3.6.96 \(\int (d x)^m (a+b x^n+c x^{2 n})^3 \, dx\) [596]

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

[Out]

3*a^2*b*x^(1+n)*(d*x)^m/(1+m+n)+3*a*(a*c+b^2)*x^(1+2*n)*(d*x)^m/(1+m+2*n)+b*(6*a*c+b^2)*x^(1+3*n)*(d*x)^m/(1+m
+3*n)+3*c*(a*c+b^2)*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 = 0.11, antiderivative size = 182, normalized size of antiderivative = 1.00, 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 = {1367, 20, 30} \begin {gather*} \frac {a^3 (d x)^{m+1}}{d (m+1)}+\frac {3 a^2 b x^{n+1} (d x)^m}{m+n+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} \end {gather*}

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 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]

Rule 1367

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]]

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*}

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Mathematica [A]
time = 0.56, size = 137, normalized size = 0.75 \begin {gather*} x (d x)^m \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^n}{1+m+n}+\frac {3 a \left (b^2+a c\right ) x^{2 n}}{1+m+2 n}+\frac {b \left (b^2+6 a c\right ) x^{3 n}}{1+m+3 n}+\frac {3 c \left (b^2+a c\right ) x^{4 n}}{1+m+4 n}+\frac {3 b c^2 x^{5 n}}{1+m+5 n}+\frac {c^3 x^{6 n}}{1+m+6 n}\right ) \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.08, size = 3798, normalized size = 20.87

method	result	size
risch	\(\text {Expression too large to display}\)	\(3798\)

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,method=_RETURNVERBOSE)

[Out]

x*(2904*a*b*c*m*n^2*(x^n)^3+b^3*(x^n)^3+726*b^3*m^2*n^2*(x^n)^3+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+2106*a*b^2*m^2*n^4*(x^n)^2+c^3*m^6*(x^n)^6+6*c^3*m^5*(x
^n)^6+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+5
1*b^2*c*(x^n)^4*n+1740*a^2*b*n^3*x^n+45*a^2*c*m^2*(x^n)^2+508*b^3*n^4*(x^n)^3+240*b^3*n^5*(x^n)^3+15*c^3*m^2*(
x^n)^6+85*c^3*n^2*(x^n)^6+15*b^3*m^4*(x^n)^3+340*c^3*m*n^2*(x^n)^6+3*a^2*c*m^6*(x^n)^2+3*a*b^2*m^6*(x^n)^2+45*
a^2*b*m^2*x^n+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+1016*b^3*m*n^4*(x^n)^3+a^3+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+176
4*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+60*a^2*b*m^3*x^n+15*c^3*m^5*n*(x^n)^
6+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+180*b^3*m^2*n*
(x^n)^3+a^3*m^6+6*a^3*m^5+1764*a^3*n^5+15*a^3*m^4+1624*a^3*n^4+15*c^3*m^4*(x^n)^6+274*c^3*n^4*(x^n)^6+b^3*m^6*
(x^n)^3+540*a*c^2*n^5*(x^n)^4+720*a^3*n^6+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+48*b*c^2*m^5*n*(x^n)^5+6*
m*c^3*(x^n)^6+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+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+225*c^3*n^3*(x^n)^6+18*b^3*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+6*a*b*c*(x^n)^3+20*c^3*m^3*(x^n)^6+18*m*a^2*b*x^n+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+60*a^2*b*n*x^n+3132*a^2*b*n^4*x^n+60*a^2*c*m^3*(x^n)^2+180*b^3*m^3*n*(x^n)^3+c^3*
(x^n)^6+120*c^3*n^5*(x^n)^6+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+20*a^3*m^3+15*a^3*m^2+175*a^3*n^2+21*a^3*n+3*(x^n)^5*b*c^2+3*(x^n)^4*c^2*a+210*a^3*m^3*n+1050*a^3*m^2
*n^2+2205*a^3*m*n^3+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+6696*a*b*c*m^2*n^3*(x^
n)^3+6096*a*b*c*m*n^4*(x^n)^3+1080*a*b*c*m^3*n*(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+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+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+285*a*b^2*m*n*(x^n)^2+3*(x^n)^2*a*b^2+540*a*b*c*m*n*(x^n)^3+6*m*a^3+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+675*c^3*m*n^3*(x^n)^6+18*a*c^2*m^5*(x^n)^4+45*a*b^2*m^2*(x^n)^2+411*a*b^2*n^2*(x^n)^2+18*m*a*b^2*(x^n)^2+4
356*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+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+972*b*c^2*n^4*(x^n)^5+150*c^3*m^2*n*(x^n)^6+210*a^3*m^2*n+700*a^3*m*n^2+105*a^3*m*n+6*b^3*m^5*(x
^n)^3+57*a*b^2*(x^n)^2*n+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+3
048*a*b*c*m^2*n^4*(x^n)^3+1440*a*b*c*m*n^5*(x^n)^3+540*a*b*c*m^4*n*(x^n)^3+2904*a*b*c*m^3*n^2*(x^n)^3+108*a*b*
c*m^5*n*(x^n)^3+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+3132*a^2*b*m^2*n^4*x^n+2160*a^2*b*m*n^5*x^n+15*c^3*(x^n)^6*n+372*b^3*n^3*
(x^n)^3+3*a^2*b*x^n+3*(x^n)^4*b^2*c+3*(x^n)^2*a^2*c+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+465*a^2*b*n^2*x^n+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+735*a^3
*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+1080*a*b^2*m*n^5*(x^n)^2+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+1
644*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+1860*a^2*b*m*n^2*x^n+90*a*b*c*m^
4*(x^n)^3+3048*a*b*c*n^4*(x^n)^3+300*a^2*b*m*n*x^n+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+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*...

