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

Optimal. Leaf size=117 \[ \frac {a^2 (d x)^{m+1}}{d (m+1)}+\frac {x^{2 n+1} \left (2 a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac {2 a b x^{n+1} (d x)^m}{m+n+1}+\frac {2 b c x^{3 n+1} (d x)^m}{m+3 n+1}+\frac {c^2 x^{4 n+1} (d x)^m}{m+4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1353, 20, 30} \[ \frac {a^2 (d x)^{m+1}}{d (m+1)}+\frac {x^{2 n+1} \left (2 a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac {2 a b x^{n+1} (d x)^m}{m+n+1}+\frac {2 b c x^{3 n+1} (d x)^m}{m+3 n+1}+\frac {c^2 x^{4 n+1} (d x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 86, normalized size = 0.74 \[ x (d x)^m \left (\frac {a^2}{m+1}+\frac {x^{2 n} \left (2 a c+b^2\right )}{m+2 n+1}+\frac {2 a b x^n}{m+n+1}+\frac {2 b c x^{3 n}}{m+3 n+1}+\frac {c^2 x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.94, size = 706, normalized size = 6.03 \[ \frac {{\left (c^{2} m^{4} + 4 \, c^{2} m^{3} + 6 \, c^{2} m^{2} + 6 \, {\left (c^{2} m + c^{2}\right )} n^{3} + 4 \, c^{2} m + 11 \, {\left (c^{2} m^{2} + 2 \, c^{2} m + c^{2}\right )} n^{2} + c^{2} + 6 \, {\left (c^{2} m^{3} + 3 \, c^{2} m^{2} + 3 \, c^{2} m + c^{2}\right )} n\right )} x x^{4 \, n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} + 2 \, {\left (b c m^{4} + 4 \, b c m^{3} + 6 \, b c m^{2} + 8 \, {\left (b c m + b c\right )} n^{3} + 4 \, b c m + 14 \, {\left (b c m^{2} + 2 \, b c m + b c\right )} n^{2} + b c + 7 \, {\left (b c m^{3} + 3 \, b c m^{2} + 3 \, b c m + b c\right )} n\right )} x x^{3 \, n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 12 \, {\left (b^{2} + 2 \, a c + {\left (b^{2} + 2 \, a c\right )} m\right )} n^{3} + 6 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 19 \, {\left ({\left (b^{2} + 2 \, a c\right )} m^{2} + b^{2} + 2 \, a c + 2 \, {\left (b^{2} + 2 \, a c\right )} m\right )} n^{2} + b^{2} + 2 \, a c + 4 \, {\left (b^{2} + 2 \, a c\right )} m + 8 \, {\left ({\left (b^{2} + 2 \, a c\right )} m^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + b^{2} + 2 \, a c + 3 \, {\left (b^{2} + 2 \, a c\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} + 2 \, {\left (a b m^{4} + 4 \, a b m^{3} + 6 \, a b m^{2} + 24 \, {\left (a b m + a b\right )} n^{3} + 4 \, a b m + 26 \, {\left (a b m^{2} + 2 \, a b m + a b\right )} n^{2} + a b + 9 \, {\left (a b m^{3} + 3 \, a b m^{2} + 3 \, a b m + a b\right )} n\right )} x x^{n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} + {\left (a^{2} m^{4} + 24 \, a^{2} n^{4} + 4 \, a^{2} m^{3} + 6 \, a^{2} m^{2} + 50 \, {\left (a^{2} m + a^{2}\right )} n^{3} + 4 \, a^{2} m + 35 \, {\left (a^{2} m^{2} + 2 \, a^{2} m + a^{2}\right )} n^{2} + a^{2} + 10 \, {\left (a^{2} m^{3} + 3 \, a^{2} m^{2} + 3 \, a^{2} m + a^{2}\right )} n\right )} x e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{m^{5} + 24 \, {\left (m + 1\right )} n^{4} + 5 \, m^{4} + 50 \, {\left (m^{2} + 2 \, m + 1\right )} n^{3} + 10 \, m^{3} + 35 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n^{2} + 10 \, m^{2} + 10 \, {\left (m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1\right )} n + 5 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c^2*m^4 + 4*c^2*m^3 + 6*c^2*m^2 + 6*(c^2*m + c^2)*n^3 + 4*c^2*m + 11*(c^2*m^2 + 2*c^2*m + c^2)*n^2 + c^2 + 6
*(c^2*m^3 + 3*c^2*m^2 + 3*c^2*m + c^2)*n)*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 2*(b*c*m^4 + 4*b*c*m^3 + 6*b*c*m
^2 + 8*(b*c*m + b*c)*n^3 + 4*b*c*m + 14*(b*c*m^2 + 2*b*c*m + b*c)*n^2 + b*c + 7*(b*c*m^3 + 3*b*c*m^2 + 3*b*c*m
+ b*c)*n)*x*x^(3*n)*e^(m*log(d) + m*log(x)) + ((b^2 + 2*a*c)*m^4 + 4*(b^2 + 2*a*c)*m^3 + 12*(b^2 + 2*a*c + (b
^2 + 2*a*c)*m)*n^3 + 6*(b^2 + 2*a*c)*m^2 + 19*((b^2 + 2*a*c)*m^2 + b^2 + 2*a*c + 2*(b^2 + 2*a*c)*m)*n^2 + b^2
+ 2*a*c + 4*(b^2 + 2*a*c)*m + 8*((b^2 + 2*a*c)*m^3 + 3*(b^2 + 2*a*c)*m^2 + b^2 + 2*a*c + 3*(b^2 + 2*a*c)*m)*n)
*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 2*(a*b*m^4 + 4*a*b*m^3 + 6*a*b*m^2 + 24*(a*b*m + a*b)*n^3 + 4*a*b*m + 26*
(a*b*m^2 + 2*a*b*m + a*b)*n^2 + a*b + 9*(a*b*m^3 + 3*a*b*m^2 + 3*a*b*m + a*b)*n)*x*x^n*e^(m*log(d) + m*log(x))
+ (a^2*m^4 + 24*a^2*n^4 + 4*a^2*m^3 + 6*a^2*m^2 + 50*(a^2*m + a^2)*n^3 + 4*a^2*m + 35*(a^2*m^2 + 2*a^2*m + a^
2)*n^2 + a^2 + 10*(a^2*m^3 + 3*a^2*m^2 + 3*a^2*m + a^2)*n)*x*e^(m*log(d) + m*log(x)))/(m^5 + 24*(m + 1)*n^4 +
5*m^4 + 50*(m^2 + 2*m + 1)*n^3 + 10*m^3 + 35*(m^3 + 3*m^2 + 3*m + 1)*n^2 + 10*m^2 + 10*(m^4 + 4*m^3 + 6*m^2 +
4*m + 1)*n + 5*m + 1)

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giac [B]  time = 0.80, size = 5454, normalized size = 46.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*m^4*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 6*c^2*m^3*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 11*c^2*m^2*n^2*x*
x^(4*n)*e^(m*log(d) + m*log(x)) + 6*c^2*m*n^3*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 2*b*c*m^4*x*x^(3*n)*e^(m*log
(d) + m*log(x)) + c^2*m^4*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 14*b*c*m^3*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) +
6*c^2*m^3*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 28*b*c*m^2*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 11*c^2*m^2
*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 16*b*c*m*n^3*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 6*c^2*m*n^3*x*x^(3*n
)*e^(m*log(d) + m*log(x)) + b^2*m^4*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 2*a*c*m^4*x*x^(2*n)*e^(m*log(d) + m*lo
g(x)) + 2*b*c*m^4*x*x^(2*n)*e^(m*log(d) + m*log(x)) + c^2*m^4*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*b^2*m^3*n*
x*x^(2*n)*e^(m*log(d) + m*log(x)) + 16*a*c*m^3*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 14*b*c*m^3*n*x*x^(2*n)*e^
