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

Optimal. Leaf size=103 \[ \frac {1}{6} \left (d^2 x^6+\frac {12 d e x^{r+6}}{r+6}+\frac {3 e^2 x^{2 (r+3)}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^2 n x^6-\frac {2 b d e n x^{r+6}}{(r+6)^2}-\frac {b e^2 n x^{2 (r+3)}}{4 (r+3)^2} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 103, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{6} \left (d^2 x^6+\frac {12 d e x^{r+6}}{r+6}+\frac {3 e^2 x^{2 (r+3)}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^2 n x^6-\frac {2 b d e n x^{r+6}}{(r+6)^2}-\frac {b e^2 n x^{2 (r+3)}}{4 (r+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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])

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 118, normalized size = 1.15 \[ \frac {1}{36} x^6 \left (6 a \left (d^2+\frac {12 d e x^r}{r+6}+\frac {3 e^2 x^{2 r}}{r+3}\right )+6 b \log \left (c x^n\right ) \left (d^2+\frac {12 d e x^r}{r+6}+\frac {3 e^2 x^{2 r}}{r+3}\right )+b n \left (-d^2-\frac {72 d e x^r}{(r+6)^2}-\frac {9 e^2 x^{2 r}}{(r+3)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^6*(b*n*(-d^2 - (72*d*e*x^r)/(6 + r)^2 - (9*e^2*x^(2*r))/(3 + r)^2) + 6*a*(d^2 + (12*d*e*x^r)/(6 + r) + (3*e
^2*x^(2*r))/(3 + r)) + 6*b*(d^2 + (12*d*e*x^r)/(6 + r) + (3*e^2*x^(2*r))/(3 + r))*Log[c*x^n]))/36

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fricas [B]  time = 0.47, size = 489, normalized size = 4.75 \[ \frac {6 \, {\left (b d^{2} r^{4} + 18 \, b d^{2} r^{3} + 117 \, b d^{2} r^{2} + 324 \, b d^{2} r + 324 \, b d^{2}\right )} x^{6} \log \relax (c) + 6 \, {\left (b d^{2} n r^{4} + 18 \, b d^{2} n r^{3} + 117 \, b d^{2} n r^{2} + 324 \, b d^{2} n r + 324 \, b d^{2} n\right )} x^{6} \log \relax (x) - {\left ({\left (b d^{2} n - 6 \, a d^{2}\right )} r^{4} + 324 \, b d^{2} n + 18 \, {\left (b d^{2} n - 6 \, a d^{2}\right )} r^{3} - 1944 \, a d^{2} + 117 \, {\left (b d^{2} n - 6 \, a d^{2}\right )} r^{2} + 324 \, {\left (b d^{2} n - 6 \, a d^{2}\right )} r\right )} x^{6} + 9 \, {\left (2 \, {\left (b e^{2} r^{3} + 15 \, b e^{2} r^{2} + 72 \, b e^{2} r + 108 \, b e^{2}\right )} x^{6} \log \relax (c) + 2 \, {\left (b e^{2} n r^{3} + 15 \, b e^{2} n r^{2} + 72 \, b e^{2} n r + 108 \, b e^{2} n\right )} x^{6} \log \relax (x) + {\left (2 \, a e^{2} r^{3} - 36 \, b e^{2} n + 216 \, a e^{2} - {\left (b e^{2} n - 30 \, a e^{2}\right )} r^{2} - 12 \, {\left (b e^{2} n - 12 \, a e^{2}\right )} r\right )} x^{6}\right )} x^{2 \, r} + 72 \, {\left ({\left (b d e r^{3} + 12 \, b d e r^{2} + 45 \, b d e r + 54 \, b d e\right )} x^{6} \log \relax (c) + {\left (b d e n r^{3} + 12 \, b d e n r^{2} + 45 \, b d e n r + 54 \, b d e n\right )} x^{6} \log \relax (x) + {\left (a d e r^{3} - 9 \, b d e n + 54 \, a d e - {\left (b d e n - 12 \, a d e\right )} r^{2} - 3 \, {\left (2 \, b d e n - 15 \, a d e\right )} r\right )} x^{6}\right )} x^{r}}{36 \, {\left (r^{4} + 18 \, r^{3} + 117 \, r^{2} + 324 \, r + 324\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(6*(b*d^2*r^4 + 18*b*d^2*r^3 + 117*b*d^2*r^2 + 324*b*d^2*r + 324*b*d^2)*x^6*log(c) + 6*(b*d^2*n*r^4 + 18*
