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

Optimal. Leaf size=100 \[ \frac{1}{140} \left (84 d^2 e x^5+35 d^3 x^4+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{25} b d^2 e n x^5-\frac{1}{16} b d^3 n x^4-\frac{1}{12} b d e^2 n x^6-\frac{1}{49} b e^3 n x^7 \]

[Out]

-(b*d^3*n*x^4)/16 - (3*b*d^2*e*n*x^5)/25 - (b*d*e^2*n*x^6)/12 - (b*e^3*n*x^7)/49 + ((35*d^3*x^4 + 84*d^2*e*x^5
+ 70*d*e^2*x^6 + 20*e^3*x^7)*(a + b*Log[c*x^n]))/140

________________________________________________________________________________________

Rubi [A]  time = 0.105569, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {43, 2334, 12, 14} \[ \frac{1}{140} \left (84 d^2 e x^5+35 d^3 x^4+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{25} b d^2 e n x^5-\frac{1}{16} b d^3 n x^4-\frac{1}{12} b d e^2 n x^6-\frac{1}{49} b e^3 n x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^3*n*x^4)/16 - (3*b*d^2*e*n*x^5)/25 - (b*d*e^2*n*x^6)/12 - (b*e^3*n*x^7)/49 + ((35*d^3*x^4 + 84*d^2*e*x^5
+ 70*d*e^2*x^6 + 20*e^3*x^7)*(a + b*Log[c*x^n]))/140

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^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{140} x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right ) \, dx\\ &=\frac{1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{140} (b n) \int x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right ) \, dx\\ &=\frac{1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{140} (b n) \int \left (35 d^3 x^3+84 d^2 e x^4+70 d e^2 x^5+20 e^3 x^6\right ) \, dx\\ &=-\frac{1}{16} b d^3 n x^4-\frac{3}{25} b d^2 e n x^5-\frac{1}{12} b d e^2 n x^6-\frac{1}{49} b e^3 n x^7+\frac{1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0582264, size = 133, normalized size = 1.33 \[ \frac{3}{5} d^2 e x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} d^3 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} d e^2 x^6 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{25} b d^2 e n x^5-\frac{1}{16} b d^3 n x^4-\frac{1}{12} b d e^2 n x^6-\frac{1}{49} b e^3 n x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^3*n*x^4)/16 - (3*b*d^2*e*n*x^5)/25 - (b*d*e^2*n*x^6)/12 - (b*e^3*n*x^7)/49 + (d^3*x^4*(a + b*Log[c*x^n])
)/4 + (3*d^2*e*x^5*(a + b*Log[c*x^n]))/5 + (d*e^2*x^6*(a + b*Log[c*x^n]))/2 + (e^3*x^7*(a + b*Log[c*x^n]))/7

________________________________________________________________________________________

Maple [C]  time = 0.217, size = 600, 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^3*(e*x+d)^3*(a+b*ln(c*x^n)),x)

[Out]

3/5*a*d^2*e*x^5+1/2*a*d*e^2*x^6-1/14*I*Pi*b*e^3*x^7*csgn(I*c*x^n)^3-1/8*I*Pi*b*d^3*x^4*csgn(I*c*x^n)^3+1/140*b
*x^4*(20*e^3*x^3+70*d*e^2*x^2+84*d^2*e*x+35*d^3)*ln(x^n)+1/2*ln(c)*b*d*e^2*x^6+3/5*ln(c)*b*d^2*e*x^5+3/10*I*Pi
*b*d^2*e*x^5*csgn(I*c*x^n)^2*csgn(I*c)-1/8*I*Pi*b*d^3*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/10*I*Pi*b*d^2*
e*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-1/14*I*Pi*b*e^3*x^7*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/4*I*Pi*b*d*e^2*x^6
*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I*Pi*b*d*e^2*x^6*csgn(I*c*x^n)^2*csgn(I*c)-3/10*I*Pi*b*d^2*e*x^5*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)+1/7*a*e^3*x^7+1/4*ln(c)*b*d^3*x^4+1/7*ln(c)*b*e^3*x^7-1/4*I*Pi*b*d*e^2*x^6*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)+1/4*a*d^3*x^4+1/8*I*Pi*b*d^3*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/14*I*Pi*b*e^3*x^7*csgn(I
*x^n)*csgn(I*c*x^n)^2+1/14*I*Pi*b*e^3*x^7*csgn(I*c*x^n)^2*csgn(I*c)+1/8*I*Pi*b*d^3*x^4*csgn(I*x^n)*csgn(I*c*x^
n)^2-3/10*I*Pi*b*d^2*e*x^5*csgn(I*c*x^n)^3-1/4*I*Pi*b*d*e^2*x^6*csgn(I*c*x^n)^3-1/16*b*d^3*n*x^4-1/49*b*e^3*n*
x^7-3/25*b*d^2*e*n*x^5-1/12*b*d*e^2*n*x^6