(m*log(d) + m*log(x)) + 6*c^2*m^3*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 19*b^2*m^2*n^2*x*x^(2*n)*e^(m*log(d) +
m*log(x)) + 38*a*c*m^2*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 28*b*c*m^2*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x
)) + 11*c^2*m^2*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 12*b^2*m*n^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 24*a*
c*m*n^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 16*b*c*m*n^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 6*c^2*m*n^3*x*x^(
2*n)*e^(m*log(d) + m*log(x)) + 2*a*b*m^4*x*x^n*e^(m*log(d) + m*log(x)) + b^2*m^4*x*x^n*e^(m*log(d) + m*log(x))
+ 2*a*c*m^4*x*x^n*e^(m*log(d) + m*log(x)) + 2*b*c*m^4*x*x^n*e^(m*log(d) + m*log(x)) + c^2*m^4*x*x^n*e^(m*log(
d) + m*log(x)) + 18*a*b*m^3*n*x*x^n*e^(m*log(d) + m*log(x)) + 8*b^2*m^3*n*x*x^n*e^(m*log(d) + m*log(x)) + 16*a
*c*m^3*n*x*x^n*e^(m*log(d) + m*log(x)) + 14*b*c*m^3*n*x*x^n*e^(m*log(d) + m*log(x)) + 6*c^2*m^3*n*x*x^n*e^(m*l
og(d) + m*log(x)) + 52*a*b*m^2*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 19*b^2*m^2*n^2*x*x^n*e^(m*log(d) + m*log(x)
) + 38*a*c*m^2*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 28*b*c*m^2*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 11*c^2*m^2*n
^2*x*x^n*e^(m*log(d) + m*log(x)) + 48*a*b*m*n^3*x*x^n*e^(m*log(d) + m*log(x)) + 12*b^2*m*n^3*x*x^n*e^(m*log(d)
+ m*log(x)) + 24*a*c*m*n^3*x*x^n*e^(m*log(d) + m*log(x)) + 16*b*c*m*n^3*x*x^n*e^(m*log(d) + m*log(x)) + 6*c^2
*m*n^3*x*x^n*e^(m*log(d) + m*log(x)) + a^2*m^4*x*e^(m*log(d) + m*log(x)) + 2*a*b*m^4*x*e^(m*log(d) + m*log(x))
+ b^2*m^4*x*e^(m*log(d) + m*log(x)) + 2*a*c*m^4*x*e^(m*log(d) + m*log(x)) + 2*b*c*m^4*x*e^(m*log(d) + m*log(x
)) + c^2*m^4*x*e^(m*log(d) + m*log(x)) + 10*a^2*m^3*n*x*e^(m*log(d) + m*log(x)) + 18*a*b*m^3*n*x*e^(m*log(d) +
m*log(x)) + 8*b^2*m^3*n*x*e^(m*log(d) + m*log(x)) + 16*a*c*m^3*n*x*e^(m*log(d) + m*log(x)) + 14*b*c*m^3*n*x*e
^(m*log(d) + m*log(x)) + 6*c^2*m^3*n*x*e^(m*log(d) + m*log(x)) + 35*a^2*m^2*n^2*x*e^(m*log(d) + m*log(x)) + 52
*a*b*m^2*n^2*x*e^(m*log(d) + m*log(x)) + 19*b^2*m^2*n^2*x*e^(m*log(d) + m*log(x)) + 38*a*c*m^2*n^2*x*e^(m*log(
d) + m*log(x)) + 28*b*c*m^2*n^2*x*e^(m*log(d) + m*log(x)) + 11*c^2*m^2*n^2*x*e^(m*log(d) + m*log(x)) + 50*a^2*
m*n^3*x*e^(m*log(d) + m*log(x)) + 48*a*b*m*n^3*x*e^(m*log(d) + m*log(x)) + 12*b^2*m*n^3*x*e^(m*log(d) + m*log(
x)) + 24*a*c*m*n^3*x*e^(m*log(d) + m*log(x)) + 16*b*c*m*n^3*x*e^(m*log(d) + m*log(x)) + 6*c^2*m*n^3*x*e^(m*log
(d) + m*log(x)) + 24*a^2*n^4*x*e^(m*log(d) + m*log(x)) + 4*c^2*m^3*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 18*c^2*
m^2*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 22*c^2*m*n^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 6*c^2*n^3*x*x^(4*n)
*e^(m*log(d) + m*log(x)) + 8*b*c*m^3*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 4*c^2*m^3*x*x^(3*n)*e^(m*log(d) + m*l
