3.402 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^4} \, dx\)
Optimal. Leaf size=191 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac {3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \]
[Out]
-1/9*b*d^3*n/x^3-1/9*b*e^3*n/(1-r)^2/(x^(3-3*r))-3*b*d^2*e*n*x^(-3+r)/(3-r)^2-3*b*d*e^2*n*x^(-3+2*r)/(3-2*r)^2
-1/3*d^3*(a+b*ln(c*x^n))/x^3-1/3*e^3*(a+b*ln(c*x^n))/(1-r)/(x^(3-3*r))-3*d^2*e*x^(-3+r)*(a+b*ln(c*x^n))/(3-r)-
3*d*e^2*x^(-3+2*r)*(a+b*ln(c*x^n))/(3-2*r)
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Rubi [A] time = 0.39, antiderivative size = 160, normalized size of antiderivative = 0.84,
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} \[ -\frac {1}{3} \left (\frac {9 d^2 e x^{r-3}}{3-r}+\frac {d^3}{x^3}+\frac {9 d e^2 x^{2 r-3}}{3-2 r}+\frac {e^3 x^{-3 (1-r)}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac {b d^3 n}{9 x^3}-\frac {3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]
[Out]
-(b*d^3*n)/(9*x^3) - (b*e^3*n)/(9*(1 - r)^2*x^(3*(1 - r))) - (3*b*d^2*e*n*x^(-3 + r))/(3 - r)^2 - (3*b*d*e^2*n
*x^(-3 + 2*r))/(3 - 2*r)^2 - ((d^3/x^3 + e^3/((1 - r)*x^(3*(1 - r))) + (9*d^2*e*x^(-3 + r))/(3 - r) + (9*d*e^2
*x^(-3 + 2*r))/(3 - 2*r))*(a + b*Log[c*x^n]))/3
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 \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}}{3 x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \frac {-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}}{x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (-\frac {d^3}{x^4}+\frac {9 d^2 e x^{-4+r}}{-3+r}+\frac {9 d e^2 x^{2 (-2+r)}}{-3+2 r}+\frac {e^3 x^{-4+3 r}}{-1+r}\right ) \, dx\\ &=-\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.39, size = 180, normalized size = 0.94 \[ \frac {3 a \left (-d^3+\frac {9 d^2 e x^r}{r-3}+\frac {9 d e^2 x^{2 r}}{2 r-3}+\frac {e^3 x^{3 r}}{r-1}\right )+3 b \log \left (c x^n\right ) \left (-d^3+\frac {9 d^2 e x^r}{r-3}+\frac {9 d e^2 x^{2 r}}{2 r-3}+\frac {e^3 x^{3 r}}{r-1}\right )+b n \left (-d^3-\frac {27 d^2 e x^r}{(r-3)^2}-\frac {27 d e^2 x^{2 r}}{(3-2 r)^2}-\frac {e^3 x^{3 r}}{(r-1)^2}\right )}{9 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]
[Out]
(b*n*(-d^3 - (27*d^2*e*x^r)/(-3 + r)^2 - (27*d*e^2*x^(2*r))/(3 - 2*r)^2 - (e^3*x^(3*r))/(-1 + r)^2) + 3*a*(-d^
3 + (9*d^2*e*x^r)/(-3 + r) + (9*d*e^2*x^(2*r))/(-3 + 2*r) + (e^3*x^(3*r))/(-1 + r)) + 3*b*(-d^3 + (9*d^2*e*x^r
)/(-3 + r) + (9*d*e^2*x^(2*r))/(-3 + 2*r) + (e^3*x^(3*r))/(-1 + r))*Log[c*x^n])/(9*x^3)
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fricas [B] time = 0.48, size = 980, normalized size = 5.13 \[ -\frac {4 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{6} - 44 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{5} + 81 \, b d^{3} n + 193 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{4} + 243 \, a d^{3} - 432 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{3} + 522 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{2} - 324 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r - {\left (12 \, a e^{3} r^{5} - 81 \, b e^{3} n - 4 \, {\left (b e^{3} n + 30 \, a e^{3}\right )} r^{4} - 243 \, a e^{3} + 9 \, {\left (4 \, b e^{3} n + 51 \, a e^{3}\right )} r^{3} - 9 \, {\left (13 \, b e^{3} n + 93 \, a e^{3}\right )} r^{2} + 81 \, {\left (2 \, b e^{3} n + 9 \, a e^{3}\right )} r + 3 \, {\left (4 \, b e^{3} r^{5} - 40 \, b e^{3} r^{4} + 153 \, b e^{3} r^{3} - 279 \, b e^{3} r^{2} + 243 \, b e^{3} r - 81 \, b e^{3}\right )} \log \relax (c) + 3 \, {\left (4 \, b e^{3} n r^{5} - 40 \, b e^{3} n r^{4} + 153 \, b e^{3} n r^{3} - 279 \, b e^{3} n r^{2} + 243 \, b e^{3} n r - 81 \, b