________________________________________________________________________________________

Maxima [A]  time = 1.09337, size = 193, normalized size = 1.93 \begin{align*} -\frac{1}{49} \, b e^{3} n x^{7} + \frac{1}{7} \, b e^{3} x^{7} \log \left (c x^{n}\right ) - \frac{1}{12} \, b d e^{2} n x^{6} + \frac{1}{7} \, a e^{3} x^{7} + \frac{1}{2} \, b d e^{2} x^{6} \log \left (c x^{n}\right ) - \frac{3}{25} \, b d^{2} e n x^{5} + \frac{1}{2} \, a d e^{2} x^{6} + \frac{3}{5} \, b d^{2} e x^{5} \log \left (c x^{n}\right ) - \frac{1}{16} \, b d^{3} n x^{4} + \frac{3}{5} \, a d^{2} e x^{5} + \frac{1}{4} \, b d^{3} x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49*b*e^3*n*x^7 + 1/7*b*e^3*x^7*log(c*x^n) - 1/12*b*d*e^2*n*x^6 + 1/7*a*e^3*x^7 + 1/2*b*d*e^2*x^6*log(c*x^n)
- 3/25*b*d^2*e*n*x^5 + 1/2*a*d*e^2*x^6 + 3/5*b*d^2*e*x^5*log(c*x^n) - 1/16*b*d^3*n*x^4 + 3/5*a*d^2*e*x^5 + 1/
4*b*d^3*x^4*log(c*x^n) + 1/4*a*d^3*x^4

________________________________________________________________________________________

Fricas [A]  time = 0.987537, size = 402, normalized size = 4.02 \begin{align*} -\frac{1}{49} \,{\left (b e^{3} n - 7 \, a e^{3}\right )} x^{7} - \frac{1}{12} \,{\left (b d e^{2} n - 6 \, a d e^{2}\right )} x^{6} - \frac{3}{25} \,{\left (b d^{2} e n - 5 \, a d^{2} e\right )} x^{5} - \frac{1}{16} \,{\left (b d^{3} n - 4 \, a d^{3}\right )} x^{4} + \frac{1}{140} \,{\left (20 \, b e^{3} x^{7} + 70 \, b d e^{2} x^{6} + 84 \, b d^{2} e x^{5} + 35 \, b d^{3} x^{4}\right )} \log \left (c\right ) + \frac{1}{140} \,{\left (20 \, b e^{3} n x^{7} + 70 \, b d e^{2} n x^{6} + 84 \, b d^{2} e n x^{5} + 35 \, b d^{3} 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)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/49*(b*e^3*n - 7*a*e^3)*x^7 - 1/12*(b*d*e^2*n - 6*a*d*e^2)*x^6 - 3/25*(b*d^2*e*n - 5*a*d^2*e)*x^5 - 1/16*(b*
d^3*n - 4*a*d^3)*x^4 + 1/140*(20*b*e^3*x^7 + 70*b*d*e^2*x^6 + 84*b*d^2*e*x^5 + 35*b*d^3*x^4)*log(c) + 1/140*(2
0*b*e^3*n*x^7 + 70*b*d*e^2*n*x^6 + 84*b*d^2*e*n*x^5 + 35*b*d^3*n*x^4)*log(x)