b*d^2*n*r^3 + 117*b*d^2*n*r^2 + 324*b*d^2*n*r + 324*b*d^2*n)*x^6*log(x) - ((b*d^2*n - 6*a*d^2)*r^4 + 324*b*d^2
*n + 18*(b*d^2*n - 6*a*d^2)*r^3 - 1944*a*d^2 + 117*(b*d^2*n - 6*a*d^2)*r^2 + 324*(b*d^2*n - 6*a*d^2)*r)*x^6 +
9*(2*(b*e^2*r^3 + 15*b*e^2*r^2 + 72*b*e^2*r + 108*b*e^2)*x^6*log(c) + 2*(b*e^2*n*r^3 + 15*b*e^2*n*r^2 + 72*b*e
^2*n*r + 108*b*e^2*n)*x^6*log(x) + (2*a*e^2*r^3 - 36*b*e^2*n + 216*a*e^2 - (b*e^2*n - 30*a*e^2)*r^2 - 12*(b*e^
2*n - 12*a*e^2)*r)*x^6)*x^(2*r) + 72*((b*d*e*r^3 + 12*b*d*e*r^2 + 45*b*d*e*r + 54*b*d*e)*x^6*log(c) + (b*d*e*n
*r^3 + 12*b*d*e*n*r^2 + 45*b*d*e*n*r + 54*b*d*e*n)*x^6*log(x) + (a*d*e*r^3 - 9*b*d*e*n + 54*a*d*e - (b*d*e*n -
12*a*d*e)*r^2 - 3*(2*b*d*e*n - 15*a*d*e)*r)*x^6)*x^r)/(r^4 + 18*r^3 + 117*r^2 + 324*r + 324)

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giac [B]  time = 0.50, size = 744, normalized size = 7.22 \[ \frac {6 \, b d^{2} n r^{4} x^{6} \log \relax (x) + 72 \, b d n r^{3} x^{6} x^{r} e \log \relax (x) - b d^{2} n r^{4} x^{6} + 6 \, b d^{2} r^{4} x^{6} \log \relax (c) + 72 \, b d r^{3} x^{6} x^{r} e \log \relax (c) + 108 \, b d^{2} n r^{3} x^{6} \log \relax (x) + 18 \, b n r^{3} x^{6} x^{2 \, r} e^{2} \log \relax (x) + 864 \, b d n r^{2} x^{6} x^{r} e \log \relax (x) - 18 \, b d^{2} n r^{3} x^{6} + 6 \, a d^{2} r^{4} x^{6} - 72 \, b d n r^{2} x^{6} x^{r} e + 72 \, a d r^{3} x^{6} x^{r} e + 108 \, b d^{2} r^{3} x^{6} \log \relax (c) + 18 \, b r^{3} x^{6} x^{2 \, r} e^{2} \log \relax (c) + 864 \, b d r^{2} x^{6} x^{r} e \log \relax (c) + 702 \, b d^{2} n r^{2} x^{6} \log \relax (x) + 270 \, b n r^{2} x^{6} x^{2 \, r} e^{2} \log \relax (x) + 3240 \, b d n r x^{6} x^{r} e \log \relax (x) - 117 \, b d^{2} n r^{2} x^{6} + 108 \, a d^{2} r^{3} x^{6} - 9 \, b n r^{2} x^{6} x^{2 \, r} e^{2} + 18 \, a r^{3} x^{6} x^{2 \, r} e^{2} - 432 \, b d n r x^{6} x^{r} e + 864 \, a d r^{2} x^{6} x^{r} e + 702 \, b d^{2} r^{2} x^{6} \log \relax (c) + 270 \, b r^{2} x^{6} x^{2 \, r} e^{2} \log \relax (c) + 3240 \, b d r x^{6} x^{r} e \log \relax (c) + 1944 \, b d^{2} n r x^{6} \log \relax (x) + 1296 \, b n r x^{6} x^{2 \, r} e^{2} \log \relax (x) + 3888 \, b d n x^{6} x^{r} e \log \relax (x) - 324 \, b d^{2} n r x^{6} + 702 \, a d^{2} r^{2} x^{6} - 108 \, b n r x^{6} x^{2 \, r} e^{2} + 270 \, a r^{2} x^{6} x^{2 \, r} e^{2} - 648 \, b d n x^{6} x^{r} e + 3240 \, a d r x^{6} x^{r} e + 1944 \, b d^{2} r x^{6} \log \relax (c) + 1296 \, b r x^{6} x^{2 \, r} e^{2} \log \relax (c) + 3888 \, b d x^{6} x^{r} e \log \relax (c) + 1944 \, b d^{2} n x^{6} \log \relax (x) + 1944 \, b n x^{6} x^{2 \, r} e^{2} \log \relax (x) - 324 \, b d^{2} n x^{6} + 1944 \, a d^{2} r x^{6} - 324 \, b n x^{6} x^{2 \, r} e^{2} + 1296 \, a r x^{6} x^{2 \, r} e^{2} + 3888 \, a d x^{6} x^{r} e + 1944 \, b d^{2} x^{6} \log \relax (c) + 1944 \, b x^{6} x^{2 \, r} e^{2} \log \relax (c) + 1944 \, a d^{2} x^{6} + 1944 \, a x^{6} x^{2 \, r} e^{2}}{36 \, {\left (r^{4} + 18 \, r^{3} + 117 \, r^{2} + 324 \, r + 324\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(6*b*d^2*n*r^4*x^6*log(x) + 72*b*d*n*r^3*x^6*x^r*e*log(x) - b*d^2*n*r^4*x^6 + 6*b*d^2*r^4*x^6*log(c) + 72