e^{3} n\right )} \log \relax (x)\right )} x^{3 \, r} - 27 \, {\left (2 \, a d e^{2} r^{5} - 9 \, b d e^{2} n - {\left (b d e^{2} n + 19 \, a d e^{2}\right )} r^{4} - 27 \, a d e^{2} + 4 \, {\left (2 \, b d e^{2} n + 17 \, a d e^{2}\right )} r^{3} - 2 \, {\left (11 \, b d e^{2} n + 57 \, a d e^{2}\right )} r^{2} + 6 \, {\left (4 \, b d e^{2} n + 15 \, a d e^{2}\right )} r + {\left (2 \, b d e^{2} r^{5} - 19 \, b d e^{2} r^{4} + 68 \, b d e^{2} r^{3} - 114 \, b d e^{2} r^{2} + 90 \, b d e^{2} r - 27 \, b d e^{2}\right )} \log \relax (c) + {\left (2 \, b d e^{2} n r^{5} - 19 \, b d e^{2} n r^{4} + 68 \, b d e^{2} n r^{3} - 114 \, b d e^{2} n r^{2} + 90 \, b d e^{2} n r - 27 \, b d e^{2} n\right )} \log \relax (x)\right )} x^{2 \, r} - 27 \, {\left (4 \, a d^{2} e r^{5} - 9 \, b d^{2} e n - 4 \, {\left (b d^{2} e n + 8 \, a d^{2} e\right )} r^{4} - 27 \, a d^{2} e + {\left (20 \, b d^{2} e n + 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n + 141 \, a d^{2} e\right )} r^{2} + 3 \, {\left (10 \, b d^{2} e n + 33 \, a d^{2} e\right )} r + {\left (4 \, b d^{2} e r^{5} - 32 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} - 141 \, b d^{2} e r^{2} + 99 \, b d^{2} e r - 27 \, b d^{2} e\right )} \log \relax (c) + {\left (4 \, b d^{2} e n r^{5} - 32 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} - 141 \, b d^{2} e n r^{2} + 99 \, b d^{2} e n r - 27 \, b d^{2} e n\right )} \log \relax (x)\right )} x^{r} + 3 \, {\left (4 \, b d^{3} r^{6} - 44 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} - 432 \, b d^{3} r^{3} + 522 \, b d^{3} r^{2} - 324 \, b d^{3} r + 81 \, b d^{3}\right )} \log \relax (c) + 3 \, {\left (4 \, b d^{3} n r^{6} - 44 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} - 432 \, b d^{3} n r^{3} + 522 \, b d^{3} n r^{2} - 324 \, b d^{3} n r + 81 \, b d^{3} n\right )} \log \relax (x)}{9 \, {\left (4 \, r^{6} - 44 \, r^{5} + 193 \, r^{4} - 432 \, r^{3} + 522 \, r^{2} - 324 \, r + 81\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")
[Out]
-1/9*(4*(b*d^3*n + 3*a*d^3)*r^6 - 44*(b*d^3*n + 3*a*d^3)*r^5 + 81*b*d^3*n + 193*(b*d^3*n + 3*a*d^3)*r^4 + 243*
a*d^3 - 432*(b*d^3*n + 3*a*d^3)*r^3 + 522*(b*d^3*n + 3*a*d^3)*r^2 - 324*(b*d^3*n + 3*a*d^3)*r - (12*a*e^3*r^5
- 81*b*e^3*n - 4*(b*e^3*n + 30*a*e^3)*r^4 - 243*a*e^3 + 9*(4*b*e^3*n + 51*a*e^3)*r^3 - 9*(13*b*e^3*n + 93*a*e^
3)*r^2 + 81*(2*b*e^3*n + 9*a*e^3)*r + 3*(4*b*e^3*r^5 - 40*b*e^3*r^4 + 153*b*e^3*r^3 - 279*b*e^3*r^2 + 243*b*e^
3*r - 81*b*e^3)*log(c) + 3*(4*b*e^3*n*r^5 - 40*b*e^3*n*r^4 + 153*b*e^3*n*r^3 - 279*b*e^3*n*r^2 + 243*b*e^3*n*r
- 81*b*e^3*n)*log(x))*x^(3*r) - 27*(2*a*d*e^2*r^5 - 9*b*d*e^2*n - (b*d*e^2*n + 19*a*d*e^2)*r^4 - 27*a*d*e^2 +
4*(2*b*d*e^2*n + 17*a*d*e^2)*r^3 - 2*(11*b*d*e^2*n + 57*a*d*e^2)*r^2 + 6*(4*b*d*e^2*n + 15*a*d*e^2)*r + (2*b*
d*e^2*r^5 - 19*b*d*e^2*r^4 + 68*b*d*e^2*r^3 - 114*b*d*e^2*r^2 + 90*b*d*e^2*r - 27*b*d*e^2)*log(c) + (2*b*d*e^2
*n*r^5 - 19*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 - 114*b*d*e^2*n*r^2 + 90*b*d*e^2*n*r - 27*b*d*e^2*n)*log(x))*x^(2
*r) - 27*(4*a*d^2*e*r^5 - 9*b*d^2*e*n - 4*(b*d^2*e*n + 8*a*d^2*e)*r^4 - 27*a*d^2*e + (20*b*d^2*e*n + 97*a*d^2*
e)*r^3 - (37*b*d^2*e*n + 141*a*d^2*e)*r^2 + 3*(10*b*d^2*e*n + 33*a*d^2*e)*r + (4*b*d^2*e*r^5 - 32*b*d^2*e*r^4
+ 97*b*d^2*e*r^3 - 141*b*d^2*e*r^2 + 99*b*d^2*e*r - 27*b*d^2*e)*log(c) + (4*b*d^2*e*n*r^5 - 32*b*d^2*e*n*r^4 +
97*b*d^2*e*n*r^3 - 141*b*d^2*e*n*r^2 + 99*b*d^2*e*n*r - 27*b*d^2*e*n)*log(x))*x^r + 3*(4*b*d^3*r^6 - 44*b*d^3
*r^5 + 193*b*d^3*r^4 - 432*b*d^3*r^3 + 522*b*d^3*r^2 - 324*b*d^3*r + 81*b*d^3)*log(c) + 3*(4*b*d^3*n*r^6 - 44*
