3.4.59 \(\int (f x)^{-1+m} (d+e x^m)^3 (a+b \log (c x^n))^2 \, dx\) [359]

Optimal. Leaf size=372 \[ \frac {2 b^2 d^3 n^2 x (f x)^{-1+m}}{m^3}+\frac {3 b^2 d^2 e n^2 x^{1+m} (f x)^{-1+m}}{4 m^3}+\frac {2 b^2 d e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{9 m^3}+\frac {b^2 e^3 n^2 x^{1+3 m} (f x)^{-1+m}}{32 m^3}+\frac {b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m} \]

[Out]

2*b^2*d^3*n^2*x*(f*x)^(-1+m)/m^3+3/4*b^2*d^2*e*n^2*x^(1+m)*(f*x)^(-1+m)/m^3+2/9*b^2*d*e^2*n^2*x^(1+2*m)*(f*x)^
(-1+m)/m^3+1/32*b^2*e^3*n^2*x^(1+3*m)*(f*x)^(-1+m)/m^3+1/4*b^2*d^4*n^2*x^(1-m)*(f*x)^(-1+m)*ln(x)^2/e/m-2*b*d^
3*n*x*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-3/2*b*d^2*e*n*x^(1+m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-2/3*b*d*e^2*n*x^
(1+2*m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-1/8*b*e^3*n*x^(1+3*m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-1/2*b*d^4*n*x^
(1-m)*(f*x)^(-1+m)*ln(x)*(a+b*ln(c*x^n))/e/m+1/4*x^(1-m)*(f*x)^(-1+m)*(d+e*x^m)^4*(a+b*ln(c*x^n))^2/e/m

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Rubi [A]
time = 0.34, antiderivative size = 372, normalized size of antiderivative = 1.00, 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 = {2377, 2376, 272, 45, 2372, 14, 2338} \begin {gather*} -\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 {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 {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 {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 {3 b^2 d^2 e n^2 x^{m+1} (f x)^{m-1}}{4 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} \end {gather*}

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) - (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)

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 45

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 272

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 2338

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]

Rule 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 2376

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 2377

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])

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*}

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Mathematica [A]
time = 0.17, size = 285, normalized size = 0.77 \begin {gather*} \frac {(f x)^m \left (72 a^2 m^2 \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-12 a b m n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )+b^2 n^2 \left (576 d^3+216 d^2 e x^m+64 d e^2 x^{2 m}+9 e^3 x^{3 m}\right )+12 b m \left (12 a m \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )\right ) \log \left (c x^n\right )+72 b^2 m^2 \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right ) \log ^2\left (c x^n\right )\right )}{288 f m^3} \end {gather*}

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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.44, size = 4156, normalized size = 11.17

method	result	size
risch	\(\text {Expression too large to display}\)	\(4156\)

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,method=_RETURNVERBOSE)

