3.360 \(\int (f x)^{-1+m} (d+e x^m)^2 (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=298 \[ -\frac{2 b d^3 n x^{1-m} \log (x) (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 e m}-\frac{2 b d^2 n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{b d e n x^{m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{m^2}+\frac{x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac{2 b e^2 n x^{2 m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{9 m^2}+\frac{b^2 d^3 n^2 x^{1-m} \log ^2(x) (f x)^{m-1}}{3 e m}+\frac{2 b^2 d^2 n^2 x (f x)^{m-1}}{m^3}+\frac{b^2 d e n^2 x^{m+1} (f x)^{m-1}}{2 m^3}+\frac{2 b^2 e^2 n^2 x^{2 m+1} (f x)^{m-1}}{27 m^3} \]

[Out]

(2*b^2*d^2*n^2*x*(f*x)^(-1 + m))/m^3 + (b^2*d*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(2*m^3) + (2*b^2*e^2*n^2*x^(1 +
2*m)*(f*x)^(-1 + m))/(27*m^3) + (b^2*d^3*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[x]^2)/(3*e*m) - (2*b*d^2*n*x*(f*x)^(
-1 + m)*(a + b*Log[c*x^n]))/m^2 - (b*d*e*n*x^(1 + m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (2*b*e^2*n*x^(1
+ 2*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(9*m^2) - (2*b*d^3*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x]*(a + b*Log[c*x^
n]))/(3*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^3*(a + b*Log[c*x^n])^2)/(3*e*m)

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Rubi [A]  time = 0.439831, antiderivative size = 245, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2339, 2338, 266, 43, 2334, 12, 14, 2301} \[ -\frac{b n x^{1-m} (f x)^{m-1} \left (\frac{18 d^2 e x^m}{m}+6 d^3 \log (x)+\frac{9 d e^2 x^{2 m}}{m}+\frac{2 e^3 x^{3 m}}{m}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac{x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac{b^2 d^3 n^2 x^{1-m} \log ^2(x) (f x)^{m-1}}{3 e m}+\frac{2 b^2 d^2 n^2 x (f x)^{m-1}}{m^3}+\frac{b^2 d e n^2 x^{m+1} (f x)^{m-1}}{2 m^3}+\frac{2 b^2 e^2 n^2 x^{2 m+1} (f x)^{m-1}}{27 m^3} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^(-1 + m)*(d + e*x^m)^2*(a + b*Log[c*x^n])^2,x]

[Out]

(2*b^2*d^2*n^2*x*(f*x)^(-1 + m))/m^3 + (b^2*d*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(2*m^3) + (2*b^2*e^2*n^2*x^(1 +
2*m)*(f*x)^(-1 + m))/(27*m^3) + (b^2*d^3*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[x]^2)/(3*e*m) - (b*n*x^(1 - m)*(f*x)
^(-1 + m)*((18*d^2*e*x^m)/m + (9*d*e^2*x^(2*m))/m + (2*e^3*x^(3*m))/m + 6*d^3*Log[x])*(a + b*Log[c*x^n]))/(9*e
*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^3*(a + b*Log[c*x^n])^2)/(3*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 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 (f x)^{-1+m} \left (d+e x^m\right )^2 \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 )^2 \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 )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac{\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{3 e m}\\ &=-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{18 d^2 e x^m}{m}+\frac{9 d e^2 x^{2 m}}{m}+\frac{2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac{\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{e x^m \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+6 d^3 m \log (x)}{6 m x} \, dx}{3 e m}\\ &=-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{18 d^2 e x^m}{m}+\frac{9 d e^2 x^{2 m}}{m}+\frac{2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{e x^m \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+6 d^3 m \log (x)}{x} \, dx}{9 e m^2}\\ &=-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{18 d^2 e x^m}{m}+\frac{9 d e^2 x^{2 m}}{m}+\frac{2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \left (18 d^2 e x^{-1+m}+9 d e^2 x^{-1+2 m}+2 e^3 x^{-1+3 m}+\frac{6 d^3 m \log (x)}{x}\right ) \, dx}{9 e m^2}\\ &=\frac{2 b^2 d^2 n^2 x (f x)^{-1+m}}{m^3}+\frac{b^2 d e n^2 x^{1+m} (f x)^{-1+m}}{2 m^3}+\frac{2 b^2 e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{27 m^3}-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{18 d^2 e x^m}{m}+\frac{9 d e^2 x^{2 m}}{m}+\frac{2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac{\left (2 b^2 d^3 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{\log (x)}{x} \, dx}{3 e m}\\ &=\frac{2 b^2 d^2 n^2 x (f x)^{-1+m}}{m^3}+\frac{b^2 d e n^2 x^{1+m} (f x)^{-1+m}}{2 m^3}+\frac{2 b^2 e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{27 m^3}+\frac{b^2 d^3 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{3 e m}-\frac{b n x^{1-m} (f x)^{-1+m} \left (\frac{18 d^2 e x^m}{m}+\frac{9 d e^2 x^{2 m}}{m}+\frac{2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}\\ \end{align*}

