3.319 \(\int (f x)^m (d+e x^2)^2 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=153 \[ \frac{d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}-\frac{b d^2 n (f x)^{m+1}}{f (m+1)^2}-\frac{2 b d e n (f x)^{m+3}}{f^3 (m+3)^2}-\frac{b e^2 n (f x)^{m+5}}{f^5 (m+5)^2} \]

[Out]

-((b*d^2*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (2*b*d*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (b*e^2*n*(f*x)^(5 + m))
/(f^5*(5 + m)^2) + (d^2*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]
))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n]))/(f^5*(5 + m))

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Rubi [A]  time = 0.179994, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {270, 2350, 12, 14} \[ \frac{d^2 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}-\frac{b d^2 n (f x)^{m+1}}{f (m+1)^2}-\frac{2 b d e n (f x)^{m+3}}{f^3 (m+3)^2}-\frac{b e^2 n (f x)^{m+5}}{f^5 (m+5)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d^2*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (2*b*d*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (b*e^2*n*(f*x)^(5 + m))
/(f^5*(5 + m)^2) + (d^2*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]
))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n]))/(f^5*(5 + m))

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 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*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] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

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]]

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{d^2 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}-(b n) \int \frac{(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{(1+m) (3+m) (5+m)} \, dx\\ &=\frac{d^2 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}-\frac{(b n) \int (f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right ) \, dx}{15+23 m+9 m^2+m^3}\\ &=\frac{d^2 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}-\frac{(b n) \int \left (d^2 (3+m) (5+m) (f x)^m+\frac{2 d e (1+m) (5+m) (f x)^{2+m}}{f^2}+\frac{e^2 (1+m) (3+m) (f x)^{4+m}}{f^4}\right ) \, dx}{15+23 m+9 m^2+m^3}\\ &=-\frac{b d^2 n (f x)^{1+m}}{f (1+m)^2}-\frac{2 b d e n (f x)^{3+m}}{f^3 (3+m)^2}-\frac{b e^2 n (f x)^{5+m}}{f^5 (5+m)^2}+\frac{d^2 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}\\ \end{align*}

Mathematica [A]  time = 0.134363, size = 112, normalized size = 0.73 \[ x (f x)^m \left (\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{m+1}+\frac{2 d e x^2 \left (a+b \log \left (c x^n\right )\right )}{m+3}+\frac{e^2 x^4 \left (a+b \log \left (c x^n\right )\right )}{m+5}-\frac{b d^2 n}{(m+1)^2}-\frac{2 b d e n x^2}{(m+3)^2}-\frac{b e^2 n x^4}{(m+5)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(f*x)^m*(-((b*d^2*n)/(1 + m)^2) - (2*b*d*e*n*x^2)/(3 + m)^2 - (b*e^2*n*x^4)/(5 + m)^2 + (d^2*(a + b*Log[c*x^