[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*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(x)+2*ln(f)))*ln(x^n)^2+1
/24*b*(24*I*Pi*b*d^3*csgn(I*c)*csgn(I*c*x^n)^2*m+24*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2*m+12*ln(c)*b*e^3*(x
^m)^3*m+48*a*d*e^2*(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+48*a*d^3*m+48*ln(c)*b*d^3*
m-48*b*d^3*n+24*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^m)^2*m-3*b*e^3*n*(x^m)^3+12*a*e^3*(x^m)^3*m-36*I*Pi*
b*d^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^m*m-24*I*Pi*b*d^3*csgn(I*c*x^n)^3*m+6*I*Pi*b*e^3*csgn(I*x^n)*csg
n(I*c*x^n)^2*(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*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*
c*x^n)*(x^m)^2*m+24*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m+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)*csgn(I*x^n)*csgn(I*c*x^n)*(x^m)^3*m+36*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*c*x^n)^
2*x^m*m+48*ln(c)*b*d*e^2*(x^m)^2*m+72*ln(c)*b*d^2*e*x^m*m-24*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*m+
6*I*Pi*b*e^3*csgn(I*c)*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-36*I*Pi*b*d^2*e*csgn(I
*c*x^n)^3*x^m*m)*x/m^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*csg
n(I*x)-I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)+2*ln(x)+2*ln(f)))*ln(x^n)+1/288*(18*I*Pi*b^2*e^3*m*n*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)*(x^m)^3+288*I*Pi*ln(c)*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m^2-72*I*Pi*a*b*e^
3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^m)^3*m^2-72*Pi^2*b^2*d*e^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*
(x^m)^2*m^2+144*Pi^2*b^2*d*e^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3*(x^m)^2*m^2+96*I*Pi*b^2*d*e^2*m*n*csgn(
I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^m)^2-432*I*Pi*a*b*d^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^m*m^2-576*a*b*
d^3*m*n+72*a^2*e^3*(x^m)^3*m^2+9*b^2*e^3*n^2*(x^m)^3-576*ln(c)*b^2*d^3*m*n+576*ln(c)*a*b*d^3*m^2-288*I*Pi*ln(c
)*b^2*d*e^2*csgn(I*c*x^n)^3*(x^m)^2*m^2+216*I*Pi*b^2*d^2*e*m*n*csgn(I*c*x^n)^3*x^m-432*I*Pi*ln(c)*b^2*d^2*e*cs
gn(I*c*x^n)^3*x^m*m^2+144*Pi^2*b^2*d^3*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3*m^2+288*a^2*d^3*m^2-72*I*Pi*ln(
c)*b^2*e^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^m)^3*m^2+288*I*Pi*ln(c)*b^2*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*
(x^m)^2*m^2+36*Pi^2*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)^5*(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+288*a^2*d*e^2*(x^m)^2*m^2+432*a^2*d^2*e*x^m*m^2-192*ln(c)*b^2*d*e^2*m*n*(x
^m)^2-432*ln(c)*b^2*d^2*e*m*n*x^m+864*ln(c)*a*b*d^2*e*x^m*m^2+576*b^2*d^3*n^2+432*I*Pi*ln(c)*b^2*d^2*e*csgn(I*
c)*csgn(I*c*x^n)^2*x^m*m^2+432*I*Pi*ln(c)*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m^2-72*I*Pi*ln(c)*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+216*Pi^2*b^2*d^2*e*csgn(I*x^n)*csgn(I*
c*x^n)^5*x^m*m^2-108*Pi^2*b^2*d^2*e*csgn(I*c)^2*csgn(I*c*x^n)^4*x^m*m^2-288*I*Pi*a*b*d^3*csgn(I*c*x^n)^3*m^2+2
88*I*Pi*b^2*d^3*m*n*csgn(I*c*x^n)^3+288*I*Pi*a*b*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^m)^2*m^2-288*I*Pi*b^2*d^3*
m*n*csgn(I*c)*csgn(I*c*x^n)^2-288*I*Pi*b^2*d^3*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2+18*I*Pi*b^2*e^3*m*n*csgn(I*c*x^
n)^3*(x^m)^3-432*Pi^2*b^2*d^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4*x^m*m^2-108*Pi^2*b^2*d^2*e*csgn(I*c)^2*c
sgn(I*x^n)^2*csgn(I*c*x^n)^2*x^m*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
*e^2*(x^m)^2*m^2-288*Pi^2*b^2*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4*m^2+288*I*Pi*a*b*d^3*csgn(I*x^n)*csgn(
I*c*x^n)^2*m^2+36*Pi^2*b^2*e^3*csgn(I*c)*csgn(I*c*x^n)^5*(x^m)^3*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*a*b*e^3*m*n*(x^m)^3-36*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+144*ln(
c)*a*b*e^3*(x^m)^3*m^2+432*ln(c)^2*b^2*d^2*e*x^m*m^2-432*I*Pi*a*b*d^2*e*csgn(I*c*x^n)^3*x^m*m^2+72*I*Pi*a*b*e^
3*csgn(I*c)*csgn(I*c*x^n)^2*(x^m)^3*m^2-288*I*Pi*ln(c)*b^2*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*m^2+216*Pi^
2*b^2*d^2*e*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3*x^m*m^2+144*Pi^2*b^2*d*e^2*csgn(I*c)*csgn(I*c*x^n)^5*(x^m)
^2*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+36*Pi^2*b^2*e^3*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3*(x^m)^3*m^2-72*Pi^2*b^2*e^3*csgn(I*c)*
csgn(I*x^n)*csgn(I*c*x^n)^4*(x^m)^3*m^2-108*Pi^2*b^2*d^2*e*csgn(I*c*x^n)^6*x^m*m^2-72*Pi^2*b^2*d^3*csgn(I*c)^2
*csgn(I*c*x^n)^4*m^2+144*Pi^2*b^2*d^3*csgn(I*c)*csgn(I*c*x^n)^5*m^2-72*Pi^2*b^2*d^3*csgn(I*x^n)^2*csgn(I*c*x^n
)^4*m^2-216*I*Pi*b^2*d^2*e*m*n*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m+144*Pi^2*b^2*d*e^2*csgn(I*c)*csgn(I*x^n)^2*csgn
(I*c*x^n)^3*(x^m)^2*m^2+216*I*Pi*b^2*d^2*e*m*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^m-288*I*Pi*ln(c)*b^2*d*e^
2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^m)^2*m^2-432*I*Pi*ln(c)*b^2*d^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)
*x^m*m^2-18*I*Pi*b^2*e^3*m*n*csgn(I*c)*csgn(I*c*x^n)^2*(x^m)^3-288*I*Pi*a*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)*m^2+216*Pi^2*b^2*d^2*e*csgn(I*c)*csgn(I*x...

