3.359 \(\int (f x)^{-1+m} (d+e x^m)^3 (a+b \log (c x^n))^2 \, dx\)
Optimal. Leaf size=372 \[ -\frac{3 b d^2 e n x^{m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac{b d^4 n x^{1-m} \log (x) (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{2 e m}-\frac{2 b d^3 n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{2 b d e^2 n x^{2 m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}+\frac{x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac{b e^3 n x^{3 m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}+\frac{3 b^2 d^2 e n^2 x^{m+1} (f x)^{m-1}}{4 m^3}+\frac{b^2 d^4 n^2 x^{1-m} \log ^2(x) (f x)^{m-1}}{4 e m}+\frac{2 b^2 d^3 n^2 x (f x)^{m-1}}{m^3}+\frac{2 b^2 d e^2 n^2 x^{2 m+1} (f x)^{m-1}}{9 m^3}+\frac{b^2 e^3 n^2 x^{3 m+1} (f x)^{m-1}}{32 m^3} \]
[Out]
(2*b^2*d^3*n^2*x*(f*x)^(-1 + m))/m^3 + (3*b^2*d^2*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(4*m^3) + (2*b^2*d*e^2*n^2*x
^(1 + 2*m)*(f*x)^(-1 + m))/(9*m^3) + (b^2*e^3*n^2*x^(1 + 3*m)*(f*x)^(-1 + m))/(32*m^3) + (b^2*d^4*n^2*x^(1 - m
)*(f*x)^(-1 + m)*Log[x]^2)/(4*e*m) - (2*b*d^3*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (3*b*d^2*e*n*x^(1 +
m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(2*m^2) - (2*b*d*e^2*n*x^(1 + 2*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(
3*m^2) - (b*e^3*n*x^(1 + 3*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(8*m^2) - (b*d^4*n*x^(1 - m)*(f*x)^(-1 + m)*L
og[x]*(a + b*Log[c*x^n]))/(2*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^4*(a + b*Log[c*x^n])^2)/(4*e*m)
________________________________________________________________________________________
Rubi [A] time = 0.480505, antiderivative size = 294, normalized size of antiderivative =
0.79, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules
used = {2339, 2338, 266, 43, 2334, 14, 2301} \[ -\frac{b n x^{1-m} (f x)^{m-1} \left (\frac{36 d^2 e^2 x^{2 m}}{m}+\frac{48 d^3 e x^m}{m}+12 d^4 \log (x)+\frac{16 d e^3 x^{3 m}}{m}+\frac{3 e^4 x^{4 m}}{m}\right ) \left (a+b \log \left (c x^n\right )\right )}{24 e m}+\frac{x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac{3 b^2 d^2 e n^2 x^{m+1} (f x)^{m-1}}{4 m^3}+\frac{b^2 d^4 n^2 x^{1-m} \log ^2(x) (f x)^{m-1}}{4 e m}+\frac{2 b^2 d^3 n^2 x (f x)^{m-1}}{m^3}+\frac{2 b^2 d e^2 n^2 x^{2 m+1} (f x)^{m-1}}{9 m^3}+\frac{b^2 e^3 n^2 x^{3 m+1} (f x)^{m-1}}{32 m^3} \]
Antiderivative was successfully verified.
