3.196 \(\int x^5 (d+e x^2)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{120} \left (45 d^2 e x^8+20 d^3 x^6+36 d e^2 x^{10}+10 e^3 x^{12}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{64} b d^2 e n x^8-\frac{1}{36} b d^3 n x^6-\frac{3}{100} b d e^2 n x^{10}-\frac{1}{144} b e^3 n x^{12} \]

[Out]

-(b*d^3*n*x^6)/36 - (3*b*d^2*e*n*x^8)/64 - (3*b*d*e^2*n*x^10)/100 - (b*e^3*n*x^12)/144 + ((20*d^3*x^6 + 45*d^2
*e*x^8 + 36*d*e^2*x^10 + 10*e^3*x^12)*(a + b*Log[c*x^n]))/120

________________________________________________________________________________________

Rubi [A]  time = 0.106148, antiderivative size = 100, 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} \[ \frac{1}{120} \left (45 d^2 e x^8+20 d^3 x^6+36 d e^2 x^{10}+10 e^3 x^{12}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{64} b d^2 e n x^8-\frac{1}{36} b d^3 n x^6-\frac{3}{100} b d e^2 n x^{10}-\frac{1}{144} b e^3 n x^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^6)/36 - (3*b*d^2*e*n*x^8)/64 - (3*b*d*e^2*n*x^10)/100 - (b*e^3*n*x^12)/144 + ((20*d^3*x^6 + 45*d^2
*e*x^8 + 36*d*e^2*x^10 + 10*e^3*x^12)*(a + b*Log[c*x^n]))/120

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^5 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{120} \left (20 d^3 x^6+45 d^2 e x^8+36 d e^2 x^{10}+10 e^3 x^{12}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{120} x^5 \left (20 d^3+45 d^2 e x^2+36 d e^2 x^4+10 e^3 x^6\right ) \, dx\\ &=\frac{1}{120} \left (20 d^3 x^6+45 d^2 e x^8+36 d e^2 x^{10}+10 e^3 x^{12}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{120} (b n) \int x^5 \left (20 d^3+45 d^2 e x^2+36 d e^2 x^4+10 e^3 x^6\right ) \, dx\\ &=\frac{1}{120} \left (20 d^3 x^6+45 d^2 e x^8+36 d e^2 x^{10}+10 e^3 x^{12}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{120} (b n) \int \left (20 d^3 x^5+45 d^2 e x^7+36 d e^2 x^9+10 e^3 x^{11}\right ) \, dx\\ &=-\frac{1}{36} b d^3 n x^6-\frac{3}{64} b d^2 e n x^8-\frac{3}{100} b d e^2 n x^{10}-\frac{1}{144} b e^3 n x^{12}+\frac{1}{120} \left (20 d^3 x^6+45 d^2 e x^8+36 d e^2 x^{10}+10 e^3 x^{12}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0525838, size = 120, normalized size = 1.2 \[ \frac{x^6 \left (120 a \left (45 d^2 e x^2+20 d^3+36 d e^2 x^4+10 e^3 x^6\right )+120 b \left (45 d^2 e x^2+20 d^3+36 d e^2 x^4+10 e^3 x^6\right ) \log \left (c x^n\right )-b n \left (675 d^2 e x^2+400 d^3+432 d e^2 x^4+100 e^3 x^6\right )\right )}{14400} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

(x^6*(120*a*(20*d^3 + 45*d^2*e*x^2 + 36*d*e^2*x^4 + 10*e^3*x^6) - b*n*(400*d^3 + 675*d^2*e*x^2 + 432*d*e^2*x^4
+ 100*e^3*x^6) + 120*b*(20*d^3 + 45*d^2*e*x^2 + 36*d*e^2*x^4 + 10*e^3*x^6)*Log[c*x^n]))/14400

________________________________________________________________________________________

Maple [C]  time = 0.209, size = 602, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x^2+d)^3*(a+b*ln(c*x^n)),x)

[Out]

