3.385 \(\int x^4 (d+e x^r)^2 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=105 \[ \frac{1}{5} \left (d^2 x^5+\frac{10 d e x^{r+5}}{r+5}+\frac{5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d^2 n x^5-\frac{2 b d e n x^{r+5}}{(r+5)^2}-\frac{b e^2 n x^{2 r+5}}{(2 r+5)^2} \]

[Out]

-(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^(5 + r))/(5 + r)^2 - (b*e^2*n*x^(5 + 2*r))/(5 + 2*r)^2 + ((d^2*x^5 + (10*d*e*
x^(5 + r))/(5 + r) + (5*e^2*x^(5 + 2*r))/(5 + 2*r))*(a + b*Log[c*x^n]))/5

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Rubi [A]  time = 0.160261, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ \frac{1}{5} \left (d^2 x^5+\frac{10 d e x^{r+5}}{r+5}+\frac{5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d^2 n x^5-\frac{2 b d e n x^{r+5}}{(r+5)^2}-\frac{b e^2 n x^{2 r+5}}{(2 r+5)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^(5 + r))/(5 + r)^2 - (b*e^2*n*x^(5 + 2*r))/(5 + 2*r)^2 + ((d^2*x^5 + (10*d*e*
x^(5 + r))/(5 + r) + (5*e^2*x^(5 + 2*r))/(5 + 2*r))*(a + b*Log[c*x^n]))/5

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{5} \left (d^2 x^5+\frac{10 d e x^{5+r}}{5+r}+\frac{5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{5} x^4 \left (d^2+\frac{10 d e x^r}{5+r}+\frac{5 e^2 x^{2 r}}{5+2 r}\right ) \, dx\\ &=\frac{1}{5} \left (d^2 x^5+\frac{10 d e x^{5+r}}{5+r}+\frac{5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{5} (b n) \int x^4 \left (d^2+\frac{10 d e x^r}{5+r}+\frac{5 e^2 x^{2 r}}{5+2 r}\right ) \, dx\\ &=\frac{1}{5} \left (d^2 x^5+\frac{10 d e x^{5+r}}{5+r}+\frac{5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{5} (b n) \int \left (d^2 x^4+\frac{5 e^2 x^{2 (2+r)}}{5+2 r}+\frac{10 d e x^{4+r}}{5+r}\right ) \, dx\\ &=-\frac{1}{25} b d^2 n x^5-\frac{2 b d e n x^{5+r}}{(5+r)^2}-\frac{b e^2 n x^{5+2 r}}{(5+2 r)^2}+\frac{1}{5} \left (d^2 x^5+\frac{10 d e x^{5+r}}{5+r}+\frac{5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.255053, size = 124, normalized size = 1.18 \[ \frac{1}{25} x^5 \left (5 a \left (d^2+\frac{10 d e x^r}{r+5}+\frac{5 e^2 x^{2 r}}{2 r+5}\right )+5 b \log \left (c x^n\right ) \left (d^2+\frac{10 d e x^r}{r+5}+\frac{5 e^2 x^{2 r}}{2 r+5}\right )+b n \left (-d^2-\frac{50 d e x^r}{(r+5)^2}-\frac{25 e^2 x^{2 r}}{(2 r+5)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

(x^5*(b*n*(-d^2 - (50*d*e*x^r)/(5 + r)^2 - (25*e^2*x^(2*r))/(5 + 2*r)^2) + 5*a*(d^2 + (10*d*e*x^r)/(5 + r) + (
5*e^2*x^(2*r))/(5 + 2*r)) + 5*b*(d^2 + (10*d*e*x^r)/(5 + r) + (5*e^2*x^(2*r))/(5 + 2*r))*Log[c*x^n]))/25

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Maple [C]  time = 0.294, size = 1930, normalized size = 18.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d+e*x^r)^2*(a+b*ln(c*x^n)),x)

[Out]

