3.596 \(\int (d x)^m (a+b x^n+c x^{2 n})^3 \, dx\)

Optimal. Leaf size=182 \[ \frac{3 a^2 b x^{n+1} (d x)^m}{m+n+1}+\frac{a^3 (d x)^{m+1}}{d (m+1)}+\frac{3 a x^{2 n+1} \left (a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac{b x^{3 n+1} \left (6 a c+b^2\right ) (d x)^m}{m+3 n+1}+\frac{3 c x^{4 n+1} \left (a c+b^2\right ) (d x)^m}{m+4 n+1}+\frac{3 b c^2 x^{5 n+1} (d x)^m}{m+5 n+1}+\frac{c^3 x^{6 n+1} (d x)^m}{m+6 n+1} \]

[Out]

(3*a^2*b*x^(1 + n)*(d*x)^m)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(1 + 2*n)*(d*x)^m)/(1 + m + 2*n) + (b*(b^2 + 6*a*
c)*x^(1 + 3*n)*(d*x)^m)/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(1 + 4*n)*(d*x)^m)/(1 + m + 4*n) + (3*b*c^2*x^(1 +
5*n)*(d*x)^m)/(1 + m + 5*n) + (c^3*x^(1 + 6*n)*(d*x)^m)/(1 + m + 6*n) + (a^3*(d*x)^(1 + m))/(d*(1 + m))

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Rubi [A]  time = 0.155761, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1353, 20, 30} \[ \frac{3 a^2 b x^{n+1} (d x)^m}{m+n+1}+\frac{a^3 (d x)^{m+1}}{d (m+1)}+\frac{3 a x^{2 n+1} \left (a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac{b x^{3 n+1} \left (6 a c+b^2\right ) (d x)^m}{m+3 n+1}+\frac{3 c x^{4 n+1} \left (a c+b^2\right ) (d x)^m}{m+4 n+1}+\frac{3 b c^2 x^{5 n+1} (d x)^m}{m+5 n+1}+\frac{c^3 x^{6 n+1} (d x)^m}{m+6 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^3,x]

[Out]

(3*a^2*b*x^(1 + n)*(d*x)^m)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(1 + 2*n)*(d*x)^m)/(1 + m + 2*n) + (b*(b^2 + 6*a*
c)*x^(1 + 3*n)*(d*x)^m)/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(1 + 4*n)*(d*x)^m)/(1 + m + 4*n) + (3*b*c^2*x^(1 +
5*n)*(d*x)^m)/(1 + m + 5*n) + (c^3*x^(1 + 6*n)*(d*x)^m)/(1 + m + 6*n) + (a^3*(d*x)^(1 + m))/(d*(1 + m))

Rule 1353

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d
*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] &&  !Int
egerQ[Simplify[(m + 1)/n]]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.32275, size = 137, normalized size = 0.75 \[ x (d x)^m \left (\frac{3 a^2 b x^n}{m+n+1}+\frac{a^3}{m+1}+\frac{3 a x^{2 n} \left (a c+b^2\right )}{m+2 n+1}+\frac{b x^{3 n} \left (6 a c+b^2\right )}{m+3 n+1}+\frac{3 c x^{4 n} \left (a c+b^2\right )}{m+4 n+1}+\frac{3 b c^2 x^{5 n}}{m+5 n+1}+\frac{c^3 x^{6 n}}{m+6 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n))^3,x]

[Out]

x*(d*x)^m*(a^3/(1 + m) + (3*a^2*b*x^n)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(2*n))/(1 + m + 2*n) + (b*(b^2 + 6*a*c
)*x^(3*n))/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(4*n))/(1 + m + 4*n) + (3*b*c^2*x^(5*n))/(1 + m + 5*n) + (c^3*x^
(6*n))/(1 + m + 6*n))

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Maple [C]  time = 0.105, size = 3798, normalized size = 20.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*x^n+c*x^(2*n))^3,x)