b*d^3*n*r^5 + 193*b*d^3*n*r^4 - 432*b*d^3*n*r^3 + 522*b*d^3*n*r^2 - 324*b*d^3*n*r + 81*b*d^3*n)*log(x))/((4*r^
6 - 44*r^5 + 193*r^4 - 432*r^3 + 522*r^2 - 324*r + 81)*x^3)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="giac")
[Out]
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^4, x)
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maple [C] time = 0.49, size = 4027, normalized size = 21.08 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d+e*x^r)^3*(b*ln(c*x^n)+a)/x^4,x)
[Out]
-1/3*b*(-2*e^3*r^2*(x^r)^3-9*d*e^2*r^2*(x^r)^2+9*e^3*r*(x^r)^3+2*d^3*r^3-18*d^2*e*r^2*x^r+36*d*e^2*r*(x^r)^2-9
*e^3*(x^r)^3-11*d^3*r^2+45*d^2*e*r*x^r-27*d*e^2*(x^r)^2+18*d^3*r-27*d^2*e*x^r-9*d^3)/x^3/(r-1)/(2*r-3)/(r-3)*l
n(x^n)-1/18*(3078*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*c
sgn(I*c)*(x^r)^2+3807*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-120*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)*(x^r)^3+513*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+513*I*Pi*b*d*e^2*r^4*csgn(I*c
*x^n)^2*csgn(I*c)*(x^r)^2+864*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-729*I*Pi*b*d^2*e*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*x^r-264*a*d^3*r^5+1158*a*d^3*r^4+8*b*d^3*n*r^6-88*b*d^3*n*r^5+386*b*d^3*n*r^4+486*a*e^3*
(x^r)^3+486*a*d^3+24*ln(c)*b*d^3*r^6-264*ln(c)*b*d^3*r^5+1158*ln(c)*b*d^3*r^4-2592*ln(c)*b*d^3*r^3+3132*ln(c)*
b*d^3*r^2-1944*ln(c)*b*d^3*r+24*a*d^3*r^6+162*b*d^3*n-24*a*e^3*r^5*(x^r)^3+240*a*e^3*r^4*(x^r)^3+486*ln(c)*b*e
^3*(x^r)^3+162*b*e^3*n*(x^r)^3-918*a*e^3*r^3*(x^r)^3+1674*a*e^3*r^2*(x^r)^3-1458*a*e^3*r*(x^r)^3+1458*a*d*e^2*
(x^r)^2+1458*a*d^2*e*x^r+486*b*d^3*ln(c)-864*b*d^3*n*r^3+1044*b*d^3*n*r^2-648*b*d^3*n*r-132*I*Pi*b*d^3*r^5*csg
n(I*x^n)*csgn(I*c*x^n)^2+729*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-2592*a*d^3*r^3+3132*a*d^3*r^2-1944*a*d^3*r+1
32*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3+1296*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-4860*ln(c)*b*d*e^2*r*(x^r)^2-5238*ln(c)*
b*d^2*e*r^3*x^r+7614*ln(c)*b*d^2*e*r^2*x^r-5346*ln(c)*b*d^2*e*r*x^r-3672*ln(c)*b*d*e^2*r^3*(x^r)^2+6156*ln(c)*
b*d*e^2*r^2*(x^r)^2-1620*b*d^2*e*n*r*x^r+1188*b*d*e^2*n*r^2*(x^r)^2+1998*b*d^2*e*n*r^2*x^r+1566*I*Pi*b*d^3*r^2
*csgn(I*c*x^n)^2*csgn(I*c)+12*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-729*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^
2-1566*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-243*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+3807*I*Pi*b*d^2*e*r^2*csgn(I*c*x^
n)^2*csgn(I*c)*x^r-2430*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+972*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+57
9*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+579*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)+243*I*Pi*b*e^3*csgn(
I*c*x^n)^2*csgn(I*c)*(x^r)^3+1566*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+2430*I*Pi*b*d*e^2*r*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)*(x^r)^2+2673*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+2619*I*Pi*b*d^2*e*r
^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1296*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)-243*I*Pi*b*d^3*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+243*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*
c*x^n)^2+243*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3-579*I*Pi*b*d^3*r^4*csgn(I*
c*x^n)^3+486*b*d^2*e*n*x^r-3672*a*d*e^2*r^3*(x^r)^2+6156*a*d*e^2*r^2*(x^r)^2-4860*a*d*e^2*r*(x^r)^2-5238*a*d^2