[In]
Int[(f*x)^(-1 + m)*(d + e*x^m)^3*(a + b*Log[c*x^n])^2,x]
[Out]
(2*b^2*d^3*n^2*x*(f*x)^(-1 + m))/m^3 + (3*b^2*d^2*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(4*m^3) + (2*b^2*d*e^2*n^2*x
^(1 + 2*m)*(f*x)^(-1 + m))/(9*m^3) + (b^2*e^3*n^2*x^(1 + 3*m)*(f*x)^(-1 + m))/(32*m^3) + (b^2*d^4*n^2*x^(1 - m
)*(f*x)^(-1 + m)*Log[x]^2)/(4*e*m) - (b*n*x^(1 - m)*(f*x)^(-1 + m)*((48*d^3*e*x^m)/m + (36*d^2*e^2*x^(2*m))/m
+ (16*d*e^3*x^(3*m))/m + (3*e^4*x^(4*m))/m + 12*d^4*Log[x])*(a + b*Log[c*x^n]))/(24*e*m) + (x^(1 - m)*(f*x)^(-
1 + m)*(d + e*x^m)^4*(a + b*Log[c*x^n])^2)/(4*e*m)
Rule 2339
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :>
Dist[(f*x)^m/x^m, Int[x^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r},
x] && EqQ[m, r - 1] && IGtQ[p, 0] && !(IntegerQ[m] || GtQ[f, 0])
Rule 2338
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[(f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*r*(q + 1)), x] - Dist[(b*f^m*n*p)/(e*r*(q + 1)), Int[
((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
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 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 (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac{\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e m}\\ &=-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{48 d^3 e x^m}{m}+\frac{36 d^2 e^2 x^{2 m}}{m}+\frac{16 d e^3 x^{3 m}}{m}+\frac{3 e^4 x^{4 m}}{m}+12 d^4 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{24 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \left (\frac{e x^{-1+m} \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )}{12 m}+\frac{d^4 \log (x)}{x}\right ) \, dx}{2 e m}\\ &=-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{48 d^3 e x^m}{m}+\frac{36 d^2 e^2 x^{2 m}}{m}+\frac{16 d e^3 x^{3 m}}{m}+\frac{3 e^4 x^{4 m}}{m}+12 d^4 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{24 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right ) \, dx}{24 m^2}+\frac{\left (b^2 d^4 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{\log (x)}{x} \, dx}{2 e m}\\ &=\frac{b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{48 d^3 e x^m}{m}+\frac{36 d^2 e^2 x^{2 m}}{m}+\frac{16 d e^3 x^{3 m}}{m}+\frac{3 e^4 x^{4 m}}{m}+12 d^4 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{24 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \left (48 d^3 x^{-1+m}+36 d^2 e x^{-1+2 m}+16 d e^2 x^{-1+3 m}+3 e^3 x^{-1+4 m}\right ) \, dx}{24 m^2}\\ &=\frac{2 b^2 d^3 n^2 x (f x)^{-1+m}}{m^3}+\frac{3 b^2 d^2 e n^2 x^{1+m} (f x)^{-1+m}}{4 m^3}+\frac{2 b^2 d e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{9 m^3}+\frac{b^2 e^3 n^2 x^{1+3 m} (f x)^{-1+m}}{32 m^3}+\frac{b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{48 d^3 e x^m}{m}+\frac{36 d^2 e^2 x^{2 m}}{m}+\frac{16 d e^3 x^{3 m}}{m}+\frac{3 e^4 x^{4 m}}{m}+12 d^4 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{24 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}\\ \end{align*}