n]))/(1 + m) + (2*d*e*x^2*(a + b*Log[c*x^n]))/(3 + m) + (e^2*x^4*(a + b*Log[c*x^n]))/(5 + m))

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Maple [C]  time = 0.289, size = 2790, normalized size = 18.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*(e^2*m^2*x^4+4*e^2*m*x^4+2*d*e*m^2*x^2+3*e^2*x^4+12*d*e*m*x^2+d^2*m^2+10*d*e*x^2+8*d^2*m+15*d^2)/(1+m)/(3+
m)/(5+m)*exp(1/2*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)))*ln(x^n)+1/2*x*(450*ln(c)*b*d^2+90*ln(c)*b*e^2*x^4+17*I*Pi*b*d^2*m^4
*csgn(I*c*x^n)^2*csgn(I*c)+465*I*Pi*b*d^2*m*csgn(I*x^n)*csgn(I*c*x^n)^2+465*I*Pi*b*d^2*m*csgn(I*c*x^n)^2*csgn(
I*c)+450*a*d^2+2*ln(c)*b*e^2*m^5*x^4+124*ln(c)*b*e^2*m^3*x^4+268*ln(c)*b*e^2*m^2*x^4+258*ln(c)*b*e^2*m*x^4+26*
ln(c)*b*e^2*m^4*x^4+300*a*d*e*x^2-I*Pi*b*d^2*m^5*csgn(I*c*x^n)^3-2*b*e^2*m^4*n*x^4+4*a*d*e*m^5*x^2+26*a*e^2*m^
4*x^4-2*b*d^2*m^4*n+334*I*Pi*b*d^2*m^2*csgn(I*c*x^n)^2*csgn(I*c)+17*I*Pi*b*d^2*m^4*csgn(I*x^n)*csgn(I*c*x^n)^2
+110*I*Pi*b*d^2*m^3*csgn(I*c*x^n)^2*csgn(I*c)+334*I*Pi*b*d^2*m^2*csgn(I*x^n)*csgn(I*c*x^n)^2+300*ln(c)*b*d*e*x
^2-188*b*d^2*m^2*n-480*b*d^2*m*n-32*b*d^2*m^3*n-450*b*d^2*n+2*ln(c)*b*d^2*m^5+34*ln(c)*b*d^2*m^4+220*ln(c)*b*d
^2*m^3+668*ln(c)*b*d^2*m^2+930*ln(c)*b*d^2*m+2*a*e^2*m^5*x^4+124*a*e^2*m^3*x^4+268*a*e^2*m^2*x^4+258*a*e^2*m*x
^4+328*a*d*e*m^3*x^2+792*a*d*e*m^2*x^2+820*a*d*e*m*x^2+90*a*e^2*x^4-16*b*e^2*m^3*n*x^4+60*a*d*e*m^4*x^2+328*ln
(c)*b*d*e*m^3*x^2+792*ln(c)*b*d*e*m^2*x^2+820*ln(c)*b*d*e*m*x^2+60*ln(c)*b*d*e*m^4*x^2+4*ln(c)*b*d*e*m^5*x^2+3
4*a*d^2*m^4-2*I*Pi*b*d*e*m^5*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-30*I*Pi*b*d*e*m^4*x^2*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)+150*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+150*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^2*csgn(I*c)-1
50*I*Pi*b*d*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-225*I*Pi*b*d^2*csgn(I*c*x^n)^3+45*I*Pi*b*e^2*x^4*csgn(I*
x^n)*csgn(I*c*x^n)^2-62*I*Pi*b*e^2*m^3*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-134*I*Pi*b*e^2*m^2*x^4*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*e^2*m^5*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+220*a*d^2*m^3+668*a*d^2*m^2
+930*a*d^2*m-48*b*d*e*m^3*n*x^2-184*b*d*e*m^2*n*x^2-240*b*d*e*m*n*x^2-44*b*e^2*m^2*n*x^4-48*b*e^2*m*n*x^4+396*
I*Pi*b*d*e*m^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+410*I*Pi*b*d*e*m*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-410*I*Pi*b*d*e*m