Mathematica [A]  time = 0.200529, size = 207, normalized size = 0.69 \[ \frac{(f x)^m \left (18 a^2 m^2 \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )+6 b m \log \left (c x^n\right ) \left (6 a m \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )-b n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )\right )-6 a b m n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+18 b^2 m^2 \log ^2\left (c x^n\right ) \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )+b^2 n^2 \left (108 d^2+27 d e x^m+4 e^2 x^{2 m}\right )\right )}{54 f m^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^(-1 + m)*(d + e*x^m)^2*(a + b*Log[c*x^n])^2,x]

[Out]

((f*x)^m*(18*a^2*m^2*(3*d^2 + 3*d*e*x^m + e^2*x^(2*m)) - 6*a*b*m*n*(18*d^2 + 9*d*e*x^m + 2*e^2*x^(2*m)) + b^2*
n^2*(108*d^2 + 27*d*e*x^m + 4*e^2*x^(2*m)) + 6*b*m*(6*a*m*(3*d^2 + 3*d*e*x^m + e^2*x^(2*m)) - b*n*(18*d^2 + 9*
d*e*x^m + 2*e^2*x^(2*m)))*Log[c*x^n] + 18*b^2*m^2*(3*d^2 + 3*d*e*x^m + e^2*x^(2*m))*Log[c*x^n]^2))/(54*f*m^3)

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Maple [C]  time = 0.387, size = 3038, normalized size = 10.2 \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)^2*(a+b*ln(c*x^n))^2,x)

[Out]