Mathematica [A] time = 0.252766, size = 285, normalized size = 0.77 \[ \frac{(f x)^m \left (72 a^2 m^2 \left (6 d^2 e x^m+4 d^3+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )+12 b m \log \left (c x^n\right ) \left (12 a m \left (6 d^2 e x^m+4 d^3+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (36 d^2 e x^m+48 d^3+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )\right )-12 a b m n \left (36 d^2 e x^m+48 d^3+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )+72 b^2 m^2 \log ^2\left (c x^n\right ) \left (6 d^2 e x^m+4 d^3+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )+b^2 n^2 \left (216 d^2 e x^m+576 d^3+64 d e^2 x^{2 m}+9 e^3 x^{3 m}\right )\right )}{288 f m^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(f*x)^(-1 + m)*(d + e*x^m)^3*(a + b*Log[c*x^n])^2,x]
[Out]
((f*x)^m*(72*a^2*m^2*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m)) - 12*a*b*m*n*(48*d^3 + 36*d^2*e*x^m
+ 16*d*e^2*x^(2*m) + 3*e^3*x^(3*m)) + b^2*n^2*(576*d^3 + 216*d^2*e*x^m + 64*d*e^2*x^(2*m) + 9*e^3*x^(3*m)) +
12*b*m*(12*a*m*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m)) - b*n*(48*d^3 + 36*d^2*e*x^m + 16*d*e^2*x
^(2*m) + 3*e^3*x^(3*m)))*Log[c*x^n] + 72*b^2*m^2*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m))*Log[c*x
^n]^2))/(288*f*m^3)
________________________________________________________________________________________
Maple [C] time = 0.493, size = 4156, normalized size = 11.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x)^(-1+m)*(d+e*x^m)^3*(a+b*ln(c*x^n))^2,x)
[Out]
1/4*b^2*(e^3*(x^m)^3+4*d*e^2*(x^m)^2+6*d^2*e*x^m+4*d^3)*x/m*ln(x^n)^2*exp(1/2*(-1+m)*(-I*Pi*csgn(I*f*x)^3+I*Pi
*csgn(I*f*x)^2*csgn(I*f)+I*Pi*csgn(I*f*x)^2*csgn(I*x)-I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)+2*ln(f)+2*ln(x)))+1
/24*b*(48*ln(c)*b*d^3*m+48*a*d*e^2*(x^m)^2*m+24*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^2*m+72*a*d^2*e*x^
m*m-16*b*d*e^2*n*(x^m)^2-36*b*d^2*e*n*x^m+12*ln(c)*b*e^3*(x^m)^3*m+48*a*d^3*m-24*I*Pi*b*d^3*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)*m+6*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^3*m-6*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^m)^3*m-
3*b*e^3*n*(x^m)^3+12*a*e^3*(x^m)^3*m-48*b*d^3*n+36*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m+6*I*Pi*b*e^3
*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^3*m-24*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^m)^2*m-24*I*Pi*b*d^3*csgn(I*c*x^n)^3*m
+48*ln(c)*b*d*e^2*(x^m)^2*m+72*ln(c)*b*d^2*e*x^m*m-36*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^m*m-24*I*Pi*b*d*e^2*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2*m-36*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^m*m+24*I*Pi*b*d
^3*m*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*Pi*b*d^3*m*csgn(I*c*x^n)^2*csgn(I*c)+24*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*
