3.199 \(\int \frac{(d+e x^2)^3 (a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=130 \[ \frac{3}{2} d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{3}{4} d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} e^3 x^6 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b d^2 e n x^2-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3}{16} b d e^2 n x^4-\frac{1}{36} b e^3 n x^6 \]

[Out]

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

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Rubi [A]  time = 0.104462, antiderivative size = 100, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 14, 2301} \[ \frac{1}{12} \left (18 d^2 e x^2+12 d^3 \log (x)+9 d e^2 x^4+2 e^3 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b d^2 e n x^2-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3}{16} b d e^2 n x^4-\frac{1}{36} b e^3 n x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d^2*e*n*x^2)/4 - (3*b*d*e^2*n*x^4)/16 - (b*e^3*n*x^6)/36 - (b*d^3*n*Log[x]^2)/2 + ((18*d^2*e*x^2 + 9*d*e
^2*x^4 + 2*e^3*x^6 + 12*d^3*Log[x])*(a + b*Log[c*x^n]))/12

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 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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{1}{12} \left (18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{1}{12} e x \left (18 d^2+9 d e x^2+2 e^2 x^4\right )+\frac{d^3 \log (x)}{x}\right ) \, dx\\ &=\frac{1}{12} \left (18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^3 n\right ) \int \frac{\log (x)}{x} \, dx-\frac{1}{12} (b e n) \int x \left (18 d^2+9 d e x^2+2 e^2 x^4\right ) \, dx\\ &=-\frac{1}{2} b d^3 n \log ^2(x)+\frac{1}{12} \left (18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{12} (b e n) \int \left (18 d^2 x+9 d e x^3+2 e^2 x^5\right ) \, dx\\ &=-\frac{3}{4} b d^2 e n x^2-\frac{3}{16} b d e^2 n x^4-\frac{1}{36} b e^3 n x^6-\frac{1}{2} b d^3 n \log ^2(x)+\frac{1}{12} \left (18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0634043, size = 116, normalized size = 0.89 \[ \frac{1}{144} \left (216 d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{72 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+108 d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+24 e^3 x^6 \left (a+b \log \left (c x^n\right )\right )-108 b d^2 e n x^2-27 b d e^2 n x^4-4 b e^3 n x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-108*b*d^2*e*n*x^2 - 27*b*d*e^2*n*x^4 - 4*b*e^3*n*x^6 + 216*d^2*e*x^2*(a + b*Log[c*x^n]) + 108*d*e^2*x^4*(a +
b*Log[c*x^n]) + 24*e^3*x^6*(a + b*Log[c*x^n]) + (72*d^3*(a + b*Log[c*x^n])^2)/(b*n))/144

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Maple [C]  time = 0.222, size = 595, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16289, size = 180, normalized size = 1.38 \begin{align*} -\frac{1}{36} \, b e^{3} n x^{6} + \frac{1}{6} \, b e^{3} x^{6} \log \left (c x^{n}\right ) + \frac{1}{6} \, a e^{3} x^{6} - \frac{3}{16} \, b d e^{2} n x^{4} + \frac{3}{4} \, b d e^{2} x^{4} \log \left (c x^{n}\right ) + \frac{3}{4} \, a d e^{2} x^{4} - \frac{3}{4} \, b d^{2} e n x^{2} + \frac{3}{2} \, b d^{2} e x^{2} \log \left (c x^{n}\right ) + \frac{3}{2} \, a d^{2} e x^{2} + \frac{b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50386, size = 378, normalized size = 2.91 \begin{align*} -\frac{1}{36} \,{\left (b e^{3} n - 6 \, a e^{3}\right )} x^{6} + \frac{1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac{3}{16} \,{\left (b d e^{2} n - 4 \, a d e^{2}\right )} x^{4} - \frac{3}{4} \,{\left (b d^{2} e n - 2 \, a d^{2} e\right )} x^{2} + \frac{1}{12} \,{\left (2 \, b e^{3} x^{6} + 9 \, b d e^{2} x^{4} + 18 \, b d^{2} e x^{2}\right )} \log \left (c\right ) + \frac{1}{12} \,{\left (2 \, b e^{3} n x^{6} + 9 \, b d e^{2} n x^{4} + 18 \, b d^{2} e n x^{2} + 12 \, b d^{3} \log \left (c\right ) + 12 \, a d^{3}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(b*e^3*n - 6*a*e^3)*x^6 + 1/2*b*d^3*n*log(x)^2 - 3/16*(b*d*e^2*n - 4*a*d*e^2)*x^4 - 3/4*(b*d^2*e*n - 2*a
*d^2*e)*x^2 + 1/12*(2*b*e^3*x^6 + 9*b*d*e^2*x^4 + 18*b*d^2*e*x^2)*log(c) + 1/12*(2*b*e^3*n*x^6 + 9*b*d*e^2*n*x
^4 + 18*b*d^2*e*n*x^2 + 12*b*d^3*log(c) + 12*a*d^3)*log(x)

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Sympy [A]  time = 8.98491, size = 212, normalized size = 1.63 \begin{align*} a d^{3} \log{\left (x \right )} + \frac{3 a d^{2} e x^{2}}{2} + \frac{3 a d e^{2} x^{4}}{4} + \frac{a e^{3} x^{6}}{6} + \frac{b d^{3} n \log{\left (x \right )}^{2}}{2} + b d^{3} \log{\left (c \right )} \log{\left (x \right )} + \frac{3 b d^{2} e n x^{2} \log{\left (x \right )}}{2} - \frac{3 b d^{2} e n x^{2}}{4} + \frac{3 b d^{2} e x^{2} \log{\left (c \right )}}{2} + \frac{3 b d e^{2} n x^{4} \log{\left (x \right )}}{4} - \frac{3 b d e^{2} n x^{4}}{16} + \frac{3 b d e^{2} x^{4} \log{\left (c \right )}}{4} + \frac{b e^{3} n x^{6} \log{\left (x \right )}}{6} - \frac{b e^{3} n x^{6}}{36} + \frac{b e^{3} x^{6} \log{\left (c \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30301, size = 213, normalized size = 1.64 \begin{align*} \frac{1}{6} \, b n x^{6} e^{3} \log \left (x\right ) - \frac{1}{36} \, b n x^{6} e^{3} + \frac{1}{6} \, b x^{6} e^{3} \log \left (c\right ) + \frac{3}{4} \, b d n x^{4} e^{2} \log \left (x\right ) + \frac{1}{6} \, a x^{6} e^{3} - \frac{3}{16} \, b d n x^{4} e^{2} + \frac{3}{4} \, b d x^{4} e^{2} \log \left (c\right ) + \frac{3}{2} \, b d^{2} n x^{2} e \log \left (x\right ) + \frac{3}{4} \, a d x^{4} e^{2} - \frac{3}{4} \, b d^{2} n x^{2} e + \frac{3}{2} \, b d^{2} x^{2} e \log \left (c\right ) + \frac{1}{2} \, b d^{3} n \log \left (x\right )^{2} + \frac{3}{2} \, a d^{2} x^{2} e + b d^{3} \log \left (c\right ) \log \left (x\right ) + a d^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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