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Maxima [A]
time = 0.30, size = 273, normalized size = 1.50 \begin {gather*} \frac {c^{3} d^{m} x e^{\left (m \log \left (x\right ) + 6 \, n \log \left (x\right )\right )}}{m + 6 \, n + 1} + \frac {3 \, b c^{2} d^{m} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right )\right )}}{m + 5 \, n + 1} + \frac {3 \, b^{2} c d^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, a c^{2} d^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {b^{3} d^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {6 \, a b c d^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, a b^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, a^{2} c d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, a^{2} b d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2303 vs. \(2 (182) = 364\).
time = 0.50, size = 2303, normalized size = 12.65 \begin {gather*} \text {Too large to display} \end {gather*}

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)

________________________________________________________________________________________

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25656 vs. \(2 (182) = 364\).
time = 3.75, size = 25656, normalized size = 140.97 \begin {gather*} \text {Too large to display} \end {gather*}

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]

(c^3*m^6*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 15*c^3*m^5*n*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 85*c^3*m^4*n^2*x
*x^(6*n)*e^(m*log(d) + m*log(x)) + 225*c^3*m^3*n^3*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 274*c^3*m^2*n^4*x*x^(6*
n)*e^(m*log(d) + m*log(x)) + 120*c^3*m*n^5*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 3*b*c^2*m^6*x*x^(5*n)*e^(m*log(
d) + m*log(x)) + c^3*m^6*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 48*b*c^2*m^5*n*x*x^(5*n)*e^(m*log(d) + m*log(x))
+ 15*c^3*m^5*n*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 285*b*c^2*m^4*n^2*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 85*c^
3*m^4*n^2*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 780*b*c^2*m^3*n^3*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 225*c^3*m^
3*n^3*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 972*b*c^2*m^2*n^4*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 274*c^3*m^2*n^
4*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 432*b*c^2*m*n^5*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 120*c^3*m*n^5*x*x^(5
*n)*e^(m*log(d) + m*log(x)) + 3*b^2*c*m^6*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 3*a*c^2*m^6*x*x^(4*n)*e^(m*log(d
) + m*log(x)) + 3*b*c^2*m^6*x*x^(4*n)*e^(m*log(d) + m*log(x)) + c^3*m^6*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 51
*b^2*c*m^5*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 51*a*c^2*m^5*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 48*b*c^2*m
^5*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 15*c^3*m^5*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 321*b^2*c*m^4*n^2*x*
x^(4*n)*e^(m*log(d) + m*log(x)) + 321*a*c^2*m^4*n^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 285*b*c^2*m^4*n^2*x*x^
(4*n)*e^(m*log(d) + m*log(x)) + 85*c^3*m^4*n^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 921*b^2*c*m^3*n^3*x*x^(4*n)
*e^(m*log(d) + m*log(x)) + 921*a*c^2*m^3*n^3*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 780*b*c^2*m^3*n^3*x*x^(4*n)*e
^(m*log(d) + m*log(x)) + 225*c^3*m^3*n^3*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 1188*b^2*c*m^2*n^4*x*x^(4*n)*e^(m
*log(d) + m*log(x)) + 1188*a*c^2*m^2*n^4*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 972*b*c^2*m^2*n^4*x*x^(4*n)*e^(m*
