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

Optimal. Leaf size=152 \[ \frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}-\frac{3 b d^2 e n x^r}{r^2}-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3 b d e^2 n x^{2 r}}{4 r^2}-\frac{b e^3 n x^{3 r}}{9 r^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.153988, antiderivative size = 124, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {266, 43, 2334, 12, 14, 2301} \[ \frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+6 d^3 \log (x)+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^r}{r^2}-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3 b d e^2 n x^{2 r}}{4 r^2}-\frac{b e^3 n x^{3 r}}{9 r^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]]

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

Mathematica [A]  time = 0.355995, size = 132, normalized size = 0.87 \[ \frac{1}{36} \left (\frac{e x^r \left (6 a r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )-b n \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )\right )}{r^2}+\frac{6 b e x^r \log \left (c x^n\right ) \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )}{r}+\frac{18 b d^3 \log ^2\left (c x^n\right )}{n}\right )+a d^3 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^3*Log[x] + ((e*x^r*(6*a*r*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2*r)) - b*n*(108*d^2 + 27*d*e*x^r + 4*e^2*x^(2*r)
)))/r^2 + (6*b*e*x^r*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2*r))*Log[c*x^n])/r + (18*b*d^3*Log[c*x^n]^2)/n)/36

________________________________________________________________________________________

Maple [C]  time = 0.077, size = 693, 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((d+e*x^r)^3*(a+b*ln(c*x^n))/x,x)

[Out]

3/2/r*ln(c)*b*d*e^2*(x^r)^2+3/r*ln(c)*b*d^2*e*x^r-3/4/r^2*b*d*e^2*n*(x^r)^2-1/6*I/r*Pi*b*e^3*csgn(I*x^n)*csgn(
I*c*x^n)*csgn(I*c)*(x^r)^3+3/4*I/r*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+1/3/r*ln(c)*b*e^3*(x^r)^3-1/
9/r^2*b*e^3*n*(x^r)^3+3/2/r*a*d*e^2*(x^r)^2+3/r*a*d^2*e*x^r+1/3/r*a*e^3*(x^r)^3+1/6*b*(2*e^3*(x^r)^3+6*d^3*ln(
x)*r+9*d*e^2*(x^r)^2+18*d^2*e*x^r)/r*ln(x^n)-3*b*d^2*e*n*x^r/r^2-3/4*I/r*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*c)*(x^r)^2+ln(x)*ln(c)*b*d^3-3/2*I/r*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1/2*I*ln(x)*Pi*
b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/4*I/r*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+3/2*I/r*Pi*b*d^
2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+3/2*I/r*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-1/2*I*ln(x)*Pi*b*d^3*csgn
(I*c*x^n)^3-3/2*I/r*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+1/6*I/r*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+1/6*I/
r*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-3/4*I/r*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+ln(x)*a*d^3+1/2*I*ln(x
)*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*ln(x)*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-1/6*I/r*Pi*b*e^3*csgn(I*
c*x^n)^3*(x^r)^3-1/2*b*d^3*n*ln(x)^2

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.39162, size = 419, normalized size = 2.76 \begin{align*} \frac{18 \, b d^{3} n r^{2} \log \left (x\right )^{2} + 4 \,{\left (3 \, b e^{3} n r \log \left (x\right ) + 3 \, b e^{3} r \log \left (c\right ) - b e^{3} n + 3 \, a e^{3} r\right )} x^{3 \, r} + 27 \,{\left (2 \, b d e^{2} n r \log \left (x\right ) + 2 \, b d e^{2} r \log \left (c\right ) - b d e^{2} n + 2 \, a d e^{2} r\right )} x^{2 \, r} + 108 \,{\left (b d^{2} e n r \log \left (x\right ) + b d^{2} e r \log \left (c\right ) - b d^{2} e n + a d^{2} e r\right )} x^{r} + 36 \,{\left (b d^{3} r^{2} \log \left (c\right ) + a d^{3} r^{2}\right )} \log \left (x\right )}{36 \, r^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(18*b*d^3*n*r^2*log(x)^2 + 4*(3*b*e^3*n*r*log(x) + 3*b*e^3*r*log(c) - b*e^3*n + 3*a*e^3*r)*x^(3*r) + 27*(
2*b*d*e^2*n*r*log(x) + 2*b*d*e^2*r*log(c) - b*d*e^2*n + 2*a*d*e^2*r)*x^(2*r) + 108*(b*d^2*e*n*r*log(x) + b*d^2
*e*r*log(c) - b*d^2*e*n + a*d^2*e*r)*x^r + 36*(b*d^3*r^2*log(c) + a*d^3*r^2)*log(x))/r^2

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [A]  time = 1.33299, size = 284, normalized size = 1.87 \begin{align*} \frac{1}{2} \, b d^{3} n \log \left (x\right )^{2} + \frac{3 \, b d^{2} n x^{r} e \log \left (x\right )}{r} + b d^{3} \log \left (c\right ) \log \left (x\right ) + \frac{3 \, b d^{2} x^{r} e \log \left (c\right )}{r} + a d^{3} \log \left (x\right ) + \frac{3 \, b d n x^{2 \, r} e^{2} \log \left (x\right )}{2 \, r} - \frac{3 \, b d^{2} n x^{r} e}{r^{2}} + \frac{3 \, a d^{2} x^{r} e}{r} + \frac{3 \, b d x^{2 \, r} e^{2} \log \left (c\right )}{2 \, r} + \frac{b n x^{3 \, r} e^{3} \log \left (x\right )}{3 \, r} - \frac{3 \, b d n x^{2 \, r} e^{2}}{4 \, r^{2}} + \frac{3 \, a d x^{2 \, r} e^{2}}{2 \, r} + \frac{b x^{3 \, r} e^{3} \log \left (c\right )}{3 \, r} - \frac{b n x^{3 \, r} e^{3}}{9 \, r^{2}} + \frac{a x^{3 \, r} e^{3}}{3 \, r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*d^3*n*log(x)^2 + 3*b*d^2*n*x^r*e*log(x)/r + b*d^3*log(c)*log(x) + 3*b*d^2*x^r*e*log(c)/r + a*d^3*log(x)
+ 3/2*b*d*n*x^(2*r)*e^2*log(x)/r - 3*b*d^2*n*x^r*e/r^2 + 3*a*d^2*x^r*e/r + 3/2*b*d*x^(2*r)*e^2*log(c)/r + 1/3*
b*n*x^(3*r)*e^3*log(x)/r - 3/4*b*d*n*x^(2*r)*e^2/r^2 + 3/2*a*d*x^(2*r)*e^2/r + 1/3*b*x^(3*r)*e^3*log(c)/r - 1/
9*b*n*x^(3*r)*e^3/r^2 + 1/3*a*x^(3*r)*e^3/r