c*x^n)^2*(x^m)^2*m-6*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^3*m+36*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*
csgn(I*c)*x^m*m)*x/m^2*ln(x^n)*exp(1/2*(-1+m)*(-I*Pi*csgn(I*f*x)^3+I*Pi*csgn(I*f*x)^2*csgn(I*f)+I*Pi*csgn(I*f*
x)^2*csgn(I*x)-I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)+2*ln(f)+2*ln(x)))+1/288*(72*a^2*e^3*(x^m)^3*m^2+9*b^2*e^3*
n^2*(x^m)^3+288*ln(c)^2*b^2*d^3*m^2+288*I*ln(c)*Pi*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m^2+288*I*ln(
c)*Pi*b^2*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^2*m^2-72*I*Pi*a*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m
)^3*m^2+64*b^2*d*e^2*n^2*(x^m)^2+216*b^2*d^2*e*n^2*x^m+72*ln(c)^2*b^2*e^3*(x^m)^3*m^2-72*Pi^2*b^2*d^3*csgn(I*c
*x^n)^6*m^2+288*a^2*d*e^2*(x^m)^2*m^2+432*a^2*d^2*e*x^m*m^2-72*Pi^2*b^2*d*e^2*csgn(I*c*x^n)^6*(x^m)^2*m^2-108*
Pi^2*b^2*d^2*e*csgn(I*c*x^n)^6*x^m*m^2-18*Pi^2*b^2*e^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*(x^m)^3*m^2+36*Pi^2*b^2*e
^3*csgn(I*x^n)*csgn(I*c*x^n)^5*(x^m)^3*m^2-576*a*b*d^3*m*n-96*I*Pi*b^2*d*e^2*m*n*csgn(I*c*x^n)^2*csgn(I*c)*(x^
m)^2+432*I*Pi*a*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m^2+432*I*Pi*a*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^m*m
^2+288*a^2*d^3*m^2+576*b^2*d^3*n^2-72*I*ln(c)*Pi*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^3*m^2+18*I*
Pi*b^2*e^3*m*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^3+432*I*ln(c)*Pi*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^
2*x^m*m^2+432*I*ln(c)*Pi*b^2*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^m*m^2+72*I*ln(c)*Pi*b^2*e^3*csgn(I*x^n)*csgn(I*
c*x^n)^2*(x^m)^3*m^2+72*I*ln(c)*Pi*b^2*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^3*m^2-288*I*ln(c)*Pi*b^2*d*e^2*csgn
(I*c*x^n)^3*(x^m)^2*m^2+72*I*Pi*a*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^3*m^2+72*I*Pi*a*b*e^3*csgn(I*c*x^n)^
2*csgn(I*c)*(x^m)^3*m^2-18*I*Pi*b^2*e^3*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^3-18*I*Pi*b^2*e^3*m*n*csgn(I*c*x
^n)^2*csgn(I*c)*(x^m)^3-432*I*ln(c)*Pi*b^2*d^2*e*csgn(I*c*x^n)^3*x^m*m^2-216*I*Pi*b^2*d^2*e*m*n*csgn(I*x^n)*cs
gn(I*c*x^n)^2*x^m-216*I*Pi*b^2*d^2*e*m*n*csgn(I*c*x^n)^2*csgn(I*c)*x^m-18*Pi^2*b^2*e^3*csgn(I*x^n)^2*csgn(I*c*
x^n)^2*csgn(I*c)^2*(x^m)^3*m^2-72*Pi^2*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*(x^m)^3*m^2+36*Pi^2*b^2*e
^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*(x^m)^3*m^2-72*Pi^2*b^2*d*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4*(x^m)^2
*m^2+144*Pi^2*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^5*(x^m)^2*m^2+144*Pi^2*b^2*d*e^2*csgn(I*c*x^n)^5*csgn(I*c)*(
x^m)^2*m^2-288*I*Pi*a*b*d*e^2*csgn(I*c*x^n)^3*(x^m)^2*m^2+96*I*Pi*b^2*d*e^2*m*n*csgn(I*c*x^n)^3*(x^m)^2+288*I*
Pi*a*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m^2+288*I*Pi*a*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^2*m^2-
96*I*Pi*b^2*d*e^2*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2+576*ln(c)*a*b*d^3*m^2-576*ln(c)*b^2*d^3*m*n-18*Pi^2*