*x^2*csgn(I*c*x^n)^3-334*I*Pi*b*d^2*m^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+129*I*Pi*b*e^2*m*x^4*csgn(I*c*x^n)
^2*csgn(I*c)+I*Pi*b*e^2*m^5*x^4*csgn(I*c*x^n)^2*csgn(I*c)-17*I*Pi*b*d^2*m^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c
)+134*I*Pi*b*e^2*m^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)-465*I*Pi*b*d^2*m*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+13*I*P
i*b*e^2*m^4*x^4*csgn(I*c*x^n)^2*csgn(I*c)-2*I*Pi*b*d*e*m^5*x^2*csgn(I*c*x^n)^3-30*I*Pi*b*d*e*m^4*x^2*csgn(I*c*
x^n)^3+I*Pi*b*e^2*m^5*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+62*I*Pi*b*e^2*m^3*x^4*csgn(I*c*x^n)^2*csgn(I*c)+134*I*Pi
*b*e^2*m^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+13*I*Pi*b*e^2*m^4*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+129*I*Pi*b*e^2*m*
x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+62*I*Pi*b*e^2*m^3*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-45*I*Pi*b*e^2*x^4*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)-110*I*Pi*b*d^2*m^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-164*I*Pi*b*d*e*m^3*x^2*csgn(I
*c*x^n)^3-I*Pi*b*d^2*m^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-396*I*Pi*b*d*e*m^2*x^2*csgn(I*c*x^n)^3+30*I*Pi*b*
d*e*m^4*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+30*I*Pi*b*d*e*m^4*x^2*csgn(I*c*x^n)^2*csgn(I*c)+164*I*Pi*b*d*e*m^3*x^2
*csgn(I*x^n)*csgn(I*c*x^n)^2+2*I*Pi*b*d*e*m^5*x^2*csgn(I*c*x^n)^2*csgn(I*c)+2*I*Pi*b*d*e*m^5*x^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2-13*I*Pi*b*e^2*m^4*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-62*I*Pi*b*e^2*m^3*x^4*csgn(I*c*x^n)^3
-134*I*Pi*b*e^2*m^2*x^4*csgn(I*c*x^n)^3-129*I*Pi*b*e^2*m*x^4*csgn(I*c*x^n)^3-I*Pi*b*e^2*m^5*x^4*csgn(I*c*x^n)^
3+2*a*d^2*m^5-13*I*Pi*b*e^2*m^4*x^4*csgn(I*c*x^n)^3-100*b*d*e*n*x^2-150*I*Pi*b*d*e*x^2*csgn(I*c*x^n)^3-45*I*Pi
*b*e^2*x^4*csgn(I*c*x^n)^3-110*I*Pi*b*d^2*m^3*csgn(I*c*x^n)^3-164*I*Pi*b*d*e*m^3*x^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-396*I*Pi*b*d*e*m^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-334*I*Pi*b*d^2*m^2*csgn(I*c*x^n)^3-465*I
*Pi*b*d^2*m*csgn(I*c*x^n)^3+225*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+225*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c
)-17*I*Pi*b*d^2*m^4*csgn(I*c*x^n)^3+I*Pi*b*d^2*m^5*csgn(I*c*x^n)^2*csgn(I*c)-4*b*d*e*m^4*n*x^2+I*Pi*b*d^2*m^5*
csgn(I*x^n)*csgn(I*c*x^n)^2+164*I*Pi*b*d*e*m^3*x^2*csgn(I*c*x^n)^2*csgn(I*c)-129*I*Pi*b*e^2*m*x^4*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)+396*I*Pi*b*d*e*m^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-410*I*Pi*b*d*e*m*x^2*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)+410*I*Pi*b*d*e*m*x^2*csgn(I*c*x^n)^2*csgn(I*c)-18*b*e^2*n*x^4+45*I*Pi*b*e^2*x^4*csgn(I*c