og(x)) + 42*b*c*m^2*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 18*c^2*m^2*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 56*
b*c*m*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 22*c^2*m*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 16*b*c*n^3*x*x^
(3*n)*e^(m*log(d) + m*log(x)) + 6*c^2*n^3*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 4*b^2*m^3*x*x^(2*n)*e^(m*log(d)
+ m*log(x)) + 8*a*c*m^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*b*c*m^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 4*c^
2*m^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 24*b^2*m^2*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 48*a*c*m^2*n*x*x^(2
*n)*e^(m*log(d) + m*log(x)) + 42*b*c*m^2*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 18*c^2*m^2*n*x*x^(2*n)*e^(m*log
(d) + m*log(x)) + 38*b^2*m*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 76*a*c*m*n^2*x*x^(2*n)*e^(m*log(d) + m*log(
x)) + 56*b*c*m*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 22*c^2*m*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 12*b^2
*n^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 24*a*c*n^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 16*b*c*n^3*x*x^(2*n)*e
^(m*log(d) + m*log(x)) + 6*c^2*n^3*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*a*b*m^3*x*x^n*e^(m*log(d) + m*log(x))
+ 4*b^2*m^3*x*x^n*e^(m*log(d) + m*log(x)) + 8*a*c*m^3*x*x^n*e^(m*log(d) + m*log(x)) + 8*b*c*m^3*x*x^n*e^(m*lo
g(d) + m*log(x)) + 4*c^2*m^3*x*x^n*e^(m*log(d) + m*log(x)) + 54*a*b*m^2*n*x*x^n*e^(m*log(d) + m*log(x)) + 24*b
^2*m^2*n*x*x^n*e^(m*log(d) + m*log(x)) + 48*a*c*m^2*n*x*x^n*e^(m*log(d) + m*log(x)) + 42*b*c*m^2*n*x*x^n*e^(m*
log(d) + m*log(x)) + 18*c^2*m^2*n*x*x^n*e^(m*log(d) + m*log(x)) + 104*a*b*m*n^2*x*x^n*e^(m*log(d) + m*log(x))
+ 38*b^2*m*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 76*a*c*m*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 56*b*c*m*n^2*x*x^n
*e^(m*log(d) + m*log(x)) + 22*c^2*m*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 48*a*b*n^3*x*x^n*e^(m*log(d) + m*log(x
)) + 12*b^2*n^3*x*x^n*e^(m*log(d) + m*log(x)) + 24*a*c*n^3*x*x^n*e^(m*log(d) + m*log(x)) + 16*b*c*n^3*x*x^n*e^
(m*log(d) + m*log(x)) + 6*c^2*n^3*x*x^n*e^(m*log(d) + m*log(x)) + 4*a^2*m^3*x*e^(m*log(d) + m*log(x)) + 8*a*b*
m^3*x*e^(m*log(d) + m*log(x)) + 4*b^2*m^3*x*e^(m*log(d) + m*log(x)) + 8*a*c*m^3*x*e^(m*log(d) + m*log(x)) + 8*
b*c*m^3*x*e^(m*log(d) + m*log(x)) + 4*c^2*m^3*x*e^(m*log(d) + m*log(x)) + 30*a^2*m^2*n*x*e^(m*log(d) + m*log(x
)) + 54*a*b*m^2*n*x*e^(m*log(d) + m*log(x)) + 24*b^2*m^2*n*x*e^(m*log(d) + m*log(x)) + 48*a*c*m^2*n*x*e^(m*log
(d) + m*log(x)) + 42*b*c*m^2*n*x*e^(m*log(d) + m*log(x)) + 18*c^2*m^2*n*x*e^(m*log(d) + m*log(x)) + 70*a^2*m*n
^2*x*e^(m*log(d) + m*log(x)) + 104*a*b*m*n^2*x*e^(m*log(d) + m*log(x)) + 38*b^2*m*n^2*x*e^(m*log(d) + m*log(x)