b^2*e^3*csgn(I*c*x^n)^6*(x^m)^3*m^2-288*I*ln(c)*Pi*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2*m^2+3
6*Pi^2*b^2*e^3*csgn(I*c*x^n)^5*csgn(I*c)*(x^m)^3*m^2-288*I*ln(c)*Pi*b^2*d^3*csgn(I*c*x^n)^3*m^2-288*I*Pi*a*b*d
^3*csgn(I*c*x^n)^3*m^2+288*I*Pi*b^2*d^3*m*n*csgn(I*c*x^n)^3-108*Pi^2*b^2*d^2*e*csgn(I*x^n)^2*csgn(I*c*x^n)^4*x
^m*m^2+216*Pi^2*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^5*x^m*m^2+216*Pi^2*b^2*d^2*e*csgn(I*c*x^n)^5*csgn(I*c)*x^m
*m^2-108*Pi^2*b^2*d^2*e*csgn(I*c*x^n)^4*csgn(I*c)^2*x^m*m^2+36*Pi^2*b^2*e^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn
(I*c)*(x^m)^3*m^2-192*a*b*d*e^2*m*n*(x^m)^2-432*a*b*d^2*e*m*n*x^m-192*ln(c)*b^2*d*e^2*m*n*(x^m)^2-432*ln(c)*b^
2*d^2*e*m*n*x^m-288*Pi^2*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*m^2+144*Pi^2*b^2*d^3*csgn(I*x^n)*csgn(I
*c*x^n)^3*csgn(I*c)^2*m^2-18*Pi^2*b^2*e^3*csgn(I*c*x^n)^4*csgn(I*c)^2*(x^m)^3*m^2-432*I*Pi*a*b*d^2*e*csgn(I*c*
x^n)^3*x^m*m^2+216*I*Pi*b^2*d^2*e*m*n*csgn(I*c*x^n)^3*x^m-288*I*ln(c)*Pi*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)*m^2-288*I*Pi*a*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*m^2+288*I*Pi*a*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)
*m^2-288*I*Pi*b^2*d^3*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2-288*I*Pi*b^2*d^3*m*n*csgn(I*c*x^n)^2*csgn(I*c)-432*I*Pi*
a*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^m*m^2+216*I*Pi*b^2*d^2*e*m*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)*x^m-72*I*ln(c)*Pi*b^2*e^3*csgn(I*c*x^n)^3*(x^m)^3*m^2-72*I*Pi*a*b*e^3*csgn(I*c*x^n)^3*(x^m)^3*m^2+18*I*Pi*b
^2*e^3*m*n*csgn(I*c*x^n)^3*(x^m)^3-36*a*b*e^3*m*n*(x^m)^3-72*Pi^2*b^2*d*e^2*csgn(I*c*x^n)^4*csgn(I*c)^2*(x^m)^
2*m^2+288*I*ln(c)*Pi*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2*m^2+288*I*ln(c)*Pi*b^2*d^3*csgn(I*c*x^n)^2*csgn(I*c)*
m^2+288*I*Pi*a*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2*m^2-432*I*ln(c)*Pi*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)*x^m*m^2-288*I*Pi*a*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2*m^2+96*I*Pi*b^2*d*e^2*m*n*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2+576*ln(c)*a*b*d*e^2*(x^m)^2*m^2+864*ln(c)*a*b*d^2*e*x^m*m^2+144*Pi^2*b^2*d^
3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*m^2-72*Pi^2*b^2*d^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*m^2+14
4*ln(c)*a*b*e^3*(x^m)^3*m^2+432*ln(c)^2*b^2*d^2*e*x^m*m^2-36*ln(c)*b^2*e^3*m*n*(x^m)^3+288*ln(c)^2*b^2*d*e^2*(
x^m)^2*m^2-72*Pi^2*b^2*d^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*m^2+144*Pi^2*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)^5*m^2+
144*Pi^2*b^2*d^3*csgn(I*c*x^n)^5*csgn(I*c)*m^2-72*Pi^2*b^2*d^3*csgn(I*c*x^n)^4*csgn(I*c)^2*m^2+288*I*Pi*b^2*d^
3*m*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+144*Pi^2*b^2*d*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*(x^m)^2*m