*b*d*r^3*x^6*x^r*e*log(c) + 108*b*d^2*n*r^3*x^6*log(x) + 18*b*n*r^3*x^6*x^(2*r)*e^2*log(x) + 864*b*d*n*r^2*x^6
*x^r*e*log(x) - 18*b*d^2*n*r^3*x^6 + 6*a*d^2*r^4*x^6 - 72*b*d*n*r^2*x^6*x^r*e + 72*a*d*r^3*x^6*x^r*e + 108*b*d
^2*r^3*x^6*log(c) + 18*b*r^3*x^6*x^(2*r)*e^2*log(c) + 864*b*d*r^2*x^6*x^r*e*log(c) + 702*b*d^2*n*r^2*x^6*log(x
) + 270*b*n*r^2*x^6*x^(2*r)*e^2*log(x) + 3240*b*d*n*r*x^6*x^r*e*log(x) - 117*b*d^2*n*r^2*x^6 + 108*a*d^2*r^3*x
^6 - 9*b*n*r^2*x^6*x^(2*r)*e^2 + 18*a*r^3*x^6*x^(2*r)*e^2 - 432*b*d*n*r*x^6*x^r*e + 864*a*d*r^2*x^6*x^r*e + 70
2*b*d^2*r^2*x^6*log(c) + 270*b*r^2*x^6*x^(2*r)*e^2*log(c) + 3240*b*d*r*x^6*x^r*e*log(c) + 1944*b*d^2*n*r*x^6*l
og(x) + 1296*b*n*r*x^6*x^(2*r)*e^2*log(x) + 3888*b*d*n*x^6*x^r*e*log(x) - 324*b*d^2*n*r*x^6 + 702*a*d^2*r^2*x^
6 - 108*b*n*r*x^6*x^(2*r)*e^2 + 270*a*r^2*x^6*x^(2*r)*e^2 - 648*b*d*n*x^6*x^r*e + 3240*a*d*r*x^6*x^r*e + 1944*
b*d^2*r*x^6*log(c) + 1296*b*r*x^6*x^(2*r)*e^2*log(c) + 3888*b*d*x^6*x^r*e*log(c) + 1944*b*d^2*n*x^6*log(x) + 1
944*b*n*x^6*x^(2*r)*e^2*log(x) - 324*b*d^2*n*x^6 + 1944*a*d^2*r*x^6 - 324*b*n*x^6*x^(2*r)*e^2 + 1296*a*r*x^6*x
^(2*r)*e^2 + 3888*a*d*x^6*x^r*e + 1944*b*d^2*x^6*log(c) + 1944*b*x^6*x^(2*r)*e^2*log(c) + 1944*a*d^2*x^6 + 194
4*a*x^6*x^(2*r)*e^2)/(r^4 + 18*r^3 + 117*r^2 + 324*r + 324)

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maple [C]  time = 0.35, size = 1924, normalized size = 18.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b*(3*e^2*(x^r)^2*r+d^2*r^2+12*d*e*x^r*r+18*(x^r)^2*e^2+9*d^2*r+36*x^r*d*e+18*d^2)/(r+3)/(r+6)*ln(x^n)-
1/36*x^6*(36*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1944*ln(c)*b*e^2*(x^r)^2-18*a*e^2*r^3*(x^r
)^2-3888*a*d*e*x^r-270*a*e^2*r^2*(x^r)^2-1296*a*e^2*r*(x^r)^2+324*b*e^2*n*(x^r)^2-6*ln(c)*b*d^2*r^4-108*ln(c)*
b*d^2*r^3-702*ln(c)*b*d^2*r^2-1944*ln(c)*b*d^2*r-1944*a*d^2+b*d^2*n*r^4+18*b*d^2*n*r^3+324*b*d^2*n-1944*a*e^2*
(x^r)^2-1944*b*d^2*ln(c)-6*a*d^2*r^4+117*b*d^2*n*r^2+324*b*d^2*n*r-702*a*d^2*r^2-1944*a*d^2*r-108*a*d^2*r^3+3*
I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3+54*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+972*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2+972*I*
Pi*b*d^2*csgn(I*c*x^n)^3+648*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-972*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(
x^r)^2+9*b*e^2*n*r^2*(x^r)^2-72*a*d*e*r^3*x^r-864*a*d*e*r^2*x^r-3240*a*d*e*r*x^r+108*b*e^2*n*r*(x^r)^2+648*b*d
*e*n*x^r-270*ln(c)*b*e^2*r^2*(x^r)^2-1296*ln(c)*b*e^2*r*(x^r)^2-3888*ln(c)*b*d*e*x^r-18*ln(c)*b*e^2*r^3*(x^r)^
2-351*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d^2*r^4*c
sgn(I*c*x^n)^2*csgn(I*c)-972*I*Pi*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+9*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^3*(x^
r)^2-972*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-972*I*Pi*b*d^2*r*csgn(I*c*x^n)^2*csgn(I*c)+432*b*d*e*n*r*x^r