) + 76*a*c*m*n^2*x*e^(m*log(d) + m*log(x)) + 56*b*c*m*n^2*x*e^(m*log(d) + m*log(x)) + 22*c^2*m*n^2*x*e^(m*log(
d) + m*log(x)) + 50*a^2*n^3*x*e^(m*log(d) + m*log(x)) + 48*a*b*n^3*x*e^(m*log(d) + m*log(x)) + 12*b^2*n^3*x*e^
(m*log(d) + m*log(x)) + 24*a*c*n^3*x*e^(m*log(d) + m*log(x)) + 16*b*c*n^3*x*e^(m*log(d) + m*log(x)) + 6*c^2*n^
3*x*e^(m*log(d) + m*log(x)) + 6*c^2*m^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 18*c^2*m*n*x*x^(4*n)*e^(m*log(d) +
m*log(x)) + 11*c^2*n^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 12*b*c*m^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 6*c
^2*m^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 42*b*c*m*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 18*c^2*m*n*x*x^(3*n)
*e^(m*log(d) + m*log(x)) + 28*b*c*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 11*c^2*n^2*x*x^(3*n)*e^(m*log(d) + m
*log(x)) + 6*b^2*m^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 12*a*c*m^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 12*b*c
*m^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 6*c^2*m^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 24*b^2*m*n*x*x^(2*n)*e^
(m*log(d) + m*log(x)) + 48*a*c*m*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 42*b*c*m*n*x*x^(2*n)*e^(m*log(d) + m*lo
g(x)) + 18*c^2*m*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 19*b^2*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 38*a*c*n
^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 28*b*c*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 11*c^2*n^2*x*x^(2*n)*e^(
m*log(d) + m*log(x)) + 12*a*b*m^2*x*x^n*e^(m*log(d) + m*log(x)) + 6*b^2*m^2*x*x^n*e^(m*log(d) + m*log(x)) + 12
*a*c*m^2*x*x^n*e^(m*log(d) + m*log(x)) + 12*b*c*m^2*x*x^n*e^(m*log(d) + m*log(x)) + 6*c^2*m^2*x*x^n*e^(m*log(d
) + m*log(x)) + 54*a*b*m*n*x*x^n*e^(m*log(d) + m*log(x)) + 24*b^2*m*n*x*x^n*e^(m*log(d) + m*log(x)) + 48*a*c*m
*n*x*x^n*e^(m*log(d) + m*log(x)) + 42*b*c*m*n*x*x^n*e^(m*log(d) + m*log(x)) + 18*c^2*m*n*x*x^n*e^(m*log(d) + m
*log(x)) + 52*a*b*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 19*b^2*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 38*a*c*n^2*x*
x^n*e^(m*log(d) + m*log(x)) + 28*b*c*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 11*c^2*n^2*x*x^n*e^(m*log(d) + m*log(
x)) + 6*a^2*m^2*x*e^(m*log(d) + m*log(x)) + 12*a*b*m^2*x*e^(m*log(d) + m*log(x)) + 6*b^2*m^2*x*e^(m*log(d) + m
*log(x)) + 12*a*c*m^2*x*e^(m*log(d) + m*log(x)) + 12*b*c*m^2*x*e^(m*log(d) + m*log(x)) + 6*c^2*m^2*x*e^(m*log(
d) + m*log(x)) + 30*a^2*m*n*x*e^(m*log(d) + m*log(x)) + 54*a*b*m*n*x*e^(m*log(d) + m*log(x)) + 24*b^2*m*n*x*e^
(m*log(d) + m*log(x)) + 48*a*c*m*n*x*e^(m*log(d) + m*log(x)) + 42*b*c*m*n*x*e^(m*log(d) + m*log(x)) + 18*c^2*m
*n*x*e^(m*log(d) + m*log(x)) + 35*a^2*n^2*x*e^(m*log(d) + m*log(x)) + 52*a*b*n^2*x*e^(m*log(d) + m*log(x)) + 1