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Maxima [A]
time = 0.31, size = 585, normalized size = 1.57 \begin {gather*} \frac {3 \, b^{2} d^{2} f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )^{2}}{2 \, m} - 2 \, {\left (\frac {f^{m - 1} n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{m}}{m^{3}}\right )} b^{2} d^{3} - \frac {2 \, a b d^{3} f^{m - 1} n x^{m}}{m^{2}} - \frac {3}{4} \, {\left (\frac {2 \, f^{m - 1} n x^{2 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{2 \, m}}{m^{3}}\right )} b^{2} d^{2} e + \frac {3 \, a b d^{2} f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )}{m} + \frac {\left (f x\right )^{m} b^{2} d^{3} \log \left (c x^{n}\right )^{2}}{f m} + \frac {b^{2} d f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )^{2}}{m} - \frac {2}{9} \, {\left (\frac {3 \, f^{m - 1} n x^{3 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{3 \, m}}{m^{3}}\right )} b^{2} d e^{2} + \frac {3 \, a^{2} d^{2} f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )}}{2 \, m} - \frac {3 \, a b d^{2} f^{m - 1} n e^{\left (2 \, m \log \left (x\right ) + 1\right )}}{2 \, m^{2}} + \frac {2 \, \left (f x\right )^{m} a b d^{3} \log \left (c x^{n}\right )}{f m} + \frac {2 \, a b d f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )}{m} + \frac {b^{2} f^{m - 1} e^{\left (4 \, m \log \left (x\right ) + 3\right )} \log \left (c x^{n}\right )^{2}}{4 \, m} + \frac {\left (f x\right )^{m} a^{2} d^{3}}{f m} - \frac {1}{32} \, {\left (\frac {4 \, f^{m - 1} n x^{4 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{4 \, m}}{m^{3}}\right )} b^{2} e^{3} + \frac {a^{2} d f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )}}{m} - \frac {2 \, a b d f^{m - 1} n e^{\left (3 \, m \log \left (x\right ) + 2\right )}}{3 \, m^{2}} + \frac {a b f^{m - 1} e^{\left (4 \, m \log \left (x\right ) + 3\right )} \log \left (c x^{n}\right )}{2 \, m} + \frac {a^{2} f^{m - 1} e^{\left (4 \, m \log \left (x\right ) + 3\right )}}{4 \, m} - \frac {a b f^{m - 1} n e^{\left (4 \, m \log \left (x\right ) + 3\right )}}{8 \, m^{2}} \end {gather*}

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]

3/2*b^2*d^2*f^(m - 1)*e^(2*m*log(x) + 1)*log(c*x^n)^2/m - 2*(f^(m - 1)*n*x^m*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^
m/m^3)*b^2*d^3 - 2*a*b*d^3*f^(m - 1)*n*x^m/m^2 - 3/4*(2*f^(m - 1)*n*x^(2*m)*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^(
2*m)/m^3)*b^2*d^2*e + 3*a*b*d^2*f^(m - 1)*e^(2*m*log(x) + 1)*log(c*x^n)/m + (f*x)^m*b^2*d^3*log(c*x^n)^2/(f*m)
+ b^2*d*f^(m - 1)*e^(3*m*log(x) + 2)*log(c*x^n)^2/m - 2/9*(3*f^(m - 1)*n*x^(3*m)*log(c*x^n)/m^2 - f^(m - 1)*n
^2*x^(3*m)/m^3)*b^2*d*e^2 + 3/2*a^2*d^2*f^(m - 1)*e^(2*m*log(x) + 1)/m - 3/2*a*b*d^2*f^(m - 1)*n*e^(2*m*log(x)
+ 1)/m^2 + 2*(f*x)^m*a*b*d^3*log(c*x^n)/(f*m) + 2*a*b*d*f^(m - 1)*e^(3*m*log(x) + 2)*log(c*x^n)/m + 1/4*b^2*f
^(m - 1)*e^(4*m*log(x) + 3)*log(c*x^n)^2/m + (f*x)^m*a^2*d^3/(f*m) - 1/32*(4*f^(m - 1)*n*x^(4*m)*log(c*x^n)/m^
2 - f^(m - 1)*n^2*x^(4*m)/m^3)*b^2*e^3 + a^2*d*f^(m - 1)*e^(3*m*log(x) + 2)/m - 2/3*a*b*d*f^(m - 1)*n*e^(3*m*l
og(x) + 2)/m^2 + 1/2*a*b*f^(m - 1)*e^(4*m*log(x) + 3)*log(c*x^n)/m + 1/4*a^2*f^(m - 1)*e^(4*m*log(x) + 3)/m -
1/8*a*b*f^(m - 1)*n*e^(4*m*log(x) + 3)/m^2

