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

Optimal. Leaf size=74 \[ \frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{1}{18} b d e n x^6-\frac{1}{64} b e^2 n x^8 \]

[Out]

-(b*d^2*n*x^4)/16 - (b*d*e*n*x^6)/18 - (b*e^2*n*x^8)/64 + ((6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8)*(a + b*Log[c*x^
n]))/24

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Rubi [A]  time = 0.088268, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 12, 14} \[ \frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{1}{18} b d e n x^6-\frac{1}{64} b e^2 n x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x^4)/16 - (b*d*e*n*x^6)/18 - (b*e^2*n*x^8)/64 + ((6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8)*(a + b*Log[c*x^
n]))/24

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

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 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])

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]]

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{24} x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \, dx\\ &=\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{24} (b n) \int x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \, dx\\ &=\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{24} (b n) \int \left (6 d^2 x^3+8 d e x^5+3 e^2 x^7\right ) \, dx\\ &=-\frac{1}{16} b d^2 n x^4-\frac{1}{18} b d e n x^6-\frac{1}{64} b e^2 n x^8+\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0365365, size = 87, normalized size = 1.18 \[ \frac{1}{576} x^4 \left (24 a \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+24 b \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \log \left (c x^n\right )-b n \left (36 d^2+32 d e x^2+9 e^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(24*a*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) - b*n*(36*d^2 + 32*d*e*x^2 + 9*e^2*x^4) + 24*b*(6*d^2 + 8*d*e*x^2 +
 3*e^2*x^4)*Log[c*x^n]))/576

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Maple [C]  time = 0.198, size = 434, normalized size = 5.9 \begin{align*}{\frac{b{x}^{4} \left ( 3\,{e}^{2}{x}^{4}+8\,de{x}^{2}+6\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{24}}-{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,bde{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{8}}{8}}-{\frac{b{e}^{2}n{x}^{8}}{64}}+{\frac{a{e}^{2}{x}^{8}}{8}}+{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,bde{x}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{6}}\pi \,bde{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) bde{x}^{6}}{3}}-{\frac{bden{x}^{6}}{18}}+{\frac{ade{x}^{6}}{3}}+{\frac{i}{6}}\pi \,bde{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) b{d}^{2}{x}^{4}}{4}}-{\frac{b{d}^{2}n{x}^{4}}{16}}+{\frac{a{d}^{2}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b*x^4*(3*e^2*x^4+8*d*e*x^2+6*d^2)*ln(x^n)-1/16*I*Pi*b*e^2*x^8*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/8*I*P
i*b*d^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/8*I*Pi*b*d^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I*Pi*b*d*e*
x^6*csgn(I*c*x^n)^3+1/8*ln(c)*b*e^2*x^8-1/64*b*e^2*n*x^8+1/8*a*e^2*x^8+1/8*I*Pi*b*d^2*x^4*csgn(I*c*x^n)^2*csgn
(I*c)-1/6*I*Pi*b*d*e*x^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/8*I*Pi*b*d^2*x^4*csgn(I*c*x^n)^3+1/6*I*Pi*b*d*e
*x^6*csgn(I*x^n)*csgn(I*c*x^n)^2+1/3*ln(c)*b*d*e*x^6-1/18*b*d*e*n*x^6+1/3*a*d*e*x^6+1/6*I*Pi*b*d*e*x^6*csgn(I*
c*x^n)^2*csgn(I*c)-1/16*I*Pi*b*e^2*x^8*csgn(I*c*x^n)^3+1/16*I*Pi*b*e^2*x^8*csgn(I*c*x^n)^2*csgn(I*c)+1/16*I*Pi
*b*e^2*x^8*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*ln(c)*b*d^2*x^4-1/16*b*d^2*n*x^4+1/4*a*d^2*x^4

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Maxima [A]  time = 1.12407, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{64} \, b e^{2} n x^{8} + \frac{1}{8} \, b e^{2} x^{8} \log \left (c x^{n}\right ) + \frac{1}{8} \, a e^{2} x^{8} - \frac{1}{18} \, b d e n x^{6} + \frac{1}{3} \, b d e x^{6} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d e x^{6} - \frac{1}{16} \, b d^{2} n x^{4} + \frac{1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*b*e^2*n*x^8 + 1/8*b*e^2*x^8*log(c*x^n) + 1/8*a*e^2*x^8 - 1/18*b*d*e*n*x^6 + 1/3*b*d*e*x^6*log(c*x^n) + 1
/3*a*d*e*x^6 - 1/16*b*d^2*n*x^4 + 1/4*b*d^2*x^4*log(c*x^n) + 1/4*a*d^2*x^4