log(d) + m*log(x)) + 274*c^3*m^2*n^4*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 540*b^2*c*m*n^5*x*x^(4*n)*e^(m*log(d)
+ m*log(x)) + 540*a*c^2*m*n^5*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 432*b*c^2*m*n^5*x*x^(4*n)*e^(m*log(d) + m*l
og(x)) + 120*c^3*m*n^5*x*x^(4*n)*e^(m*log(d) + m*log(x)) + b^3*m^6*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 6*a*b*c
*m^6*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 3*b^2*c*m^6*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 3*a*c^2*m^6*x*x^(3*n)
*e^(m*log(d) + m*log(x)) + 3*b*c^2*m^6*x*x^(3*n)*e^(m*log(d) + m*log(x)) + c^3*m^6*x*x^(3*n)*e^(m*log(d) + m*l
og(x)) + 18*b^3*m^5*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 108*a*b*c*m^5*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) +
51*b^2*c*m^5*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 51*a*c^2*m^5*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 48*b*c^2
*m^5*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 15*c^3*m^5*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 121*b^3*m^4*n^2*x*
x^(3*n)*e^(m*log(d) + m*log(x)) + 726*a*b*c*m^4*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 321*b^2*c*m^4*n^2*x*x^
(3*n)*e^(m*log(d) + m*log(x)) + 321*a*c^2*m^4*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 285*b*c^2*m^4*n^2*x*x^(3
*n)*e^(m*log(d) + m*log(x)) + 85*c^3*m^4*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 372*b^3*m^3*n^3*x*x^(3*n)*e^(
m*log(d) + m*log(x)) + 2232*a*b*c*m^3*n^3*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 921*b^2*c*m^3*n^3*x*x^(3*n)*e^(m
*log(d) + m*log(x)) + 921*a*c^2*m^3*n^3*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 780*b*c^2*m^3*n^3*x*x^(3*n)*e^(m*l
og(d) + m*log(x)) + 225*c^3*m^3*n^3*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 508*b^3*m^2*n^4*x*x^(3*n)*e^(m*log(d)
+ m*log(x)) + 3048*a*b*c*m^2*n^4*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 1188*b^2*c*m^2*n^4*x*x^(3*n)*e^(m*log(d)
+ m*log(x)) + 1188*a*c^2*m^2*n^4*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 972*b*c^2*m^2*n^4*x*x^(3*n)*e^(m*log(d) +
m*log(x)) + 274*c^3*m^2*n^4*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 240*b^3*m*n^5*x*x^(3*n)*e^(m*log(d) + m*log(x
)) + 1440*a*b*c*m*n^5*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 540*b^2*c*m*n^5*x*x^(3*n)*e^(m*log(d) + m*log(x)) +
540*a*c^2*m*n^5*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 432*b*c^2*m*n^5*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 120*c^
3*m*n^5*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 3*a*b^2*m^6*x*x^(2*n)*e^(m*log(d) + m*log(x)) + b^3*m^6*x*x^(2*n)*
e^(m*log(d) + m*log(x)) + 3*a^2*c*m^6*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 6*a*b*c*m^6*x*x^(2*n)*e^(m*log(d) +
m*log(x)) + 3*b^2*c*m^6*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 3*a*c^2*m^6*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 3*
b*c^2*m^6*x*x^(2*n)*e^(m*log(d) + m*log(x)) + c^3*m^6*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 57*a*b^2*m^5*n*x*x^(
2*n)*e^(m*log(d) + m*log(x)) + 18*b^3*m^5*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 57*a^2*c*m^5*n*x*x^(2*n)*e^(m*
log(d) + m*log(x)) + 108*a*b*c*m^5*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 51*b^2*c*m^5*n*x*x^(2*n)*e^(m*log(d)
+ m*log(x)) + 51*a*c^2*m^5*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 48*b*c^2*m^5*n*x*x^(2*n)*e^(m*log(d) + m*log(
x)) + 15*c^3*m^5*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 411*a*b^2*m^4*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 1
21*b^3*m^4*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 411*a^2*c*m^4*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 726*a
*b*c*m^4*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) ...