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Fricas [A]
time = 0.39, size = 570, normalized size = 1.53 \begin {gather*} \frac {9 \, {\left (8 \, b^{2} m^{2} n^{2} e^{3} \log \left (x\right )^{2} + 8 \, b^{2} m^{2} e^{3} \log \left (c\right )^{2} + 4 \, {\left (4 \, a b m^{2} - b^{2} m n\right )} e^{3} \log \left (c\right ) + {\left (8 \, a^{2} m^{2} - 4 \, a b m n + b^{2} n^{2}\right )} e^{3} + 4 \, {\left (4 \, b^{2} m^{2} n e^{3} \log \left (c\right ) + {\left (4 \, a b m^{2} n - b^{2} m n^{2}\right )} e^{3}\right )} \log \left (x\right )\right )} f^{m - 1} x^{4 \, m} + 32 \, {\left (9 \, b^{2} d m^{2} n^{2} e^{2} \log \left (x\right )^{2} + 9 \, b^{2} d m^{2} e^{2} \log \left (c\right )^{2} + 6 \, {\left (3 \, a b d m^{2} - b^{2} d m n\right )} e^{2} \log \left (c\right ) + {\left (9 \, a^{2} d m^{2} - 6 \, a b d m n + 2 \, b^{2} d n^{2}\right )} e^{2} + 6 \, {\left (3 \, b^{2} d m^{2} n e^{2} \log \left (c\right ) + {\left (3 \, a b d m^{2} n - b^{2} d m n^{2}\right )} e^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 216 \, {\left (2 \, b^{2} d^{2} m^{2} n^{2} e \log \left (x\right )^{2} + 2 \, b^{2} d^{2} m^{2} e \log \left (c\right )^{2} + 2 \, {\left (2 \, a b d^{2} m^{2} - b^{2} d^{2} m n\right )} e \log \left (c\right ) + {\left (2 \, a^{2} d^{2} m^{2} - 2 \, a b d^{2} m n + b^{2} d^{2} n^{2}\right )} e + 2 \, {\left (2 \, b^{2} d^{2} m^{2} n e \log \left (c\right ) + {\left (2 \, a b d^{2} m^{2} n - b^{2} d^{2} m n^{2}\right )} e\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 {gather*}