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Fricas [A]  time = 1.34008, size = 285, normalized size = 3.85 \begin{align*} -\frac{1}{64} \,{\left (b e^{2} n - 8 \, a e^{2}\right )} x^{8} - \frac{1}{18} \,{\left (b d e n - 6 \, a d e\right )} x^{6} - \frac{1}{16} \,{\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} + \frac{1}{24} \,{\left (3 \, b e^{2} x^{8} + 8 \, b d e x^{6} + 6 \, b d^{2} x^{4}\right )} \log \left (c\right ) + \frac{1}{24} \,{\left (3 \, b e^{2} n x^{8} + 8 \, b d e n x^{6} + 6 \, b d^{2} n x^{4}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(b*e^2*n - 8*a*e^2)*x^8 - 1/18*(b*d*e*n - 6*a*d*e)*x^6 - 1/16*(b*d^2*n - 4*a*d^2)*x^4 + 1/24*(3*b*e^2*x^
8 + 8*b*d*e*x^6 + 6*b*d^2*x^4)*log(c) + 1/24*(3*b*e^2*n*x^8 + 8*b*d*e*n*x^6 + 6*b*d^2*n*x^4)*log(x)

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Sympy [B]  time = 13.8342, size = 151, normalized size = 2.04 \begin{align*} \frac{a d^{2} x^{4}}{4} + \frac{a d e x^{6}}{3} + \frac{a e^{2} x^{8}}{8} + \frac{b d^{2} n x^{4} \log{\left (x \right )}}{4} - \frac{b d^{2} n x^{4}}{16} + \frac{b d^{2} x^{4} \log{\left (c \right )}}{4} + \frac{b d e n x^{6} \log{\left (x \right )}}{3} - \frac{b d e n x^{6}}{18} + \frac{b d e x^{6} \log{\left (c \right )}}{3} + \frac{b e^{2} n x^{8} \log{\left (x \right )}}{8} - \frac{b e^{2} n x^{8}}{64} + \frac{b e^{2} x^{8} \log{\left (c \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*n*x**4*log(x)/4 - b*d**2*n*x**4/16 + b*d**2*x**4*log(c)/
4 + b*d*e*n*x**6*log(x)/3 - b*d*e*n*x**6/18 + b*d*e*x**6*log(c)/3 + b*e**2*n*x**8*log(x)/8 - b*e**2*n*x**8/64
+ b*e**2*x**8*log(c)/8

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Giac [A]  time = 1.37819, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{8} \, b n x^{8} e^{2} \log \left (x\right ) - \frac{1}{64} \, b n x^{8} e^{2} + \frac{1}{8} \, b x^{8} e^{2} \log \left (c\right ) + \frac{1}{3} \, b d n x^{6} e \log \left (x\right ) + \frac{1}{8} \, a x^{8} e^{2} - \frac{1}{18} \, b d n x^{6} e + \frac{1}{3} \, b d x^{6} e \log \left (c\right ) + \frac{1}{3} \, a d x^{6} e + \frac{1}{4} \, b d^{2} n x^{4} \log \left (x\right ) - \frac{1}{16} \, b d^{2} n x^{4} + \frac{1}{4} \, b d^{2} x^{4} \log \left (c\right ) + \frac{1}{4} \, a d^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*n*x^8*e^2*log(x) - 1/64*b*n*x^8*e^2 + 1/8*b*x^8*e^2*log(c) + 1/3*b*d*n*x^6*e*log(x) + 1/8*a*x^8*e^2 - 1/
18*b*d*n*x^6*e + 1/3*b*d*x^6*e*log(c) + 1/3*a*d*x^6*e + 1/4*b*d^2*n*x^4*log(x) - 1/16*b*d^2*n*x^4 + 1/4*b*d^2*
x^4*log(c) + 1/4*a*d^2*x^4