\(\int (b d+2 c d x)^m (a+b x+c x^2)^2 \, dx\) [1424]

Optimal result

Integrand size = 24, antiderivative size = 103 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{1+m}}{32 c^3 d (1+m)}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3+m}}{16 c^3 d^3 (3+m)}+\frac {(b d+2 c d x)^{5+m}}{32 c^3 d^5 (5+m)} \]

[Out]

1/32*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(1+m)/c^3/d/(1+m)-1/16*(-4*a*c+b^2)*(2*c*d*x+b*d)^(3+m)/c^3/d^3/(3+m)+1/32*(
2*c*d*x+b*d)^(5+m)/c^3/d^5/(5+m)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{m+3}}{16 c^3 d^3 (m+3)}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+1}}{32 c^3 d (m+1)}+\frac {(b d+2 c d x)^{m+5}}{32 c^3 d^5 (m+5)} \]

[In]

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

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(1 + m))/(32*c^3*d*(1 + m)) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3 + m))/(16*c^3
*d^3*(3 + m)) + (b*d + 2*c*d*x)^(5 + m)/(32*c^3*d^5*(5 + m))

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^m}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^{2+m}}{8 c^2 d^2}+\frac {(b d+2 c d x)^{4+m}}{16 c^2 d^4}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{1+m}}{32 c^3 d (1+m)}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3+m}}{16 c^3 d^3 (3+m)}+\frac {(b d+2 c d x)^{5+m}}{32 c^3 d^5 (5+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {(b+2 c x) (d (b+2 c x))^m \left (\frac {\left (b^2-4 a c\right )^2}{1+m}-\frac {2 \left (b^2-4 a c\right ) (b+2 c x)^2}{3+m}+\frac {(b+2 c x)^4}{5+m}\right )}{32 c^3} \]

[In]

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

[Out]

((b + 2*c*x)*(d*(b + 2*c*x))^m*((b^2 - 4*a*c)^2/(1 + m) - (2*(b^2 - 4*a*c)*(b + 2*c*x)^2)/(3 + m) + (b + 2*c*x
)^4/(5 + m)))/(32*c^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(97)=194\).