9*b^2*n^2*x*e^(m*log(d) + m*log(x)) + 38*a*c*n^2*x*e^(m*log(d) + m*log(x)) + 28*b*c*n^2*x*e^(m*log(d) + m*log(
x)) + 11*c^2*n^2*x*e^(m*log(d) + m*log(x)) + 4*c^2*m*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 6*c^2*n*x*x^(4*n)*e^(
m*log(d) + m*log(x)) + 8*b*c*m*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 4*c^2*m*x*x^(3*n)*e^(m*log(d) + m*log(x)) +
14*b*c*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 6*c^2*n*x*x^(3*n)*e^(m*log(d) + m*log(x)) + 4*b^2*m*x*x^(2*n)*e^
(m*log(d) + m*log(x)) + 8*a*c*m*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*b*c*m*x*x^(2*n)*e^(m*log(d) + m*log(x))
+ 4*c^2*m*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*b^2*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 16*a*c*n*x*x^(2*n)*e
^(m*log(d) + m*log(x)) + 14*b*c*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 6*c^2*n*x*x^(2*n)*e^(m*log(d) + m*log(x)
) + 8*a*b*m*x*x^n*e^(m*log(d) + m*log(x)) + 4*b^2*m*x*x^n*e^(m*log(d) + m*log(x)) + 8*a*c*m*x*x^n*e^(m*log(d)
+ m*log(x)) + 8*b*c*m*x*x^n*e^(m*log(d) + m*log(x)) + 4*c^2*m*x*x^n*e^(m*log(d) + m*log(x)) + 18*a*b*n*x*x^n*e
^(m*log(d) + m*log(x)) + 8*b^2*n*x*x^n*e^(m*log(d) + m*log(x)) + 16*a*c*n*x*x^n*e^(m*log(d) + m*log(x)) + 14*b
*c*n*x*x^n*e^(m*log(d) + m*log(x)) + 6*c^2*n*x*x^n*e^(m*log(d) + m*log(x)) + 4*a^2*m*x*e^(m*log(d) + m*log(x))
+ 8*a*b*m*x*e^(m*log(d) + m*log(x)) + 4*b^2*m*x*e^(m*log(d) + m*log(x)) + 8*a*c*m*x*e^(m*log(d) + m*log(x)) +
8*b*c*m*x*e^(m*log(d) + m*log(x)) + 4*c^2*m*x*e^(m*log(d) + m*log(x)) + 10*a^2*n*x*e^(m*log(d) + m*log(x)) +
18*a*b*n*x*e^(m*log(d) + m*log(x)) + 8*b^2*n*x*e^(m*log(d) + m*log(x)) + 16*a*c*n*x*e^(m*log(d) + m*log(x)) +
14*b*c*n*x*e^(m*log(d) + m*log(x)) + 6*c^2*n*x*e^(m*log(d) + m*log(x)) + c^2*x*x^(4*n)*e^(m*log(d) + m*log(x))
+ 2*b*c*x*x^(3*n)*e^(m*log(d) + m*log(x)) + c^2*x*x^(3*n)*e^(m*log(d) + m*log(x)) + b^2*x*x^(2*n)*e^(m*log(d)
+ m*log(x)) + 2*a*c*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 2*b*c*x*x^(2*n)*e^(m*log(d) + m*log(x)) + c^2*x*x^(2*
n)*e^(m*log(d) + m*log(x)) + 2*a*b*x*x^n*e^(m*log(d) + m*log(x)) + b^2*x*x^n*e^(m*log(d) + m*log(x)) + 2*a*c*x
*x^n*e^(m*log(d) + m*log(x)) + 2*b*c*x*x^n*e^(m*log(d) + m*log(x)) + c^2*x*x^n*e^(m*log(d) + m*log(x)) + a^2*x
*e^(m*log(d) + m*log(x)) + 2*a*b*x*e^(m*log(d) + m*log(x)) + b^2*x*e^(m*log(d) + m*log(x)) + 2*a*c*x*e^(m*log(
d) + m*log(x)) + 2*b*c*x*e^(m*log(d) + m*log(x)) + c^2*x*e^(m*log(d) + m*log(x)))/(m^5 + 10*m^4*n + 35*m^3*n^2
+ 50*m^2*n^3 + 24*m*n^4 + 5*m^4 + 40*m^3*n + 105*m^2*n^2 + 100*m*n^3 + 24*n^4 + 10*m^3 + 60*m^2*n + 105*m*n^2
+ 50*n^3 + 10*m^2 + 40*m*n + 35*n^2 + 5*m + 10*n + 1)