1/3*b^2*(e^2*(x^m)^2+3*d*e*x^m+3*d^2)*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*csg
n(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)))+1/9*b*(9*I*Pi*b*d^2
*csgn(I*x^n)*csgn(I*c*x^n)^2*m+3*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m-9*I*Pi*b*d^2*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)*m-3*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2*m-9*I*Pi*b*d*e*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)*x^m*m-9*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^m*m+9*I*Pi*b*d^2*m*csgn(I*c*x^n)^2*csgn(I*c)+9*I*Pi*
b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^m*m-9*I*Pi*b*d^2*m*csgn(I*c*x^n)^3-3*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^m)^2*m+3*
I*Pi*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^2*m+9*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m+6*ln(c)*b*e^2*(x
^m)^2*m+18*ln(c)*b*d*e*x^m*m+6*a*e^2*(x^m)^2*m-2*b*e^2*n*(x^m)^2+18*ln(c)*b*d^2*m+18*a*d*e*x^m*m-9*b*d*e*n*x^m
+18*a*d^2*m-18*b*d^2*n)*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*cs
gn(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)))+1/108*(-27*Pi^2*b^2*d^2*csgn(I*x^
n)^2*csgn(I*c*x^n)^4*m^2+54*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^5*m^2+54*Pi^2*b^2*d^2*csgn(I*c*x^n)^5*csgn(
I*c)*m^2-27*Pi^2*b^2*d^2*csgn(I*c*x^n)^4*csgn(I*c)^2*m^2-9*Pi^2*b^2*e^2*csgn(I*c*x^n)^6*(x^m)^2*m^2+216*b^2*d^
2*n^2-36*I*Pi*a*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2*m^2+12*I*Pi*b^2*e^2*m*n*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)*(x^m)^2+108*I*Pi*a*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m^2-108*a*b*d*e*m*n*x^m+108*a^2*d^2*m
^2+36*a^2*e^2*(x^m)^2*m^2+8*b^2*e^2*n^2*(x^m)^2+108*ln(c)^2*b^2*d^2*m^2+36*I*ln(c)*Pi*b^2*e^2*csgn(I*c*x^n)^2*
csgn(I*c)*(x^m)^2*m^2-108*I*ln(c)*Pi*b^2*d*e*csgn(I*c*x^n)^3*x^m*m^2+36*I*Pi*a*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)
^2*(x^m)^2*m^2-27*Pi^2*b^2*d*e*csgn(I*c*x^n)^6*x^m*m^2+54*Pi^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)
*m^2-27*Pi^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*m^2-108*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^
4*csgn(I*c)*m^2-36*I*ln(c)*Pi*b^2*e^2*csgn(I*c*x^n)^3*(x^m)^2*m^2+18*Pi^2*b^2*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^
3*csgn(I*c)*(x^m)^2*m^2-9*Pi^2*b^2*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*(x^m)^2*m^2-27*Pi^2*b^2*d^2*c
sgn(I*c*x^n)^6*m^2+36*ln(c)^2*b^2*e^2*(x^m)^2*m^2+108*a^2*d*e*x^m*m^2+54*b^2*d*e*n^2*x^m-216*ln(c)*b^2*d^2*m*n
+216*ln(c)*a*b*d^2*m^2-216*a*b*d^2*m*n+108*I*ln(c)*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2*m^2+108*I*ln(c)*Pi*b
^2*d^2*csgn(I*c*x^n)^2*csgn(I*c)*m^2+36*I*Pi*a*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^2*m^2-12*I*Pi*b^2*e^2*m*n
*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2-12*I*Pi*b^2*e^2*m*n*csgn(I*c*x^n)^2*csgn(I*c)*(x^m)^2-108*I*Pi*a*b*d*e*cs
gn(I*c*x^n)^3*x^m*m^2+108*I*Pi*a*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2*m^2+108*I*Pi*a*b*d^2*csgn(I*c*x^n)^2*csgn(I
*c)*m^2-108*I*Pi*b^2*d^2*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2-108*I*Pi*b^2*d^2*m*n*csgn(I*c*x^n)^2*csgn(I*c)-108*I*
ln(c)*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*m^2+54*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)
^2*m^2-9*Pi^2*b^2*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4*(x^m)^2*m^2+18*Pi^2*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^5*(x
^m)^2*m^2-108*I*ln(c)*Pi*b^2*d^2*csgn(I*c*x^n)^3*m^2+54*Pi^2*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*x
^m*m^2+108*I*Pi*b^2*d^2*m*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+54*I*Pi*b^2*d*e*m*n*csgn(I*c*x^n)^3*x^m+36*I*l
n(c)*Pi*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m^2-108*I*ln(c)*Pi*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)*x^m*m^2-108*I*Pi*a*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^m*m^2-36*I*Pi*a*b*e^2*csgn(I*c*x^n)^3*(x^m
)^2*m^2+12*I*Pi*b^2*e^2*m*n*csgn(I*c*x^n)^3*(x^m)^2-36*Pi^2*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*(x^m
)^2*m^2+18*Pi^2*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*(x^m)^2*m^2-27*Pi^2*b^2*d*e*csgn(I*x^n)^2*csgn
(I*c*x^n)^4*x^m*m^2+54*Pi^2*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)^5*x^m*m^2+54*Pi^2*b^2*d*e*csgn(I*c*x^n)^5*csgn(I
*c)*x^m*m^2-27*Pi^2*b^2*d*e*csgn(I*c*x^n)^4*csgn(I*c)^2*x^m*m^2-108*I*Pi*a*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)*m^2+54*Pi^2*b^2*d*e*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*x^m*m^2-27*Pi^2*b^2*d*e*csgn(I*x^n)^2*csgn(
I*c*x^n)^2*csgn(I*c)^2*x^m*m^2-108*Pi^2*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*x^m*m^2-108*I*Pi*a*b*d^2
*csgn(I*c*x^n)^3*m^2+216*ln(c)*a*b*d*e*x^m*m^2-108*ln(c)*b^2*d*e*m*n*x^m+108*I*Pi*a*b*d*e*csgn(I*c*x^n)^2*csgn
(I*c)*x^m*m^2-54*I*Pi*b^2*d*e*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m-54*I*Pi*b^2*d*e*m*n*csgn(I*c*x^n)^2*csgn(I*c
)*x^m+108*I*Pi*b^2*d^2*m*n*csgn(I*c*x^n)^3+18*Pi^2*b^2*e^2*csgn(I*c*x^n)^5*csgn(I*c)*(x^m)^2*m^2-9*Pi^2*b^2*e^
2*csgn(I*c*x^n)^4*csgn(I*c)^2*(x^m)^2*m^2-24*a*b*e^2*m*n*(x^m)^2-24*ln(c)*b^2*e^2*m*n*(x^m)^2+108*ln(c)^2*b^2*
d*e*x^m*m^2+72*ln(c)*a*b*e^2*(x^m)^2*m^2-36*I*ln(c)*Pi*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^m)^2*m^2
+108*I*ln(c)*Pi*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m^2+108*I*ln(c)*Pi*b^2*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x
^m*m^2+54*I*Pi*b^2*d*e*m*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^m)*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)^2*(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67419, size = 944, normalized size = 3.17 \begin{align*} \frac{2 \,{\left (9 \, b^{2} e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 9 \, b^{2} e^{2} m^{2} \log \left (c\right )^{2} + 9 \, a^{2} e^{2} m^{2} - 6 \, a b e^{2} m n + 2 \, b^{2} e^{2} n^{2} + 6 \,{\left (3 \, a b e^{2} m^{2} - b^{2} e^{2} m n\right )} \log \left (c\right ) + 6 \,{\left (3 \, b^{2} e^{2} m^{2} n \log \left (c\right ) + 3 \, a b e^{2} m^{2} n - b^{2} e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 27 \,{\left (2 \, b^{2} d e m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d e m^{2} \log \left (c\right )^{2} + 2 \, a^{2} d e m^{2} - 2 \, a b d e m n + b^{2} d e n^{2} + 2 \,{\left (2 \, a b d e m^{2} - b^{2} d e m n\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} d e m^{2} n \log \left (c\right ) + 2 \, a b d e m^{2} n - b^{2} d e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 54 \,{\left (b^{2} d^{2} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{2} m^{2} \log \left (c\right )^{2} + a^{2} d^{2} m^{2} - 2 \, a b d^{2} m n + 2 \, b^{2} d^{2} n^{2} + 2 \,{\left (a b d^{2} m^{2} - b^{2} d^{2} m n\right )} \log \left (c\right ) + 2 \,{\left (b^{2} d^{2} m^{2} n \log \left (c\right ) + a b d^{2} m^{2} n - b^{2} d^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m}}{54 \, m^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(d+e*x^m)^2*(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