Time = 2.54 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.48

method	result	size
gosper	\(\frac {\left (2 c x +b \right ) \left (2 c^{4} m^{2} x^{4}+4 b \,c^{3} m^{2} x^{3}+8 c^{4} m \,x^{4}+4 a \,c^{3} m^{2} x^{2}+2 b^{2} c^{2} m^{2} x^{2}+16 b \,c^{3} m \,x^{3}+6 c^{4} x^{4}+4 a b \,c^{2} m^{2} x +24 a \,c^{3} m \,x^{2}+6 b^{2} c^{2} m \,x^{2}+12 b \,c^{3} x^{3}+2 a^{2} c^{2} m^{2}+24 a b \,c^{2} m x +20 x^{2} c^{3} a -2 b^{3} c m x +4 b^{2} c^{2} x^{2}+16 a^{2} c^{2} m -2 a \,b^{2} c m +20 a b \,c^{2} x -2 b^{3} c x +30 a^{2} c^{2}-10 a \,b^{2} c +b^{4}\right ) \left (2 c d x +b d \right )^{m}}{4 c^{3} \left (m^{3}+9 m^{2}+23 m +15\right )}\)	\(255\)
norman	\(\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{5+m}+\frac {\left (2 a c m +2 b^{2} m +10 a c +5 b^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{m^{2}+8 m +15}+\frac {5 b c \,x^{4} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 \left (5+m \right )}+\frac {b \left (2 a^{2} c^{2} m^{2}+16 a^{2} c^{2} m -2 a \,b^{2} c m +30 a^{2} c^{2}-10 a \,b^{2} c +b^{4}\right ) {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 c^{3} \left (m^{3}+9 m^{2}+23 m +15\right )}+\frac {\left (2 a^{2} c^{2} m^{2}+2 a \,b^{2} c \,m^{2}+16 a^{2} c^{2} m +10 a \,b^{2} c m -b^{4} m +30 a^{2} c^{2}\right ) x \,{\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 c^{2} \left (m^{3}+9 m^{2}+23 m +15\right )}+\frac {\left (6 a c m +b^{2} m +30 a c \right ) b \,x^{2} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 c \left (m^{2}+8 m +15\right )}\)	\(315\)
risch	\(\frac {\left (4 c^{5} m^{2} x^{5}+10 b \,c^{4} m^{2} x^{4}+16 c^{5} m \,x^{5}+8 a \,c^{4} m^{2} x^{3}+8 b^{2} c^{3} m^{2} x^{3}+40 b \,c^{4} m \,x^{4}+12 c^{5} x^{5}+12 a b \,c^{3} m^{2} x^{2}+48 a \,c^{4} m \,x^{3}+2 b^{3} c^{2} m^{2} x^{2}+28 b^{2} c^{3} m \,x^{3}+30 b \,x^{4} c^{4}+4 a^{2} c^{3} m^{2} x +4 a \,b^{2} c^{2} m^{2} x +72 a b \,c^{3} m \,x^{2}+40 a \,c^{4} x^{3}+2 b^{3} c^{2} m \,x^{2}+20 b^{2} c^{3} x^{3}+2 a^{2} b \,c^{2} m^{2}+32 a^{2} c^{3} m x +20 a \,b^{2} c^{2} m x +60 a b \,c^{3} x^{2}-2 b^{4} c m x +16 a^{2} b \,c^{2} m +60 a^{2} c^{3} x -2 a \,b^{3} c m +30 a^{2} b \,c^{2}-10 a \,b^{3} c +b^{5}\right ) \left (d \left (2 c x +b \right )\right )^{m}}{4 \left (3+m \right ) \left (5+m \right ) \left (1+m \right ) c^{3}}\)	\(331\)
parallelrisch	\(\frac {12 x^{5} \left (d \left (2 c x +b \right )\right )^{m} b \,c^{5}+30 x^{4} \left (d \left (2 c x +b \right )\right )^{m} b^{2} c^{4}+20 x^{3} \left (d \left (2 c x +b \right )\right )^{m} b^{3} c^{3}+30 \left (d \left (2 c x +b \right )\right )^{m} a^{2} b^{2} c^{2}-10 \left (d \left (2 c x +b \right )\right )^{m} a \,b^{4} c +8 x^{3} \left (d \left (2 c x +b \right )\right )^{m} a b \,c^{4} m^{2}+48 x^{3} \left (d \left (2 c x +b \right )\right )^{m} a b \,c^{4} m +12 x^{2} \left (d \left (2 c x +b \right )\right )^{m} a \,b^{2} c^{3} m^{2}+72 x^{2} \left (d \left (2 c x +b \right )\right )^{m} a \,b^{2} c^{3} m +4 x \left (d \left (2 c x +b \right )\right )^{m} a^{2} b \,c^{3} m^{2}+\left (d \left (2 c x +b \right )\right )^{m} b^{6}+4 x \left (d \left (2 c x +b \right )\right )^{m} a \,b^{3} c^{2} m^{2}+32 x \left (d \left (2 c x +b \right )\right )^{m} a^{2} b \,c^{3} m +20 x \left (d \left (2 c x +b \right )\right )^{m} a \,b^{3} c^{2} m +4 x^{5} \left (d \left (2 c x +b \right )\right )^{m} b \,c^{5} m^{2}+16 x^{5} \left (d \left (2 c x +b \right )\right )^{m} b \,c^{5} m +10 x^{4} \left (d \left (2 c x +b \right )\right )^{m} b^{2} c^{4} m^{2}+40 x^{4} \left (d \left (2 c x +b \right )\right )^{m} b^{2} c^{4} m +8 x^{3} \left (d \left (2 c x +b \right )\right )^{m} b^{3} c^{3} m^{2}+28 x^{3} \left (d \left (2 c x +b \right )\right )^{m} b^{3} c^{3} m +2 x^{2} \left (d \left (2 c x +b \right )\right )^{m} b^{4} c^{2} m^{2}+40 x^{3} \left (d \left (2 c x +b \right )\right )^{m} a b \,c^{4}+2 x^{2} \left (d \left (2 c x +b \right )\right )^{m} b^{4} c^{2} m +60 x^{2} \left (d \left (2 c x +b \right )\right )^{m} a \,b^{2} c^{3}-2 x \left (d \left (2 c x +b \right )\right )^{m} b^{5} c m +2 \left (d \left (2 c x +b \right )\right )^{m} a^{2} b^{2} c^{2} m^{2}+60 x \left (d \left (2 c x +b \right )\right )^{m} a^{2} b \,c^{3}+16 \left (d \left (2 c x +b \right )\right )^{m} a^{2} b^{2} c^{2} m -2 \left (d \left (2 c x +b \right )\right )^{m} a \,b^{4} c m}{4 \left (5+m \right ) \left (3+m \right ) b \left (1+m \right ) c^{3}}\)	\(642\)