[Out]

x*(3*a^2*c*(x^n)^2+a^3+972*b*c^2*n^4*(x^n)^5+150*c^3*m^2*n*(x^n)^6+340*c^3*m*n^2*(x^n)^6+675*c^3*m*n^3*(x^n)^6
+18*a*c^2*m^5*(x^n)^4+540*a*c^2*n^5*(x^n)^4+18*b^3*m^5*n*(x^n)^3+121*b^3*m^4*n^2*(x^n)^3+372*b^3*m^3*n^3*(x^n)
^3+508*b^3*m^2*n^4*(x^n)^3+240*b^3*m*n^5*(x^n)^3+18*b^2*c*m^5*(x^n)^4+540*b^2*c*n^5*(x^n)^4+45*b*c^2*m^4*(x^n)
^5+85*c^3*m^4*n^2*(x^n)^6+225*c^3*m^3*n^3*(x^n)^6+274*c^3*m^2*n^4*(x^n)^6+120*c^3*m*n^5*(x^n)^6+3*b*c^2*m^6*(x
^n)^5+75*c^3*m^4*n*(x^n)^6+340*c^3*m^3*n^2*(x^n)^6+675*c^3*m^2*n^3*(x^n)^6+548*c^3*m*n^4*(x^n)^6+3*a*c^2*m^6*(
x^n)^4+3*b^2*c*m^6*(x^n)^4+18*b*c^2*m^5*(x^n)^5+432*b*c^2*n^5*(x^n)^5+150*c^3*m^3*n*(x^n)^6+510*c^3*m^2*n^2*(x
^n)^6+2160*a^2*b*n^5*x^n+45*a^2*c*m^4*(x^n)^2+2106*a^2*c*n^4*(x^n)^2+45*a*b^2*m^4*(x^n)^2+2106*a*b^2*n^4*(x^n)
^2+45*a*c^2*m^2*(x^n)^4+321*a*c^2*n^2*(x^n)^4+45*b^2*c*m^2*(x^n)^4+321*b^2*c*n^2*(x^n)^4+18*m*b*c^2*(x^n)^5+48
*b*c^2*(x^n)^5*n+45*a^2*b*m^4*x^n+3132*a^2*b*n^4*x^n+60*a^2*c*m^3*(x^n)^2+15*c^3*m^5*n*(x^n)^6+1080*a^2*c*n^5*
(x^n)^2+18*a*b^2*m^5*(x^n)^2+1080*a*b^2*n^5*(x^n)^2+60*a*c^2*m^3*(x^n)^4+921*a*c^2*n^3*(x^n)^4+180*b^3*m^3*n*(
x^n)^3+726*b^3*m^2*n^2*(x^n)^3+1116*b^3*m*n^3*(x^n)^3+60*b^2*c*m^3*(x^n)^4+921*b^2*c*n^3*(x^n)^4+45*b*c^2*m^2*
(x^n)^5+285*b*c^2*n^2*(x^n)^5+18*a^2*b*m^5*x^n+6*a*b*c*(x^n)^3+3*a^2*c*m^6*(x^n)^2+3*a*b^2*m^6*(x^n)^2+45*a*c^
2*m^4*(x^n)^4+1188*a*c^2*n^4*(x^n)^4+90*b^3*m^4*n*(x^n)^3+484*b^3*m^3*n^2*(x^n)^3+1116*b^3*m^2*n^3*(x^n)^3+101
6*b^3*m*n^4*(x^n)^3+45*b^2*c*m^4*(x^n)^4+1188*b^2*c*n^4*(x^n)^4+60*b*c^2*m^3*(x^n)^5+780*b*c^2*n^3*(x^n)^5+75*
c^3*m*n*(x^n)^6+3*a^2*b*m^6*x^n+18*a^2*c*m^5*(x^n)^2+1383*a^2*c*n^3*(x^n)^2+1383*a*b^2*n^3*(x^n)^2+18*a*c^2*(x
^n)^4*m+51*a*c^2*(x^n)^4*n+18*b^2*c*(x^n)^4*m+51*b^2*c*(x^n)^4*n+1740*a^2*b*n^3*x^n+45*a^2*c*m^2*(x^n)^2+411*a
^2*c*n^2*(x^n)^2+18*a^2*c*(x^n)^2*m+57*a^2*c*(x^n)^2*n+180*b^3*m^2*n*(x^n)^3+484*b^3*m*n^2*(x^n)^3+60*a*b^2*m^
3*(x^n)^2+90*b^3*m*n*(x^n)^3+60*a^2*b*m^3*x^n+45*a*b^2*m^2*(x^n)^2+411*a*b^2*n^2*(x^n)^2+45*a^2*b*m^2*x^n+465*
