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

Optimal. Leaf size=48 \[ \frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d n x^4-\frac{1}{25} b e n x^5 \]

[Out]

-(b*d*n*x^4)/16 - (b*e*n*x^5)/25 + ((5*d*x^4 + 4*e*x^5)*(a + b*Log[c*x^n]))/20

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Rubi [A]  time = 0.0516624, 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} \[ \frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d n x^4-\frac{1}{25} b e n x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n*x^4)/16 - (b*e*n*x^5)/25 + ((5*d*x^4 + 4*e*x^5)*(a + b*Log[c*x^n]))/20

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^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{20} x^3 (5 d+4 e x) \, dx\\ &=\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{20} (b n) \int x^3 (5 d+4 e x) \, dx\\ &=\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{20} (b n) \int \left (5 d x^3+4 e x^4\right ) \, dx\\ &=-\frac{1}{16} b d n x^4-\frac{1}{25} b e n x^5+\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0244789, size = 48, normalized size = 1. \[ \frac{1}{400} x^4 \left (20 a (5 d+4 e x)+20 b (5 d+4 e x) \log \left (c x^n\right )-b n (25 d+16 e x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(20*a*(5*d + 4*e*x) - b*n*(25*d + 16*e*x) + 20*b*(5*d + 4*e*x)*Log[c*x^n]))/400

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Maple [C]  time = 0.191, size = 264, normalized size = 5.5 \begin{align*}{\frac{b{x}^{4} \left ( 4\,ex+5\,d \right ) \ln \left ({x}^{n} \right ) }{20}}+{\frac{i}{10}}\pi \,be{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{10}}\pi \,be{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{10}}\pi \,be{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{10}}\pi \,be{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{5}}{5}}-{\frac{ben{x}^{5}}{25}}+{\frac{ae{x}^{5}}{5}}+{\frac{i}{8}}\pi \,bd{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{8}}\pi \,bd{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 \,bd{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{8}}\pi \,bd{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{4}}{4}}-{\frac{bdn{x}^{4}}{16}}+{\frac{ad{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*b*x^4*(4*e*x+5*d)*ln(x^n)+1/10*I*Pi*b*e*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-1/10*I*Pi*b*e*x^5*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)-1/10*I*Pi*b*e*x^5*csgn(I*c*x^n)^3+1/10*I*Pi*b*e*x^5*csgn(I*c*x^n)^2*csgn(I*c)+1/5*ln(c)*b
*e*x^5-1/25*b*e*n*x^5+1/5*a*e*x^5+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*x^n)*cs
gn(I*c*x^n)*csgn(I*c)-1/8*I*Pi*b*d*x^4*csgn(I*c*x^n)^3+1/8*I*Pi*b*d*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/4*ln(c)*b*
d*x^4-1/16*b*d*n*x^4+1/4*a*d*x^4

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Maxima [A]  time = 1.15044, size = 77, normalized size = 1.6 \begin{align*} -\frac{1}{25} \, b e n x^{5} + \frac{1}{5} \, b e x^{5} \log \left (c x^{n}\right ) - \frac{1}{16} \, b d n x^{4} + \frac{1}{5} \, a e x^{5} + \frac{1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.985886, size = 181, normalized size = 3.77 \begin{align*} -\frac{1}{25} \,{\left (b e n - 5 \, a e\right )} x^{5} - \frac{1}{16} \,{\left (b d n - 4 \, a d\right )} x^{4} + \frac{1}{20} \,{\left (4 \, b e x^{5} + 5 \, b d x^{4}\right )} \log \left (c\right ) + \frac{1}{20} \,{\left (4 \, b e n x^{5} + 5 \, b d 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+d)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/25*(b*e*n - 5*a*e)*x^5 - 1/16*(b*d*n - 4*a*d)*x^4 + 1/20*(4*b*e*x^5 + 5*b*d*x^4)*log(c) + 1/20*(4*b*e*n*x^5
 + 5*b*d*n*x^4)*log(x)

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Sympy [B]  time = 9.23149, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{4}}{4} + \frac{a e x^{5}}{5} + \frac{b d n x^{4} \log{\left (x \right )}}{4} - \frac{b d n x^{4}}{16} + \frac{b d x^{4} \log{\left (c \right )}}{4} + \frac{b e n x^{5} \log{\left (x \right )}}{5} - \frac{b e n x^{5}}{25} + \frac{b e x^{5} \log{\left (c \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x**4/4 + a*e*x**5/5 + b*d*n*x**4*log(x)/4 - b*d*n*x**4/16 + b*d*x**4*log(c)/4 + b*e*n*x**5*log(x)/5 - b*e*
n*x**5/25 + b*e*x**5*log(c)/5

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Giac [A]  time = 1.24925, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{5} \, b n x^{5} e \log \left (x\right ) - \frac{1}{25} \, b n x^{5} e + \frac{1}{5} \, b x^{5} e \log \left (c\right ) + \frac{1}{4} \, b d n x^{4} \log \left (x\right ) - \frac{1}{16} \, b d n x^{4} + \frac{1}{5} \, a x^{5} e + \frac{1}{4} \, b d x^{4} \log \left (c\right ) + \frac{1}{4} \, a d x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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