[In]

int((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*(2*c*x+b)*(2*c^4*m^2*x^4+4*b*c^3*m^2*x^3+8*c^4*m*x^4+4*a*c^3*m^2*x^2+2*b^2*c^2*m^2*x^2+16*b*c^3*m*x^3+6*c^
4*x^4+4*a*b*c^2*m^2*x+24*a*c^3*m*x^2+6*b^2*c^2*m*x^2+12*b*c^3*x^3+2*a^2*c^2*m^2+24*a*b*c^2*m*x+20*a*c^3*x^2-2*
b^3*c*m*x+4*b^2*c^2*x^2+16*a^2*c^2*m-2*a*b^2*c*m+20*a*b*c^2*x-2*b^3*c*x+30*a^2*c^2-10*a*b^2*c+b^4)*(2*c*d*x+b*
d)^m/c^3/(m^3+9*m^2+23*m+15)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (97) = 194\).

Time = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.98 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (2 \, a^{2} b c^{2} m^{2} + 4 \, {\left (c^{5} m^{2} + 4 \, c^{5} m + 3 \, c^{5}\right )} x^{5} + b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2} + 10 \, {\left (b c^{4} m^{2} + 4 \, b c^{4} m + 3 \, b c^{4}\right )} x^{4} + 4 \, {\left (5 \, b^{2} c^{3} + 10 \, a c^{4} + 2 \, {\left (b^{2} c^{3} + a c^{4}\right )} m^{2} + {\left (7 \, b^{2} c^{3} + 12 \, a c^{4}\right )} m\right )} x^{3} + 2 \, {\left (30 \, a b c^{3} + {\left (b^{3} c^{2} + 6 \, a b c^{3}\right )} m^{2} + {\left (b^{3} c^{2} + 36 \, a b c^{3}\right )} m\right )} x^{2} - 2 \, {\left (a b^{3} c - 8 \, a^{2} b c^{2}\right )} m + 2 \, {\left (30 \, a^{2} c^{3} + 2 \, {\left (a b^{2} c^{2} + a^{2} c^{3}\right )} m^{2} - {\left (b^{4} c - 10 \, a b^{2} c^{2} - 16 \, a^{2} c^{3}\right )} m\right )} x\right )} {\left (2 \, c d x + b d\right )}^{m}}{4 \, {\left (c^{3} m^{3} + 9 \, c^{3} m^{2} + 23 \, c^{3} m + 15 \, c^{3}\right )}} \]

[In]

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

[Out]

1/4*(2*a^2*b*c^2*m^2 + 4*(c^5*m^2 + 4*c^5*m + 3*c^5)*x^5 + b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + 10*(b*c^4*m^2 + 4
*b*c^4*m + 3*b*c^4)*x^4 + 4*(5*b^2*c^3 + 10*a*c^4 + 2*(b^2*c^3 + a*c^4)*m^2 + (7*b^2*c^3 + 12*a*c^4)*m)*x^3 +
2*(30*a*b*c^3 + (b^3*c^2 + 6*a*b*c^3)*m^2 + (b^3*c^2 + 36*a*b*c^3)*m)*x^2 - 2*(a*b^3*c - 8*a^2*b*c^2)*m + 2*(3
0*a^2*c^3 + 2*(a*b^2*c^2 + a^2*c^3)*m^2 - (b^4*c - 10*a*b^2*c^2 - 16*a^2*c^3)*m)*x)*(2*c*d*x + b*d)^m/(c^3*m^3
+ 9*c^3*m^2 + 23*c^3*m + 15*c^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3196 vs. \(2 (92) = 184\).