1/54*(2*(9*b^2*e^2*m^2*n^2*log(x)^2 + 9*b^2*e^2*m^2*log(c)^2 + 9*a^2*e^2*m^2 - 6*a*b*e^2*m*n + 2*b^2*e^2*n^2 +
6*(3*a*b*e^2*m^2 - b^2*e^2*m*n)*log(c) + 6*(3*b^2*e^2*m^2*n*log(c) + 3*a*b*e^2*m^2*n - b^2*e^2*m*n^2)*log(x))
*f^(m - 1)*x^(3*m) + 27*(2*b^2*d*e*m^2*n^2*log(x)^2 + 2*b^2*d*e*m^2*log(c)^2 + 2*a^2*d*e*m^2 - 2*a*b*d*e*m*n +
b^2*d*e*n^2 + 2*(2*a*b*d*e*m^2 - b^2*d*e*m*n)*log(c) + 2*(2*b^2*d*e*m^2*n*log(c) + 2*a*b*d*e*m^2*n - b^2*d*e*
m*n^2)*log(x))*f^(m - 1)*x^(2*m) + 54*(b^2*d^2*m^2*n^2*log(x)^2 + b^2*d^2*m^2*log(c)^2 + a^2*d^2*m^2 - 2*a*b*d
^2*m*n + 2*b^2*d^2*n^2 + 2*(a*b*d^2*m^2 - b^2*d^2*m*n)*log(c) + 2*(b^2*d^2*m^2*n*log(c) + a*b*d^2*m^2*n - b^2*
d^2*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)**2*(a+b*ln(c*x**n))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.60514, size = 965, normalized size = 3.24 \begin{align*} \frac{b^{2} d^{2} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac{b^{2} d f^{m} n^{2} x^{2 \, m} e \log \left (x\right )^{2}}{f m} + \frac{2 \, b^{2} d^{2} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac{2 \, b^{2} d f^{m} n x^{2 \, m} e \log \left (c\right ) \log \left (x\right )}{f m} + \frac{b^{2} f^{m} n^{2} x^{3 \, m} e^{2} \log \left (x\right )^{2}}{3 \, f m} + \frac{b^{2} d^{2} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac{b^{2} d f^{m} x^{2 \, m} e \log \left (c\right )^{2}}{f m} + \frac{2 \, a b d^{2} f^{m} n x^{m} \log \left (x\right )}{f m} - \frac{2 \, b^{2} d^{2} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac{2 \, a b d f^{m} n x^{2 \, m} e \log \left (x\right )}{f m} - \frac{b^{2} d f^{m} n^{2} x^{2 \, m} e \log \left (x\right )}{f m^{2}} + \frac{2 \, b^{2} f^{m} n x^{3 \, m} e^{2} \log \left (c\right ) \log \left (x\right )}{3 \, f m} + \frac{2 \, a b d^{2} f^{m} x^{m} \log \left (c\right )}{f m} - \frac{2 \, b^{2} d^{2} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac{2 \, a b d f^{m} x^{2 \, m} e \log \left (c\right )}{f m} - \frac{b^{2} d f^{m} n x^{2 \, m} e \log \left (c\right )}{f m^{2}} + \frac{b^{2} f^{m} x^{3 \, m} e^{2} \log \left (c\right )^{2}}{3 \, f m} + \frac{2 \, a b f^{m} n x^{3 \, m} e^{2} \log \left (x\right )}{3 \, f m} - \frac{2 \, b^{2} f^{m} n^{2} x^{3 \, m} e^{2} \log \left (x\right )}{9 \, f m^{2}} + \frac{a^{2} d^{2} f^{m} x^{m}}{f m} - \frac{2 \, a b d^{2} f^{m} n x^{m}}{f m^{2}} + \frac{2 \, b^{2} d^{2} f^{m} n^{2} x^{m}}{f m^{3}} + \frac{a^{2} d f^{m} x^{2 \, m} e}{f m} - \frac{a b d f^{m} n x^{2 \, m} e}{f m^{2}} + \frac{b^{2} d f^{m} n^{2} x^{2 \, m} e}{2 \, f m^{3}} + \frac{2 \, a b f^{m} x^{3 \, m} e^{2} \log \left (c\right )}{3 \, f m} - \frac{2 \, b^{2} f^{m} n x^{3 \, m} e^{2} \log \left (c\right )}{9 \, f m^{2}} + \frac{a^{2} f^{m} x^{3 \, m} e^{2}}{3 \, f m} - \frac{2 \, a b f^{m} n x^{3 \, m} e^{2}}{9 \, f m^{2}} + \frac{2 \, b^{2} f^{m} n^{2} x^{3 \, m} e^{2}}{27 \, f m^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(d+e*x^m)^2*(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