*e*r^3*x^r+7614*a*d^2*e*r^2*x^r-243*I*Pi*b*d^3*csgn(I*c*x^n)^3+54*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)*(x^r)^2+108*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-729*I*Pi*b*d*e^2*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)*(x^r)^2-54*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+864*I*Pi*b*d^2*e*r^4*csgn
(I*c*x^n)^2*csgn(I*c)*x^r-837*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-579*I*Pi*b*d^3*r^4*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-513*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2+972*I*Pi*b*d^3*r*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)+132*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+459*I*Pi*b*e^3*r^3*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*(x^r)^3-1836*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+12*I*Pi*b*e^3*r^5*csgn
(I*c*x^n)^3*(x^r)^3-132*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)+1836*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)*(x^r)^2-2619*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+729*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)*(x^r)^3-2673*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-2673*I*Pi*b*d^2*e*r*csgn(I*c*x^n
)^2*csgn(I*c)*x^r+837*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+120*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*cs
gn(I*c)*(x^r)^3-2619*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-1836*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn
(I*c)*(x^r)^2-459*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-459*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I
*c)*(x^r)^3+1296*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-72*b*e^3*n*r^3*(x^r)^3-108*a*d*e^2*r^5*(x^
r)^2+1026*a*d*e^2*r^4*(x^r)^2-216*a*d^2*e*r^5*x^r+1728*a*d^2*e*r^4*x^r+234*b*e^3*n*r^2*(x^r)^3-324*b*e^3*n*r*(
x^r)^3+486*b*d*e^2*n*(x^r)^2-864*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-513*I*Pi*b*d*e^2*r^4
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-3078*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2
+8*b*e^3*n*r^4*(x^r)^3-5346*a*d^2*e*r*x^r+1458*ln(c)*b*d^2*e*x^r+1458*ln(c)*b*d*e^2*(x^r)^2-24*ln(c)*b*e^3*r^5
*(x^r)^3+240*ln(c)*b*e^3*r^4*(x^r)^3-918*ln(c)*b*e^3*r^3*(x^r)^3+1674*ln(c)*b*e^3*r^2*(x^r)^3-1458*ln(c)*b*e^3
*r*(x^r)^3+54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+108*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+2673*I*Pi*b*d^
2*e*r*csgn(I*c*x^n)^3*x^r-108*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-108*I*Pi*b*d^2*e*r^5*csgn(I*c*x
^n)^2*csgn(I*c)*x^r+12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-729*I*Pi*b*e^3*r*csgn(I*c*x^
n)^2*csgn(I*c)*(x^r)^3-1566*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+837*I*Pi*b*e^3*r^2*csgn(I*c*x^n
)^2*csgn(I*c)*(x^r)^3-1296*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-972*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^
n)^2-972*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)-3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-3807*I*Pi*b*d^2*
e*r^2*csgn(I*c*x^n)^3*x^r+120*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-864*I*Pi*b*d^2*e*r^4*csgn(I*c
*x^n)^3*x^r+459*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3-1296*b*d*e^2*n*r*(x^r)^2-108*ln(c)*b*d*e^2*r^5*(x^r)^2+
1026*ln(c)*b*d*e^2*r^4*(x^r)^2-216*ln(c)*b*d^2*e*r^5*x^r+1728*ln(c)*b*d^2*e*r^4*x^r+54*b*d*e^2*n*r^4*(x^r)^2-4
32*b*d*e^2*n*r^3*(x^r)^2+216*b*d^2*e*n*r^4*x^r-1080*b*d^2*e*n*r^3*x^r-729*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^
n)^2*(x^r)^3+2430*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+1836*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-2430*I*
Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-243*I*Pi*