+72*b*d*e*n*r^2*x^r-864*ln(c)*b*d*e*r^2*x^r-3240*ln(c)*b*d*e*r*x^r-72*ln(c)*b*d*e*r^3*x^r-972*I*Pi*b*d^2*csgn(
I*c*x^n)^2*csgn(I*c)+1620*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-54*I*Pi*b*d^2*r^3*csgn(I*x^n)*c
sgn(I*c*x^n)^2-54*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)+1944*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r+135*I*Pi*b*e^2*
r^2*csgn(I*c*x^n)^3*(x^r)^2+972*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-351*I*Pi*b*d^2*r^2*csgn(I*x^n)*
csgn(I*c*x^n)^2+351*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-9*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^2*csgn(I
*c)*(x^r)^2-1944*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-1944*I*Pi*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+3*I*
Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1620*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r+972*I*Pi*b*e^2*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-648*I*Pi*b*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+54*I*Pi*b*d^2*r^3*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)-135*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+432*I*Pi*b*d*e*r^2*csgn(I*
c*x^n)^3*x^r+36*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-648*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-135*I*
Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-9*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+972*I*Pi
*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+9*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-36*I
*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-36*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+1944*I*Pi*b*d*e*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1620*I*Pi*b*d*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+135*I*Pi*b*e^2*r^2*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+648*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-1620*I*P
i*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-432*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-432*I*Pi*b*d*e*r^
2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+351*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3+972*I*Pi*b*d^2*r*csgn(I*c*x^n)^3-972*I*Pi*b
*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+432*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r)/(r+3)^2/(r+6)^2

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maxima [A]  time = 1.09, size = 148, normalized size = 1.44 \[ -\frac {1}{36} \, b d^{2} n x^{6} + \frac {1}{6} \, b d^{2} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{2} x^{6} + \frac {b e^{2} x^{2 \, r + 6} \log \left (c x^{n}\right )}{2 \, {\left (r + 3\right )}} + \frac {2 \, b d e x^{r + 6} \log \left (c x^{n}\right )}{r + 6} - \frac {b e^{2} n x^{2 \, r + 6}}{4 \, {\left (r + 3\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 6}}{2 \, {\left (r + 3\right )}} - \frac {2 \, b d e n x^{r + 6}}{{\left (r + 6\right )}^{2}} + \frac {2 \, a d e x^{r + 6}}{r + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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