Time = 0.91 (sec) , antiderivative size = 3196, normalized size of antiderivative = 31.03 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise(((b*d)**m*(a**2*x + a*b*x**2 + b**2*x**3/3), Eq(c, 0)), (-16*a**2*c**2/(128*b**4*c**3*d**5 + 1024*b*
*3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) - 8*a*b**2*c/(128*b**
4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4)
- 64*a*b*c**2*x/(128*b**4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3
+ 2048*c**7*d**5*x**4) - 64*a*c**3*x**2/(128*b**4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**
2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) + 4*b**4*log(b/(2*c) + x)/(128*b**4*c**3*d**5 + 1024*b**3*c**
4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) + 3*b**4/(128*b**4*c**3*d**
5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) + 32*b**3*
c*x*log(b/(2*c) + x)/(128*b**4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5
*x**3 + 2048*c**7*d**5*x**4) + 16*b**3*c*x/(128*b**4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x
**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) + 96*b**2*c**2*x**2*log(b/(2*c) + x)/(128*b**4*c**3*d**5 +
1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) + 16*b**2*c**2
*x**2/(128*b**4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c*
*7*d**5*x**4) + 128*b*c**3*x**3*log(b/(2*c) + x)/(128*b**4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*
d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4) + 64*c**4*x**4*log(b/(2*c) + x)/(128*b**4*c**3*d**5 +
1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4), Eq(m, -5)),
(-8*a**2*c**2/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) + 8*a*b**2*c*log(b/(2*c) + x)/(32*b
**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) + 4*a*b**2*c/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x +
128*c**5*d**3*x**2) + 32*a*b*c**2*x*log(b/(2*c) + x)/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x*
*2) + 32*a*c**3*x**2*log(b/(2*c) + x)/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) - 2*b**4*lo
g(b/(2*c) + x)/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) - 3*b**4/(32*b**2*c**3*d**3 + 128*
b*c**4*d**3*x + 128*c**5*d**3*x**2) - 8*b**3*c*x*log(b/(2*c) + x)/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128
*c**5*d**3*x**2) - 8*b**3*c*x/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) - 8*b**2*c**2*x**2*
log(b/(2*c) + x)/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) + 16*b*c**3*x**3/(32*b**2*c**3*d
**3 + 128*b*c**4*d**3*x + 128*c**5*d**3*x**2) + 8*c**4*x**4/(32*b**2*c**3*d**3 + 128*b*c**4*d**3*x + 128*c**5*
d**3*x**2), Eq(m, -3)), (a**2*log(b/(2*c) + x)/(2*c*d) - a*b**2*log(b/(2*c) + x)/(4*c**2*d) + a*b*x/(2*c*d) +
a*x**2/(2*d) + b**4*log(b/(2*c) + x)/(32*c**3*d) - b**3*x/(16*c**2*d) + b**2*x**2/(16*c*d) + b*x**3/(4*d) + c*
x**4/(8*d), Eq(m, -1)), (2*a**2*b*c**2*m**2*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c*
*3) + 16*a**2*b*c**2*m*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 30*a**2*b*c**2*
(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 4*a**2*c**3*m**2*x*(b*d + 2*c*d*x)**m/
(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 32*a**2*c**3*m*x*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**
3*m**2 + 92*c**3*m + 60*c**3) + 60*a**2*c**3*x*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60
*c**3) - 2*a*b**3*c*m*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) - 10*a*b**3*c*(b*d
+ 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 4*a*b**2*c**2*m**2*x*(b*d + 2*c*d*x)**m/(4
*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 20*a*b**2*c**2*m*x*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**
3*m**2 + 92*c**3*m + 60*c**3) + 12*a*b*c**3*m**2*x**2*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3
*m + 60*c**3) + 72*a*b*c**3*m*x**2*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 60*
a*b*c**3*x**2*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 8*a*c**4*m**2*x**3*(b*d
+ 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 48*a*c**4*m*x**3*(b*d + 2*c*d*x)**m/(4*c**3
*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 40*a*c**4*x**3*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 +
92*c**3*m + 60*c**3) + b**5*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) - 2*b**4*c*m
*x*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 2*b**3*c**2*m**2*x**2*(b*d + 2*c*d*
x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 2*b**3*c**2*m*x**2*(b*d + 2*c*d*x)**m/(4*c**3*m**3
+ 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 8*b**2*c**3*m**2*x**3*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 +
92*c**3*m + 60*c**3) + 28*b**2*c**3*m*x**3*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c*
*3) + 20*b**2*c**3*x**3*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 10*b*c**4*m**2
*x**4*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 40*b*c**4*m*x**4*(b*d + 2*c*d*x)
**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 30*b*c**4*x**4*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c
**3*m**2 + 92*c**3*m + 60*c**3) + 4*c**5*m**2*x**5*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m
+ 60*c**3) + 16*c**5*m*x**5*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3) + 12*c**5*x*
*5*(b*d + 2*c*d*x)**m/(4*c**3*m**3 + 36*c**3*m**2 + 92*c**3*m + 60*c**3), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (97) = 194\).

