pi*b*e^3*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3/8*I*Pi*b*d^2*e*x^4*csgn(I*c*x^n)^3-3/10*I*Pi*b*d*e^2*x^5*csgn(I*c
*x^n)^3-3/16*b*d^2*e*n*x^4-3/25*b*d*e^2*n*x^5+1/6*I*Pi*b*d^3*x^3*csgn(I*c)*csgn(I*c*x^n)^2+1/6*I*Pi*b*d^3*x^3*
csgn(I*x^n)*csgn(I*c*x^n)^2+1/12*I*Pi*b*e^3*x^6*csgn(I*c)*csgn(I*c*x^n)^2-3/10*I*Pi*b*d*e^2*x^5*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)-3/8*I*Pi*b*d^2*e*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/12*I*Pi*b*e^3*x^6*csgn(I*c*x^
n)^3-1/6*I*Pi*b*d^3*x^3*csgn(I*c*x^n)^3

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Maxima [A]
time = 0.28, size = 140, normalized size = 1.40 \begin {gather*} -\frac {1}{36} \, b n x^{6} e^{3} - \frac {3}{25} \, b d n x^{5} e^{2} - \frac {3}{16} \, b d^{2} n x^{4} e + \frac {1}{6} \, b x^{6} e^{3} \log \left (c x^{n}\right ) + \frac {3}{5} \, b d x^{5} e^{2} \log \left (c x^{n}\right ) + \frac {3}{4} \, b d^{2} x^{4} e \log \left (c x^{n}\right ) - \frac {1}{9} \, b d^{3} n x^{3} + \frac {1}{6} \, a x^{6} e^{3} + \frac {3}{5} \, a d x^{5} e^{2} + \frac {3}{4} \, a d^{2} x^{4} e + \frac {1}{3} \, b d^{3} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/36*b*n*x^6*e^3 - 3/25*b*d*n*x^5*e^2 - 3/16*b*d^2*n*x^4*e + 1/6*b*x^6*e^3*log(c*x^n) + 3/5*b*d*x^5*e^2*log(c
*x^n) + 3/4*b*d^2*x^4*e*log(c*x^n) - 1/9*b*d^3*n*x^3 + 1/6*a*x^6*e^3 + 3/5*a*d*x^5*e^2 + 3/4*a*d^2*x^4*e + 1/3
*b*d^3*x^3*log(c*x^n) + 1/3*a*d^3*x^3

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Fricas [A]
time = 0.35, size = 157, normalized size = 1.57 \begin {gather*} -\frac {1}{36} \, {\left (b n - 6 \, a\right )} x^{6} e^{3} - \frac {3}{25} \, {\left (b d n - 5 \, a d\right )} x^{5} e^{2} - \frac {3}{16} \, {\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} e - \frac {1}{9} \, {\left (b d^{3} n - 3 \, a d^{3}\right )} x^{3} + \frac {1}{60} \, {\left (10 \, b x^{6} e^{3} + 36 \, b d x^{5} e^{2} + 45 \, b d^{2} x^{4} e + 20 \, b d^{3} x^{3}\right )} \log \left (c\right ) + \frac {1}{60} \, {\left (10 \, b n x^{6} e^{3} + 36 \, b d n x^{5} e^{2} + 45 \, b d^{2} n x^{4} e + 20 \, b d^{3} n x^{3}\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/36*(b*n - 6*a)*x^6*e^3 - 3/25*(b*d*n - 5*a*d)*x^5*e^2 - 3/16*(b*d^2*n - 4*a*d^2)*x^4*e - 1/9*(b*d^3*n - 3*a
*d^3)*x^3 + 1/60*(10*b*x^6*e^3 + 36*b*d*x^5*e^2 + 45*b*d^2*x^4*e + 20*b*d^3*x^3)*log(c) + 1/60*(10*b*n*x^6*e^3
+ 36*b*d*n*x^5*e^2 + 45*b*d^2*n*x^4*e + 20*b*d^3*n*x^3)*log(x)

