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

Optimal. Leaf size=48 \[ \frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d n x^3-\frac{1}{16} b e n x^4 \]

[Out]

-(b*d*n*x^3)/9 - (b*e*n*x^4)/16 + ((4*d*x^3 + 3*e*x^4)*(a + b*Log[c*x^n]))/12

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Rubi [A]  time = 0.0501997, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {43, 2334, 12} \[ \frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d n x^3-\frac{1}{16} b e n x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n*x^3)/9 - (b*e*n*x^4)/16 + ((4*d*x^3 + 3*e*x^4)*(a + b*Log[c*x^n]))/12

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]]

Rubi steps

\begin{align*} \int x^2 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{12} x^2 (4 d+3 e x) \, dx\\ &=\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{12} (b n) \int x^2 (4 d+3 e x) \, dx\\ &=\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{12} (b n) \int \left (4 d x^2+3 e x^3\right ) \, dx\\ &=-\frac{1}{9} b d n x^3-\frac{1}{16} b e n x^4+\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0237083, size = 45, normalized size = 0.94 \[ \frac{1}{144} x^3 \left (48 a d+36 a e x+12 b (4 d+3 e x) \log \left (c x^n\right )-16 b d n-9 b e n x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(48*a*d - 16*b*d*n + 36*a*e*x - 9*b*e*n*x + 12*b*(4*d + 3*e*x)*Log[c*x^n]))/144

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Maple [C]  time = 0.234, size = 264, normalized size = 5.5 \begin{align*}{\frac{b{x}^{3} \left ( 3\,ex+4\,d \right ) \ln \left ({x}^{n} \right ) }{12}}+{\frac{i}{8}}\pi \,be{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{8}}\pi \,be{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 \,be{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{8}}\pi \,be{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{4}}{4}}-{\frac{ben{x}^{4}}{16}}+{\frac{ae{x}^{4}}{4}}+{\frac{i}{6}}\pi \,bd{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,bd{x}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,bd{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{6}}\pi \,bd{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{3}}{3}}-{\frac{bdn{x}^{3}}{9}}+{\frac{ad{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b*x^3*(3*e*x+4*d)*ln(x^n)+1/8*I*Pi*b*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*I*Pi*b*e*x^4*csgn(I*x^n)*csgn(
I*c*x^n)*csgn(I*c)-1/8*I*Pi*b*e*x^4*csgn(I*c*x^n)^3+1/8*I*Pi*b*e*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/4*ln(c)*b*e*x
^4-1/16*b*e*n*x^4+1/4*a*e*x^4+1/6*I*Pi*b*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I*Pi*b*d*x^3*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)-1/6*I*Pi*b*d*x^3*csgn(I*c*x^n)^3+1/6*I*Pi*b*d*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/3*ln(c)*b*d*x^
3-1/9*b*d*n*x^3+1/3*a*d*x^3

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Maxima [A]  time = 1.16551, size = 77, normalized size = 1.6 \begin{align*} -\frac{1}{16} \, b e n x^{4} + \frac{1}{4} \, b e x^{4} \log \left (c x^{n}\right ) - \frac{1}{9} \, b d n x^{3} + \frac{1}{4} \, a e x^{4} + \frac{1}{3} \, b d x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.957587, size = 180, normalized size = 3.75 \begin{align*} -\frac{1}{16} \,{\left (b e n - 4 \, a e\right )} x^{4} - \frac{1}{9} \,{\left (b d n - 3 \, a d\right )} x^{3} + \frac{1}{12} \,{\left (3 \, b e x^{4} + 4 \, b d x^{3}\right )} \log \left (c\right ) + \frac{1}{12} \,{\left (3 \, b e n x^{4} + 4 \, b d n x^{3}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(b*e*n - 4*a*e)*x^4 - 1/9*(b*d*n - 3*a*d)*x^3 + 1/12*(3*b*e*x^4 + 4*b*d*x^3)*log(c) + 1/12*(3*b*e*n*x^4
+ 4*b*d*n*x^3)*log(x)

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Sympy [B]  time = 3.34002, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{3}}{3} + \frac{a e x^{4}}{4} + \frac{b d n x^{3} \log{\left (x \right )}}{3} - \frac{b d n x^{3}}{9} + \frac{b d x^{3} \log{\left (c \right )}}{3} + \frac{b e n x^{4} \log{\left (x \right )}}{4} - \frac{b e n x^{4}}{16} + \frac{b e x^{4} \log{\left (c \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x**3/3 + a*e*x**4/4 + b*d*n*x**3*log(x)/3 - b*d*n*x**3/9 + b*d*x**3*log(c)/3 + b*e*n*x**4*log(x)/4 - b*e*n
*x**4/16 + b*e*x**4*log(c)/4

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Giac [A]  time = 1.25848, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{4} \, b n x^{4} e \log \left (x\right ) - \frac{1}{16} \, b n x^{4} e + \frac{1}{4} \, b x^{4} e \log \left (c\right ) + \frac{1}{3} \, b d n x^{3} \log \left (x\right ) - \frac{1}{9} \, b d n x^{3} + \frac{1}{4} \, a x^{4} e + \frac{1}{3} \, b d x^{3} \log \left (c\right ) + \frac{1}{3} \, a d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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