*x^n)^2*csgn(I*c)+110*I*Pi*b*d^2*m^3*csgn(I*x^n)*csgn(I*c*x^n)^2-225*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c))/(5+m)^2/(1+m)^2/(3+m)^2*exp(1/2*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)^m*(e*x^2+d)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.37806, size = 1519, normalized size = 9.93 \begin{align*} \frac{{\left ({\left (a e^{2} m^{5} + 13 \, a e^{2} m^{4} + 62 \, a e^{2} m^{3} + 134 \, a e^{2} m^{2} + 129 \, a e^{2} m + 45 \, a e^{2} -{\left (b e^{2} m^{4} + 8 \, b e^{2} m^{3} + 22 \, b e^{2} m^{2} + 24 \, b e^{2} m + 9 \, b e^{2}\right )} n\right )} x^{5} + 2 \,{\left (a d e m^{5} + 15 \, a d e m^{4} + 82 \, a d e m^{3} + 198 \, a d e m^{2} + 205 \, a d e m + 75 \, a d e -{\left (b d e m^{4} + 12 \, b d e m^{3} + 46 \, b d e m^{2} + 60 \, b d e m + 25 \, b d e\right )} n\right )} x^{3} +{\left (a d^{2} m^{5} + 17 \, a d^{2} m^{4} + 110 \, a d^{2} m^{3} + 334 \, a d^{2} m^{2} + 465 \, a d^{2} m + 225 \, a d^{2} -{\left (b d^{2} m^{4} + 16 \, b d^{2} m^{3} + 94 \, b d^{2} m^{2} + 240 \, b d^{2} m + 225 \, b d^{2}\right )} n\right )} x +{\left ({\left (b e^{2} m^{5} + 13 \, b e^{2} m^{4} + 62 \, b e^{2} m^{3} + 134 \, b e^{2} m^{2} + 129 \, b e^{2} m + 45 \, b e^{2}\right )} x^{5} + 2 \,{\left (b d e m^{5} + 15 \, b d e m^{4} + 82 \, b d e m^{3} + 198 \, b d e m^{2} + 205 \, b d e m + 75 \, b d e\right )} x^{3} +{\left (b d^{2} m^{5} + 17 \, b d^{2} m^{4} + 110 \, b d^{2} m^{3} + 334 \, b d^{2} m^{2} + 465 \, b d^{2} m + 225 \, b d^{2}\right )} x\right )} \log \left (c\right ) +{\left ({\left (b e^{2} m^{5} + 13 \, b e^{2} m^{4} + 62 \, b e^{2} m^{3} + 134 \, b e^{2} m^{2} + 129 \, b e^{2} m + 45 \, b e^{2}\right )} n x^{5} + 2 \,{\left (b d e m^{5} + 15 \, b d e m^{4} + 82 \, b d e m^{3} + 198 \, b d e m^{2} + 205 \, b d e m + 75 \, b d e\right )} n x^{3} +{\left (b d^{2} m^{5} + 17 \, b d^{2} m^{4} + 110 \, b d^{2} m^{3} + 334 \, b d^{2} m^{2} + 465 \, b d^{2} m + 225 \, b d^{2}\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{6} + 18 \, m^{5} + 127 \, m^{4} + 444 \, m^{3} + 799 \, m^{2} + 690 \, m + 225} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*e^2*m^5 + 13*a*e^2*m^4 + 62*a*e^2*m^3 + 134*a*e^2*m^2 + 129*a*e^2*m + 45*a*e^2 - (b*e^2*m^4 + 8*b*e^2*m^3
+ 22*b*e^2*m^2 + 24*b*e^2*m + 9*b*e^2)*n)*x^5 + 2*(a*d*e*m^5 + 15*a*d*e*m^4 + 82*a*d*e*m^3 + 198*a*d*e*m^2 + 2
05*a*d*e*m + 75*a*d*e - (b*d*e*m^4 + 12*b*d*e*m^3 + 46*b*d*e*m^2 + 60*b*d*e*m + 25*b*d*e)*n)*x^3 + (a*d^2*m^5
+ 17*a*d^2*m^4 + 110*a*d^2*m^3 + 334*a*d^2*m^2 + 465*a*d^2*m + 225*a*d^2 - (b*d^2*m^4 + 16*b*d^2*m^3 + 94*b*d^