b^2*d^2*f^m*n^2*x^m*log(x)^2/(f*m) + b^2*d*f^m*n^2*x^(2*m)*e*log(x)^2/(f*m) + 2*b^2*d^2*f^m*n*x^m*log(c)*log(x
)/(f*m) + 2*b^2*d*f^m*n*x^(2*m)*e*log(c)*log(x)/(f*m) + 1/3*b^2*f^m*n^2*x^(3*m)*e^2*log(x)^2/(f*m) + b^2*d^2*f
^m*x^m*log(c)^2/(f*m) + b^2*d*f^m*x^(2*m)*e*log(c)^2/(f*m) + 2*a*b*d^2*f^m*n*x^m*log(x)/(f*m) - 2*b^2*d^2*f^m*
n^2*x^m*log(x)/(f*m^2) + 2*a*b*d*f^m*n*x^(2*m)*e*log(x)/(f*m) - b^2*d*f^m*n^2*x^(2*m)*e*log(x)/(f*m^2) + 2/3*b
^2*f^m*n*x^(3*m)*e^2*log(c)*log(x)/(f*m) + 2*a*b*d^2*f^m*x^m*log(c)/(f*m) - 2*b^2*d^2*f^m*n*x^m*log(c)/(f*m^2)
+ 2*a*b*d*f^m*x^(2*m)*e*log(c)/(f*m) - b^2*d*f^m*n*x^(2*m)*e*log(c)/(f*m^2) + 1/3*b^2*f^m*x^(3*m)*e^2*log(c)^
2/(f*m) + 2/3*a*b*f^m*n*x^(3*m)*e^2*log(x)/(f*m) - 2/9*b^2*f^m*n^2*x^(3*m)*e^2*log(x)/(f*m^2) + a^2*d^2*f^m*x^
m/(f*m) - 2*a*b*d^2*f^m*n*x^m/(f*m^2) + 2*b^2*d^2*f^m*n^2*x^m/(f*m^3) + a^2*d*f^m*x^(2*m)*e/(f*m) - a*b*d*f^m*
n*x^(2*m)*e/(f*m^2) + 1/2*b^2*d*f^m*n^2*x^(2*m)*e/(f*m^3) + 2/3*a*b*f^m*x^(3*m)*e^2*log(c)/(f*m) - 2/9*b^2*f^m
*n*x^(3*m)*e^2*log(c)/(f*m^2) + 1/3*a^2*f^m*x^(3*m)*e^2/(f*m) - 2/9*a*b*f^m*n*x^(3*m)*e^2/(f*m^2) + 2/27*b^2*f
^m*n^2*x^(3*m)*e^2/(f*m^3)