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Sympy [A]
time = 0.61, size = 175, normalized size = 1.75 \begin {gather*} \frac {a d^{3} x^{3}}{3} + \frac {3 a d^{2} e x^{4}}{4} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{6}}{6} - \frac {b d^{3} n x^{3}}{9} + \frac {b d^{3} x^{3} \log {\left (c x^{n} \right )}}{3} - \frac {3 b d^{2} e n x^{4}}{16} + \frac {3 b d^{2} e x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {3 b d e^{2} n x^{5}}{25} + \frac {3 b d e^{2} x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {b e^{3} n x^{6}}{36} + \frac {b e^{3} x^{6} \log {\left (c x^{n} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(e*x+d)**3*(a+b*ln(c*x**n)),x)

[Out]

a*d**3*x**3/3 + 3*a*d**2*e*x**4/4 + 3*a*d*e**2*x**5/5 + a*e**3*x**6/6 - b*d**3*n*x**3/9 + b*d**3*x**3*log(c*x*
*n)/3 - 3*b*d**2*e*n*x**4/16 + 3*b*d**2*e*x**4*log(c*x**n)/4 - 3*b*d*e**2*n*x**5/25 + 3*b*d*e**2*x**5*log(c*x*
*n)/5 - b*e**3*n*x**6/36 + b*e**3*x**6*log(c*x**n)/6

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Giac [A]
time = 1.77, size = 173, normalized size = 1.73 \begin {gather*} \frac {1}{6} \, b n x^{6} e^{3} \log \left (x\right ) + \frac {3}{5} \, b d n x^{5} e^{2} \log \left (x\right ) + \frac {3}{4} \, b d^{2} n x^{4} e \log \left (x\right ) - \frac {1}{36} \, b n x^{6} e^{3} - \frac {3}{25} \, b d n x^{5} e^{2} - \frac {3}{16} \, b d^{2} n x^{4} e + \frac {1}{6} \, b x^{6} e^{3} \log \left (c\right ) + \frac {3}{5} \, b d x^{5} e^{2} \log \left (c\right ) + \frac {3}{4} \, b d^{2} x^{4} e \log \left (c\right ) + \frac {1}{3} \, b d^{3} n x^{3} \log \left (x\right ) - \frac {1}{9} \, b d^{3} n x^{3} + \frac {1}{6} \, a x^{6} e^{3} + \frac {3}{5} \, a d x^{5} e^{2} + \frac {3}{4} \, a d^{2} x^{4} e + \frac {1}{3} \, b d^{3} x^{3} \log \left (c\right ) + \frac {1}{3} \, a d^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/6*b*n*x^6*e^3*log(x) + 3/5*b*d*n*x^5*e^2*log(x) + 3/4*b*d^2*n*x^4*e*log(x) - 1/36*b*n*x^6*e^3 - 3/25*b*d*n*x
^5*e^2 - 3/16*b*d^2*n*x^4*e + 1/6*b*x^6*e^3*log(c) + 3/5*b*d*x^5*e^2*log(c) + 3/4*b*d^2*x^4*e*log(c) + 1/3*b*d
^3*n*x^3*log(x) - 1/9*b*d^3*n*x^3 + 1/6*a*x^6*e^3 + 3/5*a*d*x^5*e^2 + 3/4*a*d^2*x^4*e + 1/3*b*d^3*x^3*log(c) +
1/3*a*d^3*x^3

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Mupad [B]
time = 3.64, size = 113, normalized size = 1.13 \begin {gather*} \ln \left (c\,x^n\right )\,\left (\frac {b\,d^3\,x^3}{3}+\frac {3\,b\,d^2\,e\,x^4}{4}+\frac {3\,b\,d\,e^2\,x^5}{5}+\frac {b\,e^3\,x^6}{6}\right )+\frac {d^3\,x^3\,\left (3\,a-b\,n\right )}{9}+\frac {e^3\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {3\,d^2\,e\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {3\,d\,e^2\,x^5\,\left (5\,a-b\,n\right )}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^n)*((b*d^3*x^3)/3 + (b*e^3*x^6)/6 + (3*b*d^2*e*x^4)/4 + (3*b*d*e^2*x^5)/5) + (d^3*x^3*(3*a - b*n))/9 +
(e^3*x^6*(6*a - b*n))/36 + (3*d^2*e*x^4*(4*a - b*n))/16 + (3*d*e^2*x^5*(5*a - b*n))/25

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