^2-72*Pi^2*b^2*d*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*(x^m)^2*m^2+144*Pi^2*b^2*d*e^2*csgn(I*x^n)*csgn
(I*c*x^n)^3*csgn(I*c)^2*(x^m)^2*m^2+216*Pi^2*b^2*d^2*e*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*x^m*m^2-108*Pi^
2*b^2*d^2*e*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*x^m*m^2-432*Pi^2*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^4*c
sgn(I*c)*x^m*m^2+216*Pi^2*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*x^m*m^2-288*Pi^2*b^2*d*e^2*csgn(I*
x^n)*csgn(I*c*x^n)^4*csgn(I*c)*(x^m)^2*m^2)*x/m^3*exp(1/2*(-1+m)*(-I*Pi*csgn(I*f*x)^3+I*Pi*csgn(I*f*x)^2*csgn(
I*f)+I*Pi*csgn(I*f*x)^2*csgn(I*x)-I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)+2*ln(f)+2*ln(x)))
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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((f*x)^(-1+m)*(d+e*x^m)^3*(a+b*log(c*x^n))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.63913, size = 1318, normalized size = 3.54 \begin{align*} \frac{9 \,{\left (8 \, b^{2} e^{3} m^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} e^{3} m^{2} \log \left (c\right )^{2} + 8 \, a^{2} e^{3} m^{2} - 4 \, a b e^{3} m n + b^{2} e^{3} n^{2} + 4 \,{\left (4 \, a b e^{3} m^{2} - b^{2} e^{3} m n\right )} \log \left (c\right ) + 4 \,{\left (4 \, b^{2} e^{3} m^{2} n \log \left (c\right ) + 4 \, a b e^{3} m^{2} n - b^{2} e^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{4 \, m} + 32 \,{\left (9 \, b^{2} d e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 9 \, b^{2} d e^{2} m^{2} \log \left (c\right )^{2} + 9 \, a^{2} d e^{2} m^{2} - 6 \, a b d e^{2} m n + 2 \, b^{2} d e^{2} n^{2} + 6 \,{\left (3 \, a b d e^{2} m^{2} - b^{2} d e^{2} m n\right )} \log \left (c\right ) + 6 \,{\left (3 \, b^{2} d e^{2} m^{2} n \log \left (c\right ) + 3 \, a b d e^{2} m^{2} n - b^{2} d e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 216 \,{\left (2 \, b^{2} d^{2} e m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d^{2} e m^{2} \log \left (c\right )^{2} + 2 \, a^{2} d^{2} e m^{2} - 2 \, a b d^{2} e m n + b^{2} d^{2} e n^{2} + 2 \,{\left (2 \, a b d^{2} e m^{2} - b^{2} d^{2} e m n\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} d^{2} e m^{2} n \log \left (c\right ) + 2 \, a b d^{2} e m^{2} n - b^{2} d^{2} e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 288 \,{\left (b^{2} d^{3} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{3} m^{2} \log \left (c\right )^{2} + a^{2} d^{3} m^{2} - 2 \, a b d^{3} m n + 2 \, b^{2} d^{3} n^{2} + 2 \,{\left (a b d^{3} m^{2} - b^{2} d^{3} m n\right )} \log \left (c\right ) + 2 \,{\left (b^{2} d^{3} m^{2} n \log \left (c\right ) + a b d^{3} m^{2} n - b^{2} d^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m}}{288 \, m^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^(-1+m)*(d+e*x^m)^3*(a+b*log(c*x^n))^2,x, algorithm="fricas")
[Out]
1/288*(9*(8*b^2*e^3*m^2*n^2*log(x)^2 + 8*b^2*e^3*m^2*log(c)^2 + 8*a^2*e^3*m^2 - 4*a*b*e^3*m*n + b^2*e^3*n^2 +