a^2*b*n^2*x^n+18*m*a*b^2*(x^n)^2+57*a*b^2*(x^n)^2*n+18*m*a^2*b*x^n+60*a^2*b*n*x^n+c^3*(x^n)^6+a^3*m^6+6*a^3*m^
5+1764*a^3*n^5+15*a^3*m^4+1624*a^3*n^4+720*a^3*n^6+b^3*(x^n)^3+20*a^3*m^3+15*a^3*m^2+175*a^3*n^2+21*a^3*n+735*
a^3*n^3+540*a*b*c*m^4*n*(x^n)^3+2904*a*b*c*m^3*n^2*(x^n)^3+6696*a*b*c*m^2*n^3*(x^n)^3+4356*a*b*c*m^2*n^2*(x^n)
^3+6696*a*b*c*m*n^3*(x^n)^3+1080*a*b*c*m^2*n*(x^n)^3+2904*a*b*c*m*n^2*(x^n)^3+540*a*b*c*m*n*(x^n)^3+108*a*b*c*
m^5*n*(x^n)^3+726*a*b*c*m^4*n^2*(x^n)^3+2232*a*b*c*m^3*n^3*(x^n)^3+3048*a*b*c*m^2*n^4*(x^n)^3+1440*a*b*c*m*n^5
*(x^n)^3+6096*a*b*c*m*n^4*(x^n)^3+1080*a*b*c*m^3*n*(x^n)^3+600*a^2*b*m^3*n*x^n+2790*a^2*b*m^2*n^2*x^n+5220*a^2
*b*m*n^3*x^n+570*a^2*c*m^2*n*(x^n)^2+1644*a^2*c*m*n^2*(x^n)^2+90*a*b*c*m^2*(x^n)^3+726*a*b*c*n^2*(x^n)^3+285*a
^2*c*m*n*(x^n)^2+36*a*b*c*(x^n)^3*m+108*a*b*c*(x^n)^3*n+1284*b^2*c*m*n^2*(x^n)^4+240*b*c^2*m*n*(x^n)^5+300*a^2
*b*m^4*n*x^n+1860*a^2*b*m^3*n^2*x^n+5220*a^2*b*m^2*n^3*x^n+6264*a^2*b*m*n^4*x^n+570*a^2*c*m^3*n*(x^n)^2+2466*a
^2*c*m^2*n^2*(x^n)^2+4149*a^2*c*m*n^3*(x^n)^2+570*a*b^2*m^3*n*(x^n)^2+2466*a*b^2*m^2*n^2*(x^n)^2+4149*a*b^2*m*
n^3*(x^n)^2+120*a*b*c*m^3*(x^n)^3+2232*a*b*c*n^3*(x^n)^3+255*a*c^2*m*n*(x^n)^4+255*b^2*c*m*n*(x^n)^4+3132*a^2*
b*m^2*n^4*x^n+2160*a^2*b*m*n^5*x^n+285*a^2*c*m^4*n*(x^n)^2+1644*a^2*c*m^3*n^2*(x^n)^2+4149*a^2*c*m^2*n^3*(x^n)
^2+4212*a^2*c*m*n^4*(x^n)^2+285*a*b^2*m^4*n*(x^n)^2+1644*a*b^2*m^3*n^2*(x^n)^2+4149*a*b^2*m^2*n^3*(x^n)^2+4212
*a*b^2*m*n^4*(x^n)^2+90*a*b*c*m^4*(x^n)^3+3048*a*b*c*n^4*(x^n)^3+510*a*c^2*m^2*n*(x^n)^4+1284*a*c^2*m*n^2*(x^n
)^4+510*b^2*c*m^2*n*(x^n)^4+2106*a*b^2*m^2*n^4*(x^n)^2+1080*a*b^2*m*n^5*(x^n)^2+36*a*b*c*m^5*(x^n)^3+1440*a*b*
c*n^5*(x^n)^3+510*a*c^2*m^3*n*(x^n)^4+1926*a*c^2*m^2*n^2*(x^n)^4+2763*a*c^2*m*n^3*(x^n)^4+510*b^2*c*m^3*n*(x^n
)^4+1926*b^2*c*m^2*n^2*(x^n)^4+2763*b^2*c*m*n^3*(x^n)^4+480*b*c^2*m^2*n*(x^n)^5+1140*b*c^2*m*n^2*(x^n)^5+60*a^
2*b*m^5*n*x^n+465*a^2*b*m^4*n^2*x^n+1740*a^2*b*m^3*n^3*x^n+2376*a*c^2*m*n^4*(x^n)^4+255*b^2*c*m^4*n*(x^n)^4+12
