3.2.72 \(\int x^3 (d+e x^2) (a+b \log (c x^n)) \, dx\) [172]

Optimal. Leaf size=48 \[ -\frac {1}{16} b d n x^4-\frac {1}{36} b e n x^6+\frac {1}{12} \left (3 d x^4+2 e x^6\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/16*b*d*n*x^4-1/36*b*e*n*x^6+1/12*(2*e*x^6+3*d*x^4)*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2371} \begin {gather*} \frac {1}{12} \left (3 d x^4+2 e x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d n x^4-\frac {1}{36} b e n x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(b*d*n*x^4) - (b*e*n*x^6)/36 + ((3*d*x^4 + 2*e*x^6)*(a + b*Log[c*x^n]))/12

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 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^3 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{12} \left (3 d x^4+2 e x^6\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {d x^3}{4}+\frac {e x^5}{6}\right ) \, dx\\ &=-\frac {1}{16} b d n x^4-\frac {1}{36} b e n x^6+\frac {1}{12} \left (3 d x^4+2 e x^6\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 69, normalized size = 1.44 \begin {gather*} \frac {1}{4} a d x^4-\frac {1}{16} b d n x^4+\frac {1}{6} a e x^6-\frac {1}{36} b e n x^6+\frac {1}{4} b d x^4 \log \left (c x^n\right )+\frac {1}{6} b e x^6 \log \left (c x^n\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*d*x^4)/4 - (b*d*n*x^4)/16 + (a*e*x^6)/6 - (b*e*n*x^6)/36 + (b*d*x^4*Log[c*x^n])/4 + (b*e*x^6*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 = 0.15, size = 266, normalized size = 5.54

method	result	size
risch	\(\frac {b \,x^{4} \left (2 e \,x^{2}+3 d \right ) \ln \left (x^{n}\right )}{12}-\frac {i \pi b e \,x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{12}+\frac {i \pi b e \,x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}+\frac {i \pi b e \,x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b e \,x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{12}+\frac {\ln \left (c \right ) b e \,x^{6}}{6}-\frac {b e n \,x^{6}}{36}+\frac {x^{6} a e}{6}-\frac {i \pi b d \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b d \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b d \,x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b d \,x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}+\frac {\ln \left (c \right ) b d \,x^{4}}{4}-\frac {b d n \,x^{4}}{16}+\frac {x^{4} a d}{4}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b*x^4*(2*e*x^2+3*d)*ln(x^n)-1/12*I*Pi*b*e*x^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/12*I*Pi*b*e*x^6*csgn(
I*c)*csgn(I*c*x^n)^2+1/12*I*Pi*b*e*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2-1/12*I*Pi*b*e*x^6*csgn(I*c*x^n)^3+1/6*ln(c)
*b*e*x^6-1/36*b*e*n*x^6+1/6*x^6*a*e-1/8*I*Pi*b*d*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*I*Pi*b*d*x^4*csgn
(I*c)*csgn(I*c*x^n)^2+1/8*I*Pi*b*d*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*I*Pi*b*d*x^4*csgn(I*c*x^n)^3+1/4*ln(c)*
b*d*x^4-1/16*b*d*n*x^4+1/4*x^4*a*d

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Maxima [A]
time = 0.28, size = 60, normalized size = 1.25 \begin {gather*} -\frac {1}{36} \, b n x^{6} e + \frac {1}{6} \, b x^{6} e \log \left (c x^{n}\right ) + \frac {1}{6} \, a x^{6} e - \frac {1}{16} \, b d n x^{4} + \frac {1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 71, normalized size = 1.48 \begin {gather*} -\frac {1}{36} \, {\left (b n - 6 \, a\right )} x^{6} e - \frac {1}{16} \, {\left (b d n - 4 \, a d\right )} x^{4} + \frac {1}{12} \, {\left (2 \, b x^{6} e + 3 \, b d x^{4}\right )} \log \left (c\right ) + \frac {1}{12} \, {\left (2 \, b n x^{6} e + 3 \, b d n x^{4}\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(b*n - 6*a)*x^6*e - 1/16*(b*d*n - 4*a*d)*x^4 + 1/12*(2*b*x^6*e + 3*b*d*x^4)*log(c) + 1/12*(2*b*n*x^6*e +
3*b*d*n*x^4)*log(x)

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Sympy [A]
time = 0.67, size = 66, normalized size = 1.38 \begin {gather*} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} - \frac {b d n x^{4}}{16} + \frac {b d x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {b e n x^{6}}{36} + \frac {b e x^{6} \log {\left (c x^{n} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x**4/4 + a*e*x**6/6 - b*d*n*x**4/16 + b*d*x**4*log(c*x**n)/4 - b*e*n*x**6/36 + b*e*x**6*log(c*x**n)/6

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Giac [A]
time = 2.36, size = 73, normalized size = 1.52 \begin {gather*} \frac {1}{6} \, b n x^{6} e \log \left (x\right ) - \frac {1}{36} \, b n x^{6} e + \frac {1}{6} \, b x^{6} e \log \left (c\right ) + \frac {1}{6} \, a x^{6} e + \frac {1}{4} \, b d n x^{4} \log \left (x\right ) - \frac {1}{16} \, b d n x^{4} + \frac {1}{4} \, b d x^{4} \log \left (c\right ) + \frac {1}{4} \, a d x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.39, size = 51, normalized size = 1.06 \begin {gather*} \ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^6}{6}+\frac {b\,d\,x^4}{4}\right )+\frac {d\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {e\,x^6\,\left (6\,a-b\,n\right )}{36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^n)*((b*d*x^4)/4 + (b*e*x^6)/6) + (d*x^4*(4*a - b*n))/16 + (e*x^6*(6*a - b*n))/36

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