-3/20*I*Pi*b*d*e^2*x^10*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/8*a*d^2*e*x^8+3/10*a*d*e^2*x^10-3/16*I*Pi*b*d^2*
e*x^8*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/8*ln(c)*b*d^2*e*x^8+3/10*ln(c)*b*d*e^2*x^10+3/20*I*Pi*b*d*e^2*x^10
*csgn(I*c*x^n)^2*csgn(I*c)+3/20*I*Pi*b*d*e^2*x^10*csgn(I*x^n)*csgn(I*c*x^n)^2+3/16*I*Pi*b*d^2*e*x^8*csgn(I*x^n
)*csgn(I*c*x^n)^2-1/12*I*Pi*b*d^3*x^6*csgn(I*c*x^n)^3-1/24*I*Pi*b*e^3*x^12*csgn(I*c*x^n)^3-1/12*I*Pi*b*d^3*x^6
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/6*a*d^3*x^6-1/24*I*Pi*b*e^3*x^12*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/
120*b*x^6*(10*e^3*x^6+36*d*e^2*x^4+45*d^2*e*x^2+20*d^3)*ln(x^n)+1/12*ln(c)*b*e^3*x^12+1/6*ln(c)*b*d^3*x^6+3/16
*I*Pi*b*d^2*e*x^8*csgn(I*c*x^n)^2*csgn(I*c)+1/12*a*e^3*x^12+1/24*I*Pi*b*e^3*x^12*csgn(I*x^n)*csgn(I*c*x^n)^2+1
/24*I*Pi*b*e^3*x^12*csgn(I*c*x^n)^2*csgn(I*c)-3/64*b*d^2*e*n*x^8-3/100*b*d*e^2*n*x^10+1/12*I*Pi*b*d^3*x^6*csgn
(I*x^n)*csgn(I*c*x^n)^2+1/12*I*Pi*b*d^3*x^6*csgn(I*c*x^n)^2*csgn(I*c)-3/16*I*Pi*b*d^2*e*x^8*csgn(I*c*x^n)^3-3/
20*I*Pi*b*d*e^2*x^10*csgn(I*c*x^n)^3-1/36*b*d^3*n*x^6-1/144*b*e^3*n*x^12

________________________________________________________________________________________

Maxima [A]  time = 1.12107, size = 193, normalized size = 1.93 \begin{align*} -\frac{1}{144} \, b e^{3} n x^{12} + \frac{1}{12} \, b e^{3} x^{12} \log \left (c x^{n}\right ) + \frac{1}{12} \, a e^{3} x^{12} - \frac{3}{100} \, b d e^{2} n x^{10} + \frac{3}{10} \, b d e^{2} x^{10} \log \left (c x^{n}\right ) + \frac{3}{10} \, a d e^{2} x^{10} - \frac{3}{64} \, b d^{2} e n x^{8} + \frac{3}{8} \, b d^{2} e x^{8} \log \left (c x^{n}\right ) + \frac{3}{8} \, a d^{2} e x^{8} - \frac{1}{36} \, b d^{3} n x^{6} + \frac{1}{6} \, b d^{3} x^{6} \log \left (c x^{n}\right ) + \frac{1}{6} \, a d^{3} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/144*b*e^3*n*x^12 + 1/12*b*e^3*x^12*log(c*x^n) + 1/12*a*e^3*x^12 - 3/100*b*d*e^2*n*x^10 + 3/10*b*d*e^2*x^10*
log(c*x^n) + 3/10*a*d*e^2*x^10 - 3/64*b*d^2*e*n*x^8 + 3/8*b*d^2*e*x^8*log(c*x^n) + 3/8*a*d^2*e*x^8 - 1/36*b*d^
3*n*x^6 + 1/6*b*d^3*x^6*log(c*x^n) + 1/6*a*d^3*x^6

________________________________________________________________________________________

Fricas [A]  time = 1.2941, size = 416, normalized size = 4.16 \begin{align*} -\frac{1}{144} \,{\left (b e^{3} n - 12 \, a e^{3}\right )} x^{12} - \frac{3}{100} \,{\left (b d e^{2} n - 10 \, a d e^{2}\right )} x^{10} - \frac{3}{64} \,{\left (b d^{2} e n - 8 \, a d^{2} e\right )} x^{8} - \frac{1}{36} \,{\left (b d^{3} n - 6 \, a d^{3}\right )} x^{6} + \frac{1}{120} \,{\left (10 \, b e^{3} x^{12} + 36 \, b d e^{2} x^{10} + 45 \, b d^{2} e x^{8} + 20 \, b d^{3} x^{6}\right )} \log \left (c\right ) + \frac{1}{120} \,{\left (10 \, b e^{3} n x^{12} + 36 \, b d e^{2} n x^{10} + 45 \, b d^{2} e n x^{8} + 20 \, b d^{3} n x^{6}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/144*(b*e^3*n - 12*a*e^3)*x^12 - 3/100*(b*d*e^2*n - 10*a*d*e^2)*x^10 - 3/64*(b*d^2*e*n - 8*a*d^2*e)*x^8 - 1/
36*(b*d^3*n - 6*a*d^3)*x^6 + 1/120*(10*b*e^3*x^12 + 36*b*d*e^2*x^10 + 45*b*d^2*e*x^8 + 20*b*d^3*x^6)*log(c) +
1/120*(10*b*e^3*n*x^12 + 36*b*d*e^2*n*x^10 + 45*b*d^2*e*n*x^8 + 20*b*d^3*n*x^6)*log(x)