84*b^2*c*m^3*n^2*(x^n)^4+2763*b^2*c*m^2*n^3*(x^n)^4+2376*b^2*c*m*n^4*(x^n)^4+480*b*c^2*m^3*n*(x^n)^5+1710*b*c^
2*m^2*n^2*(x^n)^5+2340*b*c^2*m*n^3*(x^n)^5+57*a^2*c*m^5*n*(x^n)^2+411*a^2*c*m^4*n^2*(x^n)^2+1383*a^2*c*m^3*n^3
*(x^n)^2+2106*a^2*c*m^2*n^4*(x^n)^2+1080*a^2*c*m*n^5*(x^n)^2+57*a*b^2*m^5*n*(x^n)^2+411*a*b^2*m^4*n^2*(x^n)^2+
1383*a*b^2*m^3*n^3*(x^n)^2+1188*a*c^2*m^2*n^4*(x^n)^4+540*a*c^2*m*n^5*(x^n)^4+51*b^2*c*m^5*n*(x^n)^4+321*b^2*c
*m^4*n^2*(x^n)^4+921*b^2*c*m^3*n^3*(x^n)^4+1188*b^2*c*m^2*n^4*(x^n)^4+540*b^2*c*m*n^5*(x^n)^4+240*b*c^2*m^4*n*
(x^n)^5+1140*b*c^2*m^3*n^2*(x^n)^5+2340*b*c^2*m^2*n^3*(x^n)^5+1944*b*c^2*m*n^4*(x^n)^5+6*a*b*c*m^6*(x^n)^3+255
*a*c^2*m^4*n*(x^n)^4+1284*a*c^2*m^3*n^2*(x^n)^4+2763*a*c^2*m^2*n^3*(x^n)^4+48*b*c^2*m^5*n*(x^n)^5+285*b*c^2*m^
4*n^2*(x^n)^5+780*b*c^2*m^3*n^3*(x^n)^5+972*b*c^2*m^2*n^4*(x^n)^5+432*b*c^2*m*n^5*(x^n)^5+51*a*c^2*m^5*n*(x^n)
^4+321*a*c^2*m^4*n^2*(x^n)^4+921*a*c^2*m^3*n^3*(x^n)^4+6*m*a^3+210*a^3*m^2*n+700*a^3*m*n^2+105*a^3*m*n+21*a^3*
m^5*n+175*a^3*m^4*n^2+735*a^3*m^3*n^3+1624*a^3*m^2*n^4+1764*a^3*m*n^5+105*a^3*m^4*n+700*a^3*m^3*n^2+2205*a^3*m
^2*n^3+3248*a^3*m*n^4+210*a^3*m^3*n+1050*a^3*m^2*n^2+2205*a^3*m*n^3+20*b^3*m^3*(x^n)^3+15*b^3*m^2*(x^n)^3+121*
b^3*n^2*(x^n)^3+6*m*b^3*(x^n)^3+18*b^3*(x^n)^3*n+3*a^2*b*x^n+3*a*b^2*(x^n)^2+508*b^3*n^4*(x^n)^3+6*m*c^3*(x^n)
^6+3*c^2*a*(x^n)^4+3*b^2*c*(x^n)^4+3*b*c^2*(x^n)^5+6*c^3*m^5*(x^n)^6+120*c^3*n^5*(x^n)^6+15*c^3*m^4*(x^n)^6+27
4*c^3*n^4*(x^n)^6+b^3*m^6*(x^n)^3+20*c^3*m^3*(x^n)^6+225*c^3*n^3*(x^n)^6+6*b^3*m^5*(x^n)^3+240*b^3*n^5*(x^n)^3
+15*c^3*m^2*(x^n)^6+c^3*m^6*(x^n)^6+15*c^3*(x^n)^6*n+372*b^3*n^3*(x^n)^3+85*c^3*n^2*(x^n)^6+15*b^3*m^4*(x^n)^3
+570*a*b^2*m^2*n*(x^n)^2+1644*a*b^2*m*n^2*(x^n)^2+600*a^2*b*m^2*n*x^n+1860*a^2*b*m*n^2*x^n+285*a*b^2*m*n*(x^n)
^2+300*a^2*b*m*n*x^n)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)/(1+m+4*n)/(1+m+5*n)/(1+m+6*n)*exp(1/2*m*(-I*csgn(I*d*x
)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+
2*ln(d)))