1/5*b*x^5*(5*e^2*(x^r)^2*r+2*d^2*r^2+20*d*e*x^r*r+25*e^2*(x^r)^2+15*d^2*r+50*d*e*x^r+25*d^2)/(5+2*r)/(5+r)*ln(
x^n)-1/50*x^5*(8*b*d^2*n*r^4+120*b*d^2*n*r^3-6250*ln(c)*b*d^2+200*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)*x^r+650*b*d^2*n*r^2+1500*b*d^2*n*r-100*a*e^2*r^3*(x^r)^2-1250*a*e^2*r^2*(x^r)^2-6250*a*d^2-12500*a*d*e*
x^r+2000*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+6250*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)*x^r-3750*I*Pi*b*d^2*r*csgn(I*c*x^n)^2*csgn(I*c)+3125*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-20
*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+3125*I*Pi*b*d^2*csgn(I*c*x^n)^3-625*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^2
*csgn(I*c)*(x^r)^2-2500*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2500*I*Pi*b*e^2*r*csgn(I*c*x^n)^2*csg
n(I*c)*(x^r)^2-6250*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+6250*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r+3125*I*Pi
*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+300*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-5000
*a*e^2*r*(x^r)^2-3250*a*d^2*r^2-7500*a*d^2*r-40*a*d^2*r^4-600*a*d^2*r^3+3750*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)+20*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-50*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^2*csgn(I
*c)*(x^r)^2+1250*b*d^2*n+2500*b*d*e*n*x^r-100*ln(c)*b*e^2*r^3*(x^r)^2-12500*ln(c)*b*d*e*x^r-1250*ln(c)*b*e^2*r
^2*(x^r)^2-5000*ln(c)*b*e^2*r*(x^r)^2-50*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-625*I*Pi*b*e^2*r^2
*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6250*ln(c)*b*e^2*(x^r)^2+1250*b*e^2*n*(x^r)^2-3250*ln(c)*b*d^2*r^2-7500*l
n(c)*b*d^2*r-6250*a*e^2*(x^r)^2-40*ln(c)*b*d^2*r^4-600*ln(c)*b*d^2*r^3+2500*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)
^2+200*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-6250*I*Pi*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+2000*I*Pi*b*d*e*r^2*cs
gn(I*c*x^n)^3*x^r+1625*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1625*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3-
3125*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+300*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+50*b*e^2*n*r^2*(x^r)^2-400*a*d*
e*r^3*x^r-4000*a*d*e*r^2*x^r-12500*a*d*e*r*x^r+500*b*e^2*n*r*(x^r)^2-300*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x
^n)^2-300*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)-1625*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)-3750*I*Pi*b*d
^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-6250*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6250*I*Pi*b*d*e*r*csgn(I*c*
x^n)^2*csgn(I*c)*x^r-200*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+50*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)*(x^r)^2+3750*I*Pi*b*d^2*r*csgn(I*c*x^n)^3-3125*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)+20*I*Pi*b*d^
2*r^4*csgn(I*c*x^n)^3+3125*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2-2000*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*
x^r-4000*ln(c)*b*d*e*r^2*x^r-12500*ln(c)*b*d*e*r*x^r-400*ln(c)*b*d*e*r^3*x^r+2500*I*Pi*b*e^2*r*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)*(x^r)^2+2000*b*d*e*n*r*x^r+400*b*d*e*n*r^2*x^r-20*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^2*csgn(I*c
)+50*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+625*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-3125*I*Pi*b*e^2*csgn(I*
x^n)*csgn(I*c*x^n)^2*(x^r)^2-1625*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-3125*I*Pi*b*e^2*csgn(I*c*x^n)^2*c
sgn(I*c)*(x^r)^2+6250*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r-200*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+625*I*
Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-2000*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+625
0*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r)/(5+2*r)^2/(5+r)^2

