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

Optimal. Leaf size=76 \[ \frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac{b d^3 n \log (x)}{6 e}-\frac{1}{4} b d^2 n x^2-\frac{1}{8} b d e n x^4-\frac{1}{36} b e^2 n x^6 \]

[Out]

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

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Rubi [A]  time = 0.0678022, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {261, 2334, 12, 266, 43} \[ \frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac{b d^3 n \log (x)}{6 e}-\frac{1}{4} b d^2 n x^2-\frac{1}{8} b d e n x^4-\frac{1}{36} b e^2 n x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 261

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

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 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])

Rubi steps

\begin{align*} \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-(b n) \int \frac{\left (d+e x^2\right )^3}{6 e x} \, dx\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac{(b n) \int \frac{\left (d+e x^2\right )^3}{x} \, dx}{6 e}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac{(b n) \operatorname{Subst}\left (\int \frac{(d+e x)^3}{x} \, dx,x,x^2\right )}{12 e}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac{(b n) \operatorname{Subst}\left (\int \left (3 d^2 e+\frac{d^3}{x}+3 d e^2 x+e^3 x^2\right ) \, dx,x,x^2\right )}{12 e}\\ &=-\frac{1}{4} b d^2 n x^2-\frac{1}{8} b d e n x^4-\frac{1}{36} b e^2 n x^6-\frac{b d^3 n \log (x)}{6 e}+\frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}\\ \end{align*}

Mathematica [A]  time = 0.035075, size = 85, normalized size = 1.12 \[ \frac{1}{72} x^2 \left (12 a \left (3 d^2+3 d e x^2+e^2 x^4\right )+12 b \left (3 d^2+3 d e x^2+e^2 x^4\right ) \log \left (c x^n\right )-b n \left (18 d^2+9 d e x^2+2 e^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(12*a*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - b*n*(18*d^2 + 9*d*e*x^2 + 2*e^2*x^4) + 12*b*(3*d^2 + 3*d*e*x^2 + e^
2*x^4)*Log[c*x^n]))/72

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Maple [C]  time = 0.204, size = 433, normalized size = 5.7 \begin{align*}{\frac{b{x}^{2} \left ({e}^{2}{x}^{4}+3\,de{x}^{2}+3\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{6}}-{\frac{i}{4}}\pi \,b{d}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{4}}\pi \,bde{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{4}}\pi \,bde{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{6}}{6}}-{\frac{b{e}^{2}n{x}^{6}}{36}}+{\frac{a{e}^{2}{x}^{6}}{6}}+{\frac{i}{4}}\pi \,b{d}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{4}}\pi \,b{d}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bde{x}^{4}}{2}}-{\frac{bden{x}^{4}}{8}}+{\frac{ade{x}^{4}}{2}}+{\frac{i}{4}}\pi \,bde{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{4}}\pi \,b{d}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{4}}\pi \,bde{x}^{4}{\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}^{2}}{2}}-{\frac{b{d}^{2}n{x}^{2}}{4}}+{\frac{a{d}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15963, size = 135, normalized size = 1.78 \begin{align*} -\frac{1}{36} \, b e^{2} n x^{6} + \frac{1}{6} \, b e^{2} x^{6} \log \left (c x^{n}\right ) + \frac{1}{6} \, a e^{2} x^{6} - \frac{1}{8} \, b d e n x^{4} + \frac{1}{2} \, b d e x^{4} \log \left (c x^{n}\right ) + \frac{1}{2} \, a d e x^{4} - \frac{1}{4} \, b d^{2} n x^{2} + \frac{1}{2} \, b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a d^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*b*e^2*n*x^6 + 1/6*b*e^2*x^6*log(c*x^n) + 1/6*a*e^2*x^6 - 1/8*b*d*e*n*x^4 + 1/2*b*d*e*x^4*log(c*x^n) + 1/
2*a*d*e*x^4 - 1/4*b*d^2*n*x^2 + 1/2*b*d^2*x^2*log(c*x^n) + 1/2*a*d^2*x^2

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Fricas [A]  time = 1.28354, size = 274, normalized size = 3.61 \begin{align*} -\frac{1}{36} \,{\left (b e^{2} n - 6 \, a e^{2}\right )} x^{6} - \frac{1}{8} \,{\left (b d e n - 4 \, a d e\right )} x^{4} - \frac{1}{4} \,{\left (b d^{2} n - 2 \, a d^{2}\right )} x^{2} + \frac{1}{6} \,{\left (b e^{2} x^{6} + 3 \, b d e x^{4} + 3 \, b d^{2} x^{2}\right )} \log \left (c\right ) + \frac{1}{6} \,{\left (b e^{2} n x^{6} + 3 \, b d e n x^{4} + 3 \, b d^{2} n x^{2}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(b*e^2*n - 6*a*e^2)*x^6 - 1/8*(b*d*e*n - 4*a*d*e)*x^4 - 1/4*(b*d^2*n - 2*a*d^2)*x^2 + 1/6*(b*e^2*x^6 + 3
*b*d*e*x^4 + 3*b*d^2*x^2)*log(c) + 1/6*(b*e^2*n*x^6 + 3*b*d*e*n*x^4 + 3*b*d^2*n*x^2)*log(x)

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Sympy [B]  time = 5.68267, size = 151, normalized size = 1.99 \begin{align*} \frac{a d^{2} x^{2}}{2} + \frac{a d e x^{4}}{2} + \frac{a e^{2} x^{6}}{6} + \frac{b d^{2} n x^{2} \log{\left (x \right )}}{2} - \frac{b d^{2} n x^{2}}{4} + \frac{b d^{2} x^{2} \log{\left (c \right )}}{2} + \frac{b d e n x^{4} \log{\left (x \right )}}{2} - \frac{b d e n x^{4}}{8} + \frac{b d e x^{4} \log{\left (c \right )}}{2} + \frac{b e^{2} n x^{6} \log{\left (x \right )}}{6} - \frac{b e^{2} n x^{6}}{36} + \frac{b e^{2} x^{6} \log{\left (c \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*n*x**2*log(x)/2 - b*d**2*n*x**2/4 + b*d**2*x**2*log(c)/2
+ b*d*e*n*x**4*log(x)/2 - b*d*e*n*x**4/8 + b*d*e*x**4*log(c)/2 + b*e**2*n*x**6*log(x)/6 - b*e**2*n*x**6/36 +
b*e**2*x**6*log(c)/6

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Giac [A]  time = 1.86026, size = 166, normalized size = 2.18 \begin{align*} \frac{1}{6} \, b n x^{6} e^{2} \log \left (x\right ) - \frac{1}{36} \, b n x^{6} e^{2} + \frac{1}{6} \, b x^{6} e^{2} \log \left (c\right ) + \frac{1}{2} \, b d n x^{4} e \log \left (x\right ) + \frac{1}{6} \, a x^{6} e^{2} - \frac{1}{8} \, b d n x^{4} e + \frac{1}{2} \, b d x^{4} e \log \left (c\right ) + \frac{1}{2} \, a d x^{4} e + \frac{1}{2} \, b d^{2} n x^{2} \log \left (x\right ) - \frac{1}{4} \, b d^{2} n x^{2} + \frac{1}{2} \, b d^{2} x^{2} \log \left (c\right ) + \frac{1}{2} \, a d^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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