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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((d*x)^m*(a+b*x^n+c*x^(2*n))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.26512, size = 5146, normalized size = 28.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*x^n+c*x^(2*n))^3,x, algorithm="fricas")

[Out]

((c^3*m^6 + 6*c^3*m^5 + 15*c^3*m^4 + 20*c^3*m^3 + 120*(c^3*m + c^3)*n^5 + 15*c^3*m^2 + 274*(c^3*m^2 + 2*c^3*m
+ c^3)*n^4 + 6*c^3*m + 225*(c^3*m^3 + 3*c^3*m^2 + 3*c^3*m + c^3)*n^3 + c^3 + 85*(c^3*m^4 + 4*c^3*m^3 + 6*c^3*m
^2 + 4*c^3*m + c^3)*n^2 + 15*(c^3*m^5 + 5*c^3*m^4 + 10*c^3*m^3 + 10*c^3*m^2 + 5*c^3*m + c^3)*n)*x*x^(6*n)*e^(m
*log(d) + m*log(x)) + 3*(b*c^2*m^6 + 6*b*c^2*m^5 + 15*b*c^2*m^4 + 20*b*c^2*m^3 + 144*(b*c^2*m + b*c^2)*n^5 + 1
5*b*c^2*m^2 + 324*(b*c^2*m^2 + 2*b*c^2*m + b*c^2)*n^4 + 6*b*c^2*m + 260*(b*c^2*m^3 + 3*b*c^2*m^2 + 3*b*c^2*m +
b*c^2)*n^3 + b*c^2 + 95*(b*c^2*m^4 + 4*b*c^2*m^3 + 6*b*c^2*m^2 + 4*b*c^2*m + b*c^2)*n^2 + 16*(b*c^2*m^5 + 5*b
*c^2*m^4 + 10*b*c^2*m^3 + 10*b*c^2*m^2 + 5*b*c^2*m + b*c^2)*n)*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 3*((b^2*c +
a*c^2)*m^6 + 6*(b^2*c + a*c^2)*m^5 + 180*(b^2*c + a*c^2 + (b^2*c + a*c^2)*m)*n^5 + 15*(b^2*c + a*c^2)*m^4 + 3
96*(b^2*c + a*c^2 + (b^2*c + a*c^2)*m^2 + 2*(b^2*c + a*c^2)*m)*n^4 + 20*(b^2*c + a*c^2)*m^3 + 307*((b^2*c + a*
c^2)*m^3 + b^2*c + a*c^2 + 3*(b^2*c + a*c^2)*m^2 + 3*(b^2*c + a*c^2)*m)*n^3 + b^2*c + a*c^2 + 15*(b^2*c + a*c^
2)*m^2 + 107*((b^2*c + a*c^2)*m^4 + 4*(b^2*c + a*c^2)*m^3 + b^2*c + a*c^2 + 6*(b^2*c + a*c^2)*m^2 + 4*(b^2*c +
a*c^2)*m)*n^2 + 6*(b^2*c + a*c^2)*m + 17*((b^2*c + a*c^2)*m^5 + 5*(b^2*c + a*c^2)*m^4 + 10*(b^2*c + a*c^2)*m^
3 + b^2*c + a*c^2 + 10*(b^2*c + a*c^2)*m^2 + 5*(b^2*c + a*c^2)*m)*n)*x*x^(4*n)*e^(m*log(d) + m*log(x)) + ((b^3
+ 6*a*b*c)*m^6 + 6*(b^3 + 6*a*b*c)*m^5 + 240*(b^3 + 6*a*b*c + (b^3 + 6*a*b*c)*m)*n^5 + 15*(b^3 + 6*a*b*c)*m^4
+ 508*(b^3 + 6*a*b*c + (b^3 + 6*a*b*c)*m^2 + 2*(b^3 + 6*a*b*c)*m)*n^4 + 20*(b^3 + 6*a*b*c)*m^3 + 372*((b^3 +
6*a*b*c)*m^3 + b^3 + 6*a*b*c + 3*(b^3 + 6*a*b*c)*m^2 + 3*(b^3 + 6*a*b*c)*m)*n^3 + b^3 + 6*a*b*c + 15*(b^3 + 6*
a*b*c)*m^2 + 121*((b^3 + 6*a*b*c)*m^4 + 4*(b^3 + 6*a*b*c)*m^3 + b^3 + 6*a*b*c + 6*(b^3 + 6*a*b*c)*m^2 + 4*(b^3
+ 6*a*b*c)*m)*n^2 + 6*(b^3 + 6*a*b*c)*m + 18*((b^3 + 6*a*b*c)*m^5 + 5*(b^3 + 6*a*b*c)*m^4 + 10*(b^3 + 6*a*b*c