________________________________________________________________________________________

Sympy [B]  time = 65.0032, size = 230, normalized size = 2.3 \begin{align*} \frac{a d^{3} x^{6}}{6} + \frac{3 a d^{2} e x^{8}}{8} + \frac{3 a d e^{2} x^{10}}{10} + \frac{a e^{3} x^{12}}{12} + \frac{b d^{3} n x^{6} \log{\left (x \right )}}{6} - \frac{b d^{3} n x^{6}}{36} + \frac{b d^{3} x^{6} \log{\left (c \right )}}{6} + \frac{3 b d^{2} e n x^{8} \log{\left (x \right )}}{8} - \frac{3 b d^{2} e n x^{8}}{64} + \frac{3 b d^{2} e x^{8} \log{\left (c \right )}}{8} + \frac{3 b d e^{2} n x^{10} \log{\left (x \right )}}{10} - \frac{3 b d e^{2} n x^{10}}{100} + \frac{3 b d e^{2} x^{10} \log{\left (c \right )}}{10} + \frac{b e^{3} n x^{12} \log{\left (x \right )}}{12} - \frac{b e^{3} n x^{12}}{144} + \frac{b e^{3} x^{12} \log{\left (c \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x**2+d)**3*(a+b*ln(c*x**n)),x)

[Out]

a*d**3*x**6/6 + 3*a*d**2*e*x**8/8 + 3*a*d*e**2*x**10/10 + a*e**3*x**12/12 + b*d**3*n*x**6*log(x)/6 - b*d**3*n*
x**6/36 + b*d**3*x**6*log(c)/6 + 3*b*d**2*e*n*x**8*log(x)/8 - 3*b*d**2*e*n*x**8/64 + 3*b*d**2*e*x**8*log(c)/8
+ 3*b*d*e**2*n*x**10*log(x)/10 - 3*b*d*e**2*n*x**10/100 + 3*b*d*e**2*x**10*log(c)/10 + b*e**3*n*x**12*log(x)/1
2 - b*e**3*n*x**12/144 + b*e**3*x**12*log(c)/12

________________________________________________________________________________________

Giac [A]  time = 1.29225, size = 234, normalized size = 2.34 \begin{align*} \frac{1}{12} \, b n x^{12} e^{3} \log \left (x\right ) - \frac{1}{144} \, b n x^{12} e^{3} + \frac{1}{12} \, b x^{12} e^{3} \log \left (c\right ) + \frac{3}{10} \, b d n x^{10} e^{2} \log \left (x\right ) + \frac{1}{12} \, a x^{12} e^{3} - \frac{3}{100} \, b d n x^{10} e^{2} + \frac{3}{10} \, b d x^{10} e^{2} \log \left (c\right ) + \frac{3}{8} \, b d^{2} n x^{8} e \log \left (x\right ) + \frac{3}{10} \, a d x^{10} e^{2} - \frac{3}{64} \, b d^{2} n x^{8} e + \frac{3}{8} \, b d^{2} x^{8} e \log \left (c\right ) + \frac{3}{8} \, a d^{2} x^{8} e + \frac{1}{6} \, b d^{3} n x^{6} \log \left (x\right ) - \frac{1}{36} \, b d^{3} n x^{6} + \frac{1}{6} \, b d^{3} x^{6} \log \left (c\right ) + \frac{1}{6} \, a d^{3} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/12*b*n*x^12*e^3*log(x) - 1/144*b*n*x^12*e^3 + 1/12*b*x^12*e^3*log(c) + 3/10*b*d*n*x^10*e^2*log(x) + 1/12*a*x
^12*e^3 - 3/100*b*d*n*x^10*e^2 + 3/10*b*d*x^10*e^2*log(c) + 3/8*b*d^2*n*x^8*e*log(x) + 3/10*a*d*x^10*e^2 - 3/6
4*b*d^2*n*x^8*e + 3/8*b*d^2*x^8*e*log(c) + 3/8*a*d^2*x^8*e + 1/6*b*d^3*n*x^6*log(x) - 1/36*b*d^3*n*x^6 + 1/6*b
*d^3*x^6*log(c) + 1/6*a*d^3*x^6