4*(4*a*b*e^3*m^2 - b^2*e^3*m*n)*log(c) + 4*(4*b^2*e^3*m^2*n*log(c) + 4*a*b*e^3*m^2*n - b^2*e^3*m*n^2)*log(x))*
f^(m - 1)*x^(4*m) + 32*(9*b^2*d*e^2*m^2*n^2*log(x)^2 + 9*b^2*d*e^2*m^2*log(c)^2 + 9*a^2*d*e^2*m^2 - 6*a*b*d*e^
2*m*n + 2*b^2*d*e^2*n^2 + 6*(3*a*b*d*e^2*m^2 - b^2*d*e^2*m*n)*log(c) + 6*(3*b^2*d*e^2*m^2*n*log(c) + 3*a*b*d*e
^2*m^2*n - b^2*d*e^2*m*n^2)*log(x))*f^(m - 1)*x^(3*m) + 216*(2*b^2*d^2*e*m^2*n^2*log(x)^2 + 2*b^2*d^2*e*m^2*lo
g(c)^2 + 2*a^2*d^2*e*m^2 - 2*a*b*d^2*e*m*n + b^2*d^2*e*n^2 + 2*(2*a*b*d^2*e*m^2 - b^2*d^2*e*m*n)*log(c) + 2*(2
*b^2*d^2*e*m^2*n*log(c) + 2*a*b*d^2*e*m^2*n - b^2*d^2*e*m*n^2)*log(x))*f^(m - 1)*x^(2*m) + 288*(b^2*d^3*m^2*n^
2*log(x)^2 + b^2*d^3*m^2*log(c)^2 + a^2*d^3*m^2 - 2*a*b*d^3*m*n + 2*b^2*d^3*n^2 + 2*(a*b*d^3*m^2 - b^2*d^3*m*n
)*log(c) + 2*(b^2*d^3*m^2*n*log(c) + a*b*d^3*m^2*n - b^2*d^3*m*n^2)*log(x))*f^(m - 1)*x^m)/m^3
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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((f*x)**(-1+m)*(d+e*x**m)**3*(a+b*ln(c*x**n))**2,x)
[Out]
Timed out
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Giac [B] time = 1.53804, size = 1330, normalized size = 3.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^(-1+m)*(d+e*x^m)^3*(a+b*log(c*x^n))^2,x, algorithm="giac")
[Out]
b^2*d^3*f^m*n^2*x^m*log(x)^2/(f*m) + 3/2*b^2*d^2*f^m*n^2*x^(2*m)*e*log(x)^2/(f*m) + 2*b^2*d^3*f^m*n*x^m*log(c)
*log(x)/(f*m) + 3*b^2*d^2*f^m*n*x^(2*m)*e*log(c)*log(x)/(f*m) + b^2*d*f^m*n^2*x^(3*m)*e^2*log(x)^2/(f*m) + b^2
*d^3*f^m*x^m*log(c)^2/(f*m) + 3/2*b^2*d^2*f^m*x^(2*m)*e*log(c)^2/(f*m) + 2*a*b*d^3*f^m*n*x^m*log(x)/(f*m) - 2*
b^2*d^3*f^m*n^2*x^m*log(x)/(f*m^2) + 3*a*b*d^2*f^m*n*x^(2*m)*e*log(x)/(f*m) - 3/2*b^2*d^2*f^m*n^2*x^(2*m)*e*lo
g(x)/(f*m^2) + 2*b^2*d*f^m*n*x^(3*m)*e^2*log(c)*log(x)/(f*m) + 1/4*b^2*f^m*n^2*x^(4*m)*e^3*log(x)^2/(f*m) + 2*
a*b*d^3*f^m*x^m*log(c)/(f*m) - 2*b^2*d^3*f^m*n*x^m*log(c)/(f*m^2) + 3*a*b*d^2*f^m*x^(2*m)*e*log(c)/(f*m) - 3/2
*b^2*d^2*f^m*n*x^(2*m)*e*log(c)/(f*m^2) + b^2*d*f^m*x^(3*m)*e^2*log(c)^2/(f*m) + 2*a*b*d*f^m*n*x^(3*m)*e^2*log
(x)/(f*m) - 2/3*b^2*d*f^m*n^2*x^(3*m)*e^2*log(x)/(f*m^2) + 1/2*b^2*f^m*n*x^(4*m)*e^3*log(c)*log(x)/(f*m) + a^2
*d^3*f^m*x^m/(f*m) - 2*a*b*d^3*f^m*n*x^m/(f*m^2) + 2*b^2*d^3*f^m*n^2*x^m/(f*m^3) + 3/2*a^2*d^2*f^m*x^(2*m)*e/(
f*m) - 3/2*a*b*d^2*f^m*n*x^(2*m)*e/(f*m^2) + 3/4*b^2*d^2*f^m*n^2*x^(2*m)*e/(f*m^3) + 2*a*b*d*f^m*x^(3*m)*e^2*l
og(c)/(f*m) - 2/3*b^2*d*f^m*n*x^(3*m)*e^2*log(c)/(f*m^2) + 1/4*b^2*f^m*x^(4*m)*e^3*log(c)^2/(f*m) + 1/2*a*b*f^
m*n*x^(4*m)*e^3*log(x)/(f*m) - 1/8*b^2*f^m*n^2*x^(4*m)*e^3*log(x)/(f*m^2) + a^2*d*f^m*x^(3*m)*e^2/(f*m) - 2/3*
a*b*d*f^m*n*x^(3*m)*e^2/(f*m^2) + 2/9*b^2*d*f^m*n^2*x^(3*m)*e^2/(f*m^3) + 1/2*a*b*f^m*x^(4*m)*e^3*log(c)/(f*m)
- 1/8*b^2*f^m*n*x^(4*m)*e^3*log(c)/(f*m^2) + 1/4*a^2*f^m*x^(4*m)*e^3/(f*m) - 1/8*a*b*f^m*n*x^(4*m)*e^3/(f*m^2
) + 1/32*b^2*f^m*n^2*x^(4*m)*e^3/(f*m^3)