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Mupad [B]
time = 2.16, size = 1734, normalized size = 9.53 \begin {gather*} \frac {a^3\,x\,{\left (d\,x\right )}^m}{m+1}+\frac {c^3\,x\,x^{6\,n}\,{\left (d\,x\right )}^m\,\left (m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {3\,a\,x\,x^{2\,n}\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^5+19\,m^4\,n+5\,m^4+137\,m^3\,n^2+76\,m^3\,n+10\,m^3+461\,m^2\,n^3+411\,m^2\,n^2+114\,m^2\,n+10\,m^2+702\,m\,n^4+922\,m\,n^3+411\,m\,n^2+76\,m\,n+5\,m+360\,n^5+702\,n^4+461\,n^3+137\,n^2+19\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {b\,x\,x^{3\,n}\,{\left (d\,x\right )}^m\,\left (b^2+6\,a\,c\right )\,\left (m^5+18\,m^4\,n+5\,m^4+121\,m^3\,n^2+72\,m^3\,n+10\,m^3+372\,m^2\,n^3+363\,m^2\,n^2+108\,m^2\,n+10\,m^2+508\,m\,n^4+744\,m\,n^3+363\,m\,n^2+72\,m\,n+5\,m+240\,n^5+508\,n^4+372\,n^3+121\,n^2+18\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {3\,c\,x\,x^{4\,n}\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^5+17\,m^4\,n+5\,m^4+107\,m^3\,n^2+68\,m^3\,n+10\,m^3+307\,m^2\,n^3+321\,m^2\,n^2+102\,m^2\,n+10\,m^2+396\,m\,n^4+614\,m\,n^3+321\,m\,n^2+68\,m\,n+5\,m+180\,n^5+396\,n^4+307\,n^3+107\,n^2+17\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {3\,a^2\,b\,x\,x^n\,{\left (d\,x\right )}^m\,\left (m^5+20\,m^4\,n+5\,m^4+155\,m^3\,n^2+80\,m^3\,n+10\,m^3+580\,m^2\,n^3+465\,m^2\,n^2+120\,m^2\,n+10\,m^2+1044\,m\,n^4+1160\,m\,n^3+465\,m\,n^2+80\,m\,n+5\,m+720\,n^5+1044\,n^4+580\,n^3+155\,n^2+20\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {3\,b\,c^2\,x\,x^{5\,n}\,{\left (d\,x\right )}^m\,\left (m^5+16\,m^4\,n+5\,m^4+95\,m^3\,n^2+64\,m^3\,n+10\,m^3+260\,m^2\,n^3+285\,m^2\,n^2+96\,m^2\,n+10\,m^2+324\,m\,n^4+520\,m\,n^3+285\,m\,n^2+64\,m\,n+5\,m+144\,n^5+324\,n^4+260\,n^3+95\,n^2+16\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1} \end {gather*}

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]

(a^3*x*(d*x)^m)/(m + 1) + (c^3*x*x^(6*n)*(d*x)^m*(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*
m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*
n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n +
3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 +
1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m
^4*n^2 + 1) + (3*a*x*x^(2*n)*(d*x)^m*(a*c + b^2)*(5*m + 19*n + 76*m*n + 411*m*n^2 + 114*m^2*n + 922*m*n^3 + 76
*m^3*n + 702*m*n^4 + 19*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 137*n^2 + 461*n^3 + 702*n^4 + 360*n^5 + 411*m^
2*n^2 + 461*m^2*n^3 + 137*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n
+ 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3
+ 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 17
5*m^4*n^2 + 1) + (b*x*x^(3*n)*(d*x)^m*(6*a*c + b^2)*(5*m + 18*n + 72*m*n + 363*m*n^2 + 108*m^2*n + 744*m*n^3 +
72*m^3*n + 508*m*n^4 + 18*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 121*n^2 + 372*n^3 + 508*n^4 + 240*n^5 + 363
*m^2*n^2 + 372*m^2*n^3 + 121*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^
3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*
n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 +
175*m^4*n^2 + 1) + (3*c*x*x^(4*n)*(d*x)^m*(a*c + b^2)*(5*m + 17*n + 68*m*n + 321*m*n^2 + 102*m^2*n + 614*m*n^
3 + 68*m^3*n + 396*m*n^4 + 17*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 107*n^2 + 307*n^3 + 396*n^4 + 180*n^5 +
321*m^2*n^2 + 307*m^2*n^3 + 107*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210
*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 7
35*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^
3 + 175*m^4*n^2 + 1) + (3*a^2*b*x*x^n*(d*x)^m*(5*m + 20*n + 80*m*n + 465*m*n^2 + 120*m^2*n + 1160*m*n^3 + 80*m
^3*n + 1044*m*n^4 + 20*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 155*n^2 + 580*n^3 + 1044*n^4 + 720*n^5 + 465*m^
2*n^2 + 580*m^2*n^3 + 155*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n
+ 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3
+ 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 17
5*m^4*n^2 + 1) + (3*b*c^2*x*x^(5*n)*(d*x)^m*(5*m + 16*n + 64*m*n + 285*m*n^2 + 96*m^2*n + 520*m*n^3 + 64*m^3*n
+ 324*m*n^4 + 16*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 95*n^2 + 260*n^3 + 324*n^4 + 144*n^5 + 285*m^2*n^2 +
260*m^2*n^3 + 95*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n + 3248*
m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 + 1624*
n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^
2 + 1)

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