________________________________________________________________________________________

Sympy [B]  time = 43.6895, size = 223, normalized size = 2.23 \begin{align*} \frac{a d^{3} x^{4}}{4} + \frac{3 a d^{2} e x^{5}}{5} + \frac{a d e^{2} x^{6}}{2} + \frac{a e^{3} x^{7}}{7} + \frac{b d^{3} n x^{4} \log{\left (x \right )}}{4} - \frac{b d^{3} n x^{4}}{16} + \frac{b d^{3} x^{4} \log{\left (c \right )}}{4} + \frac{3 b d^{2} e n x^{5} \log{\left (x \right )}}{5} - \frac{3 b d^{2} e n x^{5}}{25} + \frac{3 b d^{2} e x^{5} \log{\left (c \right )}}{5} + \frac{b d e^{2} n x^{6} \log{\left (x \right )}}{2} - \frac{b d e^{2} n x^{6}}{12} + \frac{b d e^{2} x^{6} \log{\left (c \right )}}{2} + \frac{b e^{3} n x^{7} \log{\left (x \right )}}{7} - \frac{b e^{3} n x^{7}}{49} + \frac{b e^{3} x^{7} \log{\left (c \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**3*x**4/4 + 3*a*d**2*e*x**5/5 + a*d*e**2*x**6/2 + a*e**3*x**7/7 + b*d**3*n*x**4*log(x)/4 - b*d**3*n*x**4/1
6 + b*d**3*x**4*log(c)/4 + 3*b*d**2*e*n*x**5*log(x)/5 - 3*b*d**2*e*n*x**5/25 + 3*b*d**2*e*x**5*log(c)/5 + b*d*
e**2*n*x**6*log(x)/2 - b*d*e**2*n*x**6/12 + b*d*e**2*x**6*log(c)/2 + b*e**3*n*x**7*log(x)/7 - b*e**3*n*x**7/49
+ b*e**3*x**7*log(c)/7

________________________________________________________________________________________

Giac [A]  time = 1.33537, size = 234, normalized size = 2.34 \begin{align*} \frac{1}{7} \, b n x^{7} e^{3} \log \left (x\right ) + \frac{1}{2} \, b d n x^{6} e^{2} \log \left (x\right ) + \frac{3}{5} \, b d^{2} n x^{5} e \log \left (x\right ) - \frac{1}{49} \, b n x^{7} e^{3} - \frac{1}{12} \, b d n x^{6} e^{2} - \frac{3}{25} \, b d^{2} n x^{5} e + \frac{1}{7} \, b x^{7} e^{3} \log \left (c\right ) + \frac{1}{2} \, b d x^{6} e^{2} \log \left (c\right ) + \frac{3}{5} \, b d^{2} x^{5} e \log \left (c\right ) + \frac{1}{4} \, b d^{3} n x^{4} \log \left (x\right ) - \frac{1}{16} \, b d^{3} n x^{4} + \frac{1}{7} \, a x^{7} e^{3} + \frac{1}{2} \, a d x^{6} e^{2} + \frac{3}{5} \, a d^{2} x^{5} e + \frac{1}{4} \, b d^{3} x^{4} \log \left (c\right ) + \frac{1}{4} \, a d^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b*n*x^7*e^3*log(x) + 1/2*b*d*n*x^6*e^2*log(x) + 3/5*b*d^2*n*x^5*e*log(x) - 1/49*b*n*x^7*e^3 - 1/12*b*d*n*x
^6*e^2 - 3/25*b*d^2*n*x^5*e + 1/7*b*x^7*e^3*log(c) + 1/2*b*d*x^6*e^2*log(c) + 3/5*b*d^2*x^5*e*log(c) + 1/4*b*d
^3*n*x^4*log(x) - 1/16*b*d^3*n*x^4 + 1/7*a*x^7*e^3 + 1/2*a*d*x^6*e^2 + 3/5*a*d^2*x^5*e + 1/4*b*d^3*x^4*log(c)
+ 1/4*a*d^3*x^4