2*m^2 + 240*b*d^2*m + 225*b*d^2)*n)*x + ((b*e^2*m^5 + 13*b*e^2*m^4 + 62*b*e^2*m^3 + 134*b*e^2*m^2 + 129*b*e^2*
m + 45*b*e^2)*x^5 + 2*(b*d*e*m^5 + 15*b*d*e*m^4 + 82*b*d*e*m^3 + 198*b*d*e*m^2 + 205*b*d*e*m + 75*b*d*e)*x^3 +
(b*d^2*m^5 + 17*b*d^2*m^4 + 110*b*d^2*m^3 + 334*b*d^2*m^2 + 465*b*d^2*m + 225*b*d^2)*x)*log(c) + ((b*e^2*m^5
+ 13*b*e^2*m^4 + 62*b*e^2*m^3 + 134*b*e^2*m^2 + 129*b*e^2*m + 45*b*e^2)*n*x^5 + 2*(b*d*e*m^5 + 15*b*d*e*m^4 +
82*b*d*e*m^3 + 198*b*d*e*m^2 + 205*b*d*e*m + 75*b*d*e)*n*x^3 + (b*d^2*m^5 + 17*b*d^2*m^4 + 110*b*d^2*m^3 + 334
*b*d^2*m^2 + 465*b*d^2*m + 225*b*d^2)*n*x)*log(x))*e^(m*log(f) + m*log(x))/(m^6 + 18*m^5 + 127*m^4 + 444*m^3 +
799*m^2 + 690*m + 225)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(e*x**2+d)**2*(a+b*ln(c*x**n)),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.43107, size = 535, normalized size = 3.5 \begin{align*} \frac{b f^{4} f^{m} x^{5} x^{m} e^{2} \log \left (c\right )}{f^{4} m + 5 \, f^{4}} + \frac{a f^{4} f^{m} x^{5} x^{m} e^{2}}{f^{4} m + 5 \, f^{4}} + \frac{b f^{m} m n x^{5} x^{m} e^{2} \log \left (x\right )}{m^{2} + 10 \, m + 25} + \frac{2 \, b d f^{2} f^{m} x^{3} x^{m} e \log \left (c\right )}{f^{2} m + 3 \, f^{2}} + \frac{5 \, b f^{m} n x^{5} x^{m} e^{2} \log \left (x\right )}{m^{2} + 10 \, m + 25} + \frac{2 \, b d f^{m} m n x^{3} x^{m} e \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac{b f^{m} n x^{5} x^{m} e^{2}}{m^{2} + 10 \, m + 25} + \frac{2 \, a d f^{2} f^{m} x^{3} x^{m} e}{f^{2} m + 3 \, f^{2}} + \frac{6 \, b d f^{m} n x^{3} x^{m} e \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac{2 \, b d f^{m} n x^{3} x^{m} e}{m^{2} + 6 \, m + 9} + \frac{b d^{2} f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{b d^{2} f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{b d^{2} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac{\left (f x\right )^{m} b d^{2} x \log \left (c\right )}{m + 1} + \frac{\left (f x\right )^{m} a d^{2} x}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*f^4*f^m*x^5*x^m*e^2*log(c)/(f^4*m + 5*f^4) + a*f^4*f^m*x^5*x^m*e^2/(f^4*m + 5*f^4) + b*f^m*m*n*x^5*x^m*e^2*l
og(x)/(m^2 + 10*m + 25) + 2*b*d*f^2*f^m*x^3*x^m*e*log(c)/(f^2*m + 3*f^2) + 5*b*f^m*n*x^5*x^m*e^2*log(x)/(m^2 +
10*m + 25) + 2*b*d*f^m*m*n*x^3*x^m*e*log(x)/(m^2 + 6*m + 9) - b*f^m*n*x^5*x^m*e^2/(m^2 + 10*m + 25) + 2*a*d*f
^2*f^m*x^3*x^m*e/(f^2*m + 3*f^2) + 6*b*d*f^m*n*x^3*x^m*e*log(x)/(m^2 + 6*m + 9) - 2*b*d*f^m*n*x^3*x^m*e/(m^2 +
6*m + 9) + b*d^2*f^m*m*n*x*x^m*log(x)/(m^2 + 2*m + 1) + b*d^2*f^m*n*x*x^m*log(x)/(m^2 + 2*m + 1) - b*d^2*f^m*
n*x*x^m/(m^2 + 2*m + 1) + (f*x)^m*b*d^2*x*log(c)/(m + 1) + (f*x)^m*a*d^2*x/(m + 1)