________________________________________________________________________________________

maple [C]  time = 0.07, size = 1065, normalized size = 9.10 \[ \frac {\left (2 a b \,m^{4} x^{n}+18 a b \,m^{3} n \,x^{n}+52 a b \,m^{2} n^{2} x^{n}+48 a b m \,n^{3} x^{n}+2 a c \,m^{4} x^{2 n}+16 a c \,m^{3} n \,x^{2 n}+38 a c \,m^{2} n^{2} x^{2 n}+24 a c m \,n^{3} x^{2 n}+b^{2} m^{4} x^{2 n}+8 b^{2} m^{3} n \,x^{2 n}+19 b^{2} m^{2} n^{2} x^{2 n}+12 b^{2} m \,n^{3} x^{2 n}+2 b c \,m^{4} x^{3 n}+14 b c \,m^{3} n \,x^{3 n}+28 b c \,m^{2} n^{2} x^{3 n}+16 b c m \,n^{3} x^{3 n}+c^{2} m^{4} x^{4 n}+6 c^{2} m^{3} n \,x^{4 n}+11 c^{2} m^{2} n^{2} x^{4 n}+6 c^{2} m \,n^{3} x^{4 n}+a^{2} m^{4}+10 a^{2} m^{3} n +35 a^{2} m^{2} n^{2}+50 a^{2} m \,n^{3}+24 a^{2} n^{4}+8 a b \,m^{3} x^{n}+54 a b \,m^{2} n \,x^{n}+104 a b m \,n^{2} x^{n}+48 a b \,n^{3} x^{n}+8 a c \,m^{3} x^{2 n}+48 a c \,m^{2} n \,x^{2 n}+76 a c m \,n^{2} x^{2 n}+24 a c \,n^{3} x^{2 n}+4 b^{2} m^{3} x^{2 n}+24 b^{2} m^{2} n \,x^{2 n}+38 b^{2} m \,n^{2} x^{2 n}+12 b^{2} n^{3} x^{2 n}+8 b c \,m^{3} x^{3 n}+42 b c \,m^{2} n \,x^{3 n}+56 b c m \,n^{2} x^{3 n}+16 b c \,n^{3} x^{3 n}+4 c^{2} m^{3} x^{4 n}+18 c^{2} m^{2} n \,x^{4 n}+22 c^{2} m \,n^{2} x^{4 n}+6 c^{2} n^{3} x^{4 n}+4 a^{2} m^{3}+30 a^{2} m^{2} n +70 a^{2} m \,n^{2}+50 a^{2} n^{3}+12 a b \,m^{2} x^{n}+54 a b m n \,x^{n}+52 a b \,n^{2} x^{n}+12 a c \,m^{2} x^{2 n}+48 a c m n \,x^{2 n}+38 a c \,n^{2} x^{2 n}+6 b^{2} m^{2} x^{2 n}+24 b^{2} m n \,x^{2 n}+19 b^{2} n^{2} x^{2 n}+12 b c \,m^{2} x^{3 n}+42 b c m n \,x^{3 n}+28 b c \,n^{2} x^{3 n}+6 c^{2} m^{2} x^{4 n}+18 c^{2} m n \,x^{4 n}+11 c^{2} n^{2} x^{4 n}+6 a^{2} m^{2}+30 a^{2} m n +35 a^{2} n^{2}+8 a b m \,x^{n}+18 a b n \,x^{n}+8 a c m \,x^{2 n}+16 a c n \,x^{2 n}+4 b^{2} m \,x^{2 n}+8 b^{2} n \,x^{2 n}+8 b c m \,x^{3 n}+14 b c n \,x^{3 n}+4 c^{2} m \,x^{4 n}+6 c^{2} n \,x^{4 n}+4 a^{2} m +10 a^{2} n +2 a b \,x^{n}+2 a c \,x^{2 n}+b^{2} x^{2 n}+2 b c \,x^{3 n}+c^{2} x^{4 n}+a^{2}\right ) x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) m}{2}}}{\left (m +1\right ) \left (m +n +1\right ) \left (m +2 n +1\right ) \left (m +3 n +1\right ) \left (m +4 n +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(2*(x^n)^3*b*c+a^2+c^2*(x^n)^4+a^2*m^4+4*a^2*m^3+50*a^2*n^3+6*a^2*m^2+35*a^2*n^2+24*a^2*n^4+4*a^2*m+10*a^2*n
+b^2*(x^n)^2+6*c^2*m^3*n*(x^n)^4+11*c^2*m^2*n^2*(x^n)^4+6*c^2*m*n^3*(x^n)^4+2*b*c*m^4*(x^n)^3+18*c^2*m^2*n*(x^
n)^4+22*c^2*m*n^2*(x^n)^4+52*a*b*n^2*x^n+8*a*c*(x^n)^2*m+12*a*b*m^2*x^n+12*a*c*m^2*(x^n)^2+38*a*c*n^2*(x^n)^2+
24*b^2*m*n*(x^n)^2+8*m*b*c*(x^n)^3+14*b*c*(x^n)^3*n+18*c^2*m*n*(x^n)^4+2*a*b*m^4*x^n+8*a*c*m^3*(x^n)^2+24*a*c*
n^3*(x^n)^2+24*b^2*m^2*n*(x^n)^2+38*b^2*m*n^2*(x^n)^2+12*b*c*m^2*(x^n)^3+28*b*c*n^2*(x^n)^3+8*a*b*m^3*x^n+48*a
*b*n^3*x^n+16*a*c*(x^n)^2*n+8*a*b*x^n*m+18*a*b*x^n*n+2*a*c*m^4*(x^n)^2+8*b^2*m^3*n*(x^n)^2+19*b^2*m^2*n^2*(x^n
)^2+12*b^2*m*n^3*(x^n)^2+8*b*c*m^3*(x^n)^3+16*b*c*n^3*(x^n)^3+2*(x^n)^2*a*c+c^2*m^4*(x^n)^4+4*c^2*m^3*(x^n)^4+