b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+729*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+729*I*P
i*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+729*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+2619*I*Pi*b*d^2*e*r^3
*csgn(I*c*x^n)^3*x^r-12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*c
sgn(I*c)*(x^r)^3-3807*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+729*I*Pi*b*d*e^2*csgn(I*c*x^n)^
2*csgn(I*c)*(x^r)^2-12*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+243*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*
x^n)^2*(x^r)^3-837*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3-729*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r-120*I*Pi*b*e^3*
r^4*csgn(I*c*x^n)^3*(x^r)^3)/(r-1)^2/x^3/(2*r-3)^2/(r-3)^2
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(r-4>0)', see `assume?` for mor
e details)Is r-4 equal to -1?
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^4,x)
[Out]
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^4, x)
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sympy [A] time = 178.89, size = 350, normalized size = 1.83 \[ - \frac {a d^{3}}{3 x^{3}} + 3 a d^{2} e \left (\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{3 r}}{3 r x^{3} - 3 x^{3}} & \text {for}\: r \neq 1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n}{9 x^{3}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\relax (x )} & \text {otherwise} \end {cases}}{r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 3 \\\frac {\log {\relax (x )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r - 3}}{r - 3} & \text {for}\: r - 4 \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - 3 b d e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\relax (x )} & \text {otherwise} \end {cases}}{2 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {3}{2} \\\frac {\log {\relax (x )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d e^{2} \left (\begin {cases} \frac {x^{2 r - 3}}{2 r - 3} & \text {for}\: 2 r - 4 \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{3} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{3 r}}{3 r x^{3} - 3 x^{3}} & \text {for}\: r \neq 1 \\\log {\relax (x )} & \text {otherwise} \end {cases}}{3 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 1 \\\frac {\log {\relax (x )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r - 3}}{3 r - 3} & \text {for}\: 3 r - 4 \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**4,x)
[Out]
-a*d**3/(3*x**3) + 3*a*d**2*e*Piecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)), (log(x), True)) + 3*a*d*e**2*Piece
wise((x**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (log(x), True)) + a*e**3*Piecewise((x**(3*r)/(3*r*x**3 - 3*x*
*3), Ne(r, 1)), (log(x), True)) - b*d**3*n/(9*x**3) - b*d**3*log(c*x**n)/(3*x**3) - 3*b*d**2*e*n*Piecewise((Pi
ecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)), (log(x), True))/(r - 3), (r > -oo) & (r < oo) & Ne(r, 3)), (log(x)
**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r - 3)/(r - 3), Ne(r - 4, -1)), (log(x), True))*log(c*x**n) - 3*b*d*
e**2*n*Piecewise((Piecewise((x**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (log(x), True))/(2*r - 3), (r > -oo) &
(r < oo) & Ne(r, 3/2)), (log(x)**2/2, True)) + 3*b*d*e**2*Piecewise((x**(2*r - 3)/(2*r - 3), Ne(2*r - 4, -1))
, (log(x), True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r)/(3*r*x**3 - 3*x**3), Ne(r, 1)), (log(x
), True))/(3*r - 3), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r - 3)/(
3*r - 3), Ne(3*r - 4, -1)), (log(x), True))*log(c*x**n)
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