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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(x^4*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.38302, size = 1202, normalized size = 11.45 \begin{align*} \frac{5 \,{\left (4 \, b d^{2} r^{4} + 60 \, b d^{2} r^{3} + 325 \, b d^{2} r^{2} + 750 \, b d^{2} r + 625 \, b d^{2}\right )} x^{5} \log \left (c\right ) + 5 \,{\left (4 \, b d^{2} n r^{4} + 60 \, b d^{2} n r^{3} + 325 \, b d^{2} n r^{2} + 750 \, b d^{2} n r + 625 \, b d^{2} n\right )} x^{5} \log \left (x\right ) -{\left (4 \,{\left (b d^{2} n - 5 \, a d^{2}\right )} r^{4} + 625 \, b d^{2} n + 60 \,{\left (b d^{2} n - 5 \, a d^{2}\right )} r^{3} - 3125 \, a d^{2} + 325 \,{\left (b d^{2} n - 5 \, a d^{2}\right )} r^{2} + 750 \,{\left (b d^{2} n - 5 \, a d^{2}\right )} r\right )} x^{5} + 25 \,{\left ({\left (2 \, b e^{2} r^{3} + 25 \, b e^{2} r^{2} + 100 \, b e^{2} r + 125 \, b e^{2}\right )} x^{5} \log \left (c\right ) +{\left (2 \, b e^{2} n r^{3} + 25 \, b e^{2} n r^{2} + 100 \, b e^{2} n r + 125 \, b e^{2} n\right )} x^{5} \log \left (x\right ) +{\left (2 \, a e^{2} r^{3} - 25 \, b e^{2} n + 125 \, a e^{2} -{\left (b e^{2} n - 25 \, a e^{2}\right )} r^{2} - 10 \,{\left (b e^{2} n - 10 \, a e^{2}\right )} r\right )} x^{5}\right )} x^{2 \, r} + 50 \,{\left ({\left (4 \, b d e r^{3} + 40 \, b d e r^{2} + 125 \, b d e r + 125 \, b d e\right )} x^{5} \log \left (c\right ) +{\left (4 \, b d e n r^{3} + 40 \, b d e n r^{2} + 125 \, b d e n r + 125 \, b d e n\right )} x^{5} \log \left (x\right ) +{\left (4 \, a d e r^{3} - 25 \, b d e n + 125 \, a d e - 4 \,{\left (b d e n - 10 \, a d e\right )} r^{2} - 5 \,{\left (4 \, b d e n - 25 \, a d e\right )} r\right )} x^{5}\right )} x^{r}}{25 \,{\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/25*(5*(4*b*d^2*r^4 + 60*b*d^2*r^3 + 325*b*d^2*r^2 + 750*b*d^2*r + 625*b*d^2)*x^5*log(c) + 5*(4*b*d^2*n*r^4 +
60*b*d^2*n*r^3 + 325*b*d^2*n*r^2 + 750*b*d^2*n*r + 625*b*d^2*n)*x^5*log(x) - (4*(b*d^2*n - 5*a*d^2)*r^4 + 625
*b*d^2*n + 60*(b*d^2*n - 5*a*d^2)*r^3 - 3125*a*d^2 + 325*(b*d^2*n - 5*a*d^2)*r^2 + 750*(b*d^2*n - 5*a*d^2)*r)*
x^5 + 25*((2*b*e^2*r^3 + 25*b*e^2*r^2 + 100*b*e^2*r + 125*b*e^2)*x^5*log(c) + (2*b*e^2*n*r^3 + 25*b*e^2*n*r^2
+ 100*b*e^2*n*r + 125*b*e^2*n)*x^5*log(x) + (2*a*e^2*r^3 - 25*b*e^2*n + 125*a*e^2 - (b*e^2*n - 25*a*e^2)*r^2 -
10*(b*e^2*n - 10*a*e^2)*r)*x^5)*x^(2*r) + 50*((4*b*d*e*r^3 + 40*b*d*e*r^2 + 125*b*d*e*r + 125*b*d*e)*x^5*log(
c) + (4*b*d*e*n*r^3 + 40*b*d*e*n*r^2 + 125*b*d*e*n*r + 125*b*d*e*n)*x^5*log(x) + (4*a*d*e*r^3 - 25*b*d*e*n + 1
25*a*d*e - 4*(b*d*e*n - 10*a*d*e)*r^2 - 5*(4*b*d*e*n - 25*a*d*e)*r)*x^5)*x^r)/(4*r^4 + 60*r^3 + 325*r^2 + 750*
r + 625)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d+e*x**r)**2*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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Giac [B]  time = 1.37321, size = 1007, normalized size = 9.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/25*(20*b*d^2*n*r^4*x^5*log(x) + 200*b*d*n*r^3*x^5*x^r*e*log(x) - 4*b*d^2*n*r^4*x^5 + 20*b*d^2*r^4*x^5*log(c)
+ 200*b*d*r^3*x^5*x^r*e*log(c) + 300*b*d^2*n*r^3*x^5*log(x) + 50*b*n*r^3*x^5*x^(2*r)*e^2*log(x) + 2000*b*d*n*
r^2*x^5*x^r*e*log(x) - 60*b*d^2*n*r^3*x^5 + 20*a*d^2*r^4*x^5 - 200*b*d*n*r^2*x^5*x^r*e + 200*a*d*r^3*x^5*x^r*e
+ 300*b*d^2*r^3*x^5*log(c) + 50*b*r^3*x^5*x^(2*r)*e^2*log(c) + 2000*b*d*r^2*x^5*x^r*e*log(c) + 1625*b*d^2*n*r
^2*x^5*log(x) + 625*b*n*r^2*x^5*x^(2*r)*e^2*log(x) + 6250*b*d*n*r*x^5*x^r*e*log(x) - 325*b*d^2*n*r^2*x^5 + 300
*a*d^2*r^3*x^5 - 25*b*n*r^2*x^5*x^(2*r)*e^2 + 50*a*r^3*x^5*x^(2*r)*e^2 - 1000*b*d*n*r*x^5*x^r*e + 2000*a*d*r^2
*x^5*x^r*e + 1625*b*d^2*r^2*x^5*log(c) + 625*b*r^2*x^5*x^(2*r)*e^2*log(c) + 6250*b*d*r*x^5*x^r*e*log(c) + 3750
*b*d^2*n*r*x^5*log(x) + 2500*b*n*r*x^5*x^(2*r)*e^2*log(x) + 6250*b*d*n*x^5*x^r*e*log(x) - 750*b*d^2*n*r*x^5 +
1625*a*d^2*r^2*x^5 - 250*b*n*r*x^5*x^(2*r)*e^2 + 625*a*r^2*x^5*x^(2*r)*e^2 - 1250*b*d*n*x^5*x^r*e + 6250*a*d*r
*x^5*x^r*e + 3750*b*d^2*r*x^5*log(c) + 2500*b*r*x^5*x^(2*r)*e^2*log(c) + 6250*b*d*x^5*x^r*e*log(c) + 3125*b*d^
2*n*x^5*log(x) + 3125*b*n*x^5*x^(2*r)*e^2*log(x) - 625*b*d^2*n*x^5 + 3750*a*d^2*r*x^5 - 625*b*n*x^5*x^(2*r)*e^
2 + 2500*a*r*x^5*x^(2*r)*e^2 + 6250*a*d*x^5*x^r*e + 3125*b*d^2*x^5*log(c) + 3125*b*x^5*x^(2*r)*e^2*log(c) + 31
25*a*d^2*x^5 + 3125*a*x^5*x^(2*r)*e^2)/(4*r^4 + 60*r^3 + 325*r^2 + 750*r + 625)