56*b*c*m*n^2*(x^n)^3+18*a*b*m^3*n*x^n+52*a*b*m^2*n^2*x^n+48*a*b*m*n^3*x^n+48*a*c*m^2*n*(x^n)^2+76*a*c*m*n^2*(x
^n)^2+42*b*c*m*n*(x^n)^3+54*a*b*m^2*n*x^n+104*a*b*m*n^2*x^n+48*a*c*m*n*(x^n)^2+54*a*b*m*n*x^n+6*c^2*n^3*(x^n)^
4+b^2*m^4*(x^n)^2+6*c^2*m^2*(x^n)^4+11*c^2*n^2*(x^n)^4+4*b^2*m^3*(x^n)^2+12*b^2*n^3*(x^n)^2+4*m*c^2*(x^n)^4+6*
c^2*(x^n)^4*n+6*b^2*m^2*(x^n)^2+14*b*c*m^3*n*(x^n)^3+28*b*c*m^2*n^2*(x^n)^3+16*b*c*m*n^3*(x^n)^3+16*a*c*m^3*n*
(x^n)^2+38*a*c*m^2*n^2*(x^n)^2+24*a*c*m*n^3*(x^n)^2+42*b*c*m^2*n*(x^n)^3+19*b^2*n^2*(x^n)^2+4*b^2*(x^n)^2*m+8*
b^2*(x^n)^2*n+30*a^2*m*n+2*a*b*x^n+10*a^2*m^3*n+35*a^2*m^2*n^2+50*a^2*m*n^3+30*a^2*m^2*n+70*a^2*m*n^2)/(m+1)/(
m+n+1)/(m+2*n+1)/(m+3*n+1)/(1+m+4*n)*exp(1/2*(-I*Pi*csgn(I*d)*csgn(I*x)*csgn(I*d*x)+I*Pi*csgn(I*d)*csgn(I*d*x)
^2+I*Pi*csgn(I*x)*csgn(I*d*x)^2-I*Pi*csgn(I*d*x)^3+2*ln(d)+2*ln(x))*m)

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maxima [A]  time = 1.27, size = 152, normalized size = 1.30 \[ \frac {c^{2} d^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {2 \, b c d^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {b^{2} d^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {2 \, a c d^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {2 \, a b d^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.62, size = 543, normalized size = 4.64 \[ \frac {a^2\,x\,{\left (d\,x\right )}^m}{m+1}+\frac {x\,x^{2\,n}\,{\left (d\,x\right )}^m\,\left (b^2+2\,a\,c\right )\,\left (m^3+8\,m^2\,n+3\,m^2+19\,m\,n^2+16\,m\,n+3\,m+12\,n^3+19\,n^2+8\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {c^2\,x\,x^{4\,n}\,{\left (d\,x\right )}^m\,\left (m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {2\,a\,b\,x\,x^n\,{\left (d\,x\right )}^m\,\left (m^3+9\,m^2\,n+3\,m^2+26\,m\,n^2+18\,m\,n+3\,m+24\,n^3+26\,n^2+9\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {2\,b\,c\,x\,x^{3\,n}\,{\left (d\,x\right )}^m\,\left (m^3+7\,m^2\,n+3\,m^2+14\,m\,n^2+14\,m\,n+3\,m+8\,n^3+14\,n^2+7\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x*(d*x)^m)/(m + 1) + (x*x^(2*n)*(d*x)^m*(2*a*c + b^2)*(3*m + 8*n + 16*m*n + 19*m*n^2 + 8*m^2*n + 3*m^2 +
m^3 + 19*n^2 + 12*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 +
m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (c^2*x*x^(4*n)*(d*x)^m*(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*
m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n +
6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (2*a*b*x*x^n*(d*x)^m*(3*m + 9*n + 18*m*n +
26*m*n^2 + 9*m^2*n + 3*m^2 + m^3 + 26*n^2 + 24*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3
+ 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (2*b*c*x*x^(3*n)*(d*x)^m*(3*m
+ 7*n + 14*m*n + 14*m*n^2 + 7*m^2*n + 3*m^2 + m^3 + 14*n^2 + 8*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30
*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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