)*m^3 + b^3 + 6*a*b*c + 10*(b^3 + 6*a*b*c)*m^2 + 5*(b^3 + 6*a*b*c)*m)*n)*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 3
*((a*b^2 + a^2*c)*m^6 + 6*(a*b^2 + a^2*c)*m^5 + 360*(a*b^2 + a^2*c + (a*b^2 + a^2*c)*m)*n^5 + 15*(a*b^2 + a^2*
c)*m^4 + 702*(a*b^2 + a^2*c + (a*b^2 + a^2*c)*m^2 + 2*(a*b^2 + a^2*c)*m)*n^4 + 20*(a*b^2 + a^2*c)*m^3 + 461*((
a*b^2 + a^2*c)*m^3 + a*b^2 + a^2*c + 3*(a*b^2 + a^2*c)*m^2 + 3*(a*b^2 + a^2*c)*m)*n^3 + a*b^2 + a^2*c + 15*(a*
b^2 + a^2*c)*m^2 + 137*((a*b^2 + a^2*c)*m^4 + 4*(a*b^2 + a^2*c)*m^3 + a*b^2 + a^2*c + 6*(a*b^2 + a^2*c)*m^2 +
4*(a*b^2 + a^2*c)*m)*n^2 + 6*(a*b^2 + a^2*c)*m + 19*((a*b^2 + a^2*c)*m^5 + 5*(a*b^2 + a^2*c)*m^4 + 10*(a*b^2 +
a^2*c)*m^3 + a*b^2 + a^2*c + 10*(a*b^2 + a^2*c)*m^2 + 5*(a*b^2 + a^2*c)*m)*n)*x*x^(2*n)*e^(m*log(d) + m*log(x
)) + 3*(a^2*b*m^6 + 6*a^2*b*m^5 + 15*a^2*b*m^4 + 20*a^2*b*m^3 + 720*(a^2*b*m + a^2*b)*n^5 + 15*a^2*b*m^2 + 104
4*(a^2*b*m^2 + 2*a^2*b*m + a^2*b)*n^4 + 6*a^2*b*m + 580*(a^2*b*m^3 + 3*a^2*b*m^2 + 3*a^2*b*m + a^2*b)*n^3 + a^
2*b + 155*(a^2*b*m^4 + 4*a^2*b*m^3 + 6*a^2*b*m^2 + 4*a^2*b*m + a^2*b)*n^2 + 20*(a^2*b*m^5 + 5*a^2*b*m^4 + 10*a
^2*b*m^3 + 10*a^2*b*m^2 + 5*a^2*b*m + a^2*b)*n)*x*x^n*e^(m*log(d) + m*log(x)) + (a^3*m^6 + 720*a^3*n^6 + 6*a^3
*m^5 + 15*a^3*m^4 + 20*a^3*m^3 + 1764*(a^3*m + a^3)*n^5 + 15*a^3*m^2 + 1624*(a^3*m^2 + 2*a^3*m + a^3)*n^4 + 6*
a^3*m + 735*(a^3*m^3 + 3*a^3*m^2 + 3*a^3*m + a^3)*n^3 + a^3 + 175*(a^3*m^4 + 4*a^3*m^3 + 6*a^3*m^2 + 4*a^3*m +
a^3)*n^2 + 21*(a^3*m^5 + 5*a^3*m^4 + 10*a^3*m^3 + 10*a^3*m^2 + 5*a^3*m + a^3)*n)*x*e^(m*log(d) + m*log(x)))/(
m^7 + 720*(m + 1)*n^6 + 7*m^6 + 1764*(m^2 + 2*m + 1)*n^5 + 21*m^5 + 1624*(m^3 + 3*m^2 + 3*m + 1)*n^4 + 35*m^4
+ 735*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*n^3 + 35*m^3 + 175*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*n^2 + 21*m^
2 + 21*(m^6 + 6*m^5 + 15*m^4 + 20*m^3 + 15*m^2 + 6*m + 1)*n + 7*m + 1)

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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((d*x)**m*(a+b*x**n+c*x**(2*n))**3,x)

[Out]

Timed out

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Giac [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((d*x)^m*(a+b*x^n+c*x^(2*n))^3,x, algorithm="giac")

[Out]

Timed out