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*m^2*n^2*e^3*log(x)^2 + 8*b^2*m^2*e^3*log(c)^2 + 4*(4*a*b*m^2 - b^2*m*n)*e^3*log(c) + (8*a^2*m^
2 - 4*a*b*m*n + b^2*n^2)*e^3 + 4*(4*b^2*m^2*n*e^3*log(c) + (4*a*b*m^2*n - b^2*m*n^2)*e^3)*log(x))*f^(m - 1)*x^
(4*m) + 32*(9*b^2*d*m^2*n^2*e^2*log(x)^2 + 9*b^2*d*m^2*e^2*log(c)^2 + 6*(3*a*b*d*m^2 - b^2*d*m*n)*e^2*log(c) +
(9*a^2*d*m^2 - 6*a*b*d*m*n + 2*b^2*d*n^2)*e^2 + 6*(3*b^2*d*m^2*n*e^2*log(c) + (3*a*b*d*m^2*n - b^2*d*m*n^2)*e
^2)*log(x))*f^(m - 1)*x^(3*m) + 216*(2*b^2*d^2*m^2*n^2*e*log(x)^2 + 2*b^2*d^2*m^2*e*log(c)^2 + 2*(2*a*b*d^2*m^
2 - b^2*d^2*m*n)*e*log(c) + (2*a^2*d^2*m^2 - 2*a*b*d^2*m*n + b^2*d^2*n^2)*e + 2*(2*b^2*d^2*m^2*n*e*log(c) + (2
*a*b*d^2*m^2*n - b^2*d^2*m*n^2)*e)*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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 5006 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 985 vs. \(2 (350) = 700\).
time = 7.28, size = 985, normalized size = 2.65 \begin {gather*} \frac {b^{2} d^{3} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac {3 \, b^{2} d^{2} f^{m} n^{2} x^{2 \, m} e \log \left (x\right )^{2}}{2 \, f m} + \frac {2 \, b^{2} d^{3} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {3 \, b^{2} d^{2} f^{m} n x^{2 \, m} e \log \left (c\right ) \log \left (x\right )}{f m} + \frac {b^{2} d f^{m} n^{2} x^{3 \, m} e^{2} \log \left (x\right )^{2}}{f m} + \frac {b^{2} d^{3} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac {3 \, b^{2} d^{2} f^{m} x^{2 \, m} e \log \left (c\right )^{2}}{2 \, f m} + \frac {2 \, a b d^{3} f^{m} n x^{m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d^{3} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac {3 \, a b d^{2} f^{m} n x^{2 \, m} e \log \left (x\right )}{f m} - \frac {3 \, b^{2} d^{2} f^{m} n^{2} x^{2 \, m} e \log \left (x\right )}{2 \, f m^{2}} + \frac {2 \, b^{2} d f^{m} n x^{3 \, m} e^{2} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {b^{2} f^{m} n^{2} x^{4 \, m} e^{3} \log \left (x\right )^{2}}{4 \, f m} + \frac {2 \, a b d^{3} f^{m} x^{m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d^{3} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac {3 \, a b d^{2} f^{m} x^{2 \, m} e \log \left (c\right )}{f m} - \frac {3 \, b^{2} d^{2} f^{m} n x^{2 \, m} e \log \left (c\right )}{2 \, f m^{2}} + \frac {b^{2} d f^{m} x^{3 \, m} e^{2} \log \left (c\right )^{2}}{f m} + \frac {2 \, a b d f^{m} n x^{3 \, m} e^{2} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d f^{m} n^{2} x^{3 \, m} e^{2} \log \left (x\right )}{3 \, f m^{2}} + \frac {b^{2} f^{m} n x^{4 \, m} e^{3} \log \left (c\right ) \log \left (x\right )}{2 \, f m} + \frac {a^{2} d^{3} f^{m} x^{m}}{f m} - \frac {2 \, a b d^{3} f^{m} n x^{m}}{f m^{2}} + \frac {2 \, b^{2} d^{3} f^{m} n^{2} x^{m}}{f m^{3}} + \frac {3 \, a^{2} d^{2} f^{m} x^{2 \, m} e}{2 \, f m} - \frac {3 \, a b d^{2} f^{m} n x^{2 \, m} e}{2 \, f m^{2}} + \frac {3 \, b^{2} d^{2} f^{m} n^{2} x^{2 \, m} e}{4 \, f m^{3}} + \frac {2 \, a b d f^{m} x^{3 \, m} e^{2} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d f^{m} n x^{3 \, m} e^{2} \log \left (c\right )}{3 \, f m^{2}} + \frac {b^{2} f^{m} x^{4 \, m} e^{3} \log \left (c\right )^{2}}{4 \, f m} + \frac {a b f^{m} n x^{4 \, m} e^{3} \log \left (x\right )}{2 \, f m} - \frac {b^{2} f^{m} n^{2} x^{4 \, m} e^{3} \log \left (x\right )}{8 \, f m^{2}} + \frac {a^{2} d f^{m} x^{3 \, m} e^{2}}{f m} - \frac {2 \, a b d f^{m} n x^{3 \, m} e^{2}}{3 \, f m^{2}} + \frac {2 \, b^{2} d f^{m} n^{2} x^{3 \, m} e^{2}}{9 \, f m^{3}} + \frac {a b f^{m} x^{4 \, m} e^{3} \log \left (c\right )}{2 \, f m} - \frac {b^{2} f^{m} n x^{4 \, m} e^{3} \log \left (c\right )}{8 \, f m^{2}} + \frac {a^{2} f^{m} x^{4 \, m} e^{3}}{4 \, f m} - \frac {a b f^{m} n x^{4 \, m} e^{3}}{8 \, f m^{2}} + \frac {b^{2} f^{m} n^{2} x^{4 \, m} e^{3}}{32 \, f m^{3}} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f\,x\right )}^{m-1}\,{\left (d+e\,x^m\right )}^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^(m - 1)*(d + e*x^m)^3*(a + b*log(c*x^n))^2,x)

[Out]

int((f*x)^(m - 1)*(d + e*x^m)^3*(a + b*log(c*x^n))^2, x)

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