Time = 0.23 (sec) , antiderivative size = 539, normalized size of antiderivative = 5.23 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (4 \, c^{2} d^{m} {\left (m + 1\right )} x^{2} + 2 \, b c d^{m} m x - b^{2} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a b}{2 \, {\left (m^{2} + 3 \, m + 2\right )} c^{2}} + \frac {{\left (4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{3} d^{m} x^{3} + 2 \, {\left (m^{2} + m\right )} b c^{2} d^{m} x^{2} - 2 \, b^{2} c d^{m} m x + b^{3} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b^{2}}{4 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{3}} + \frac {{\left (4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{3} d^{m} x^{3} + 2 \, {\left (m^{2} + m\right )} b c^{2} d^{m} x^{2} - 2 \, b^{2} c d^{m} m x + b^{3} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a}{2 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{2}} + \frac {{\left (2 \, c d x + b d\right )}^{m + 1} a^{2}}{2 \, c d {\left (m + 1\right )}} + \frac {{\left (8 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{4} d^{m} x^{4} + 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b c^{3} d^{m} x^{3} - 6 \, {\left (m^{2} + m\right )} b^{2} c^{2} d^{m} x^{2} + 6 \, b^{3} c d^{m} m x - 3 \, b^{4} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b}{4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{3}} + \frac {{\left (4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{5} d^{m} x^{5} + 2 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b c^{4} d^{m} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{2} c^{3} d^{m} x^{3} + 6 \, {\left (m^{2} + m\right )} b^{3} c^{2} d^{m} x^{2} - 6 \, b^{4} c d^{m} m x + 3 \, b^{5} d^{m}\right )} {\left (2 \, c x + b\right )}^{m}}{4 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{3}} \]

[In]

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

[Out]

1/2*(4*c^2*d^m*(m + 1)*x^2 + 2*b*c*d^m*m*x - b^2*d^m)*(2*c*x + b)^m*a*b/((m^2 + 3*m + 2)*c^2) + 1/4*(4*(m^2 +
3*m + 2)*c^3*d^m*x^3 + 2*(m^2 + m)*b*c^2*d^m*x^2 - 2*b^2*c*d^m*m*x + b^3*d^m)*(2*c*x + b)^m*b^2/((m^3 + 6*m^2
+ 11*m + 6)*c^3) + 1/2*(4*(m^2 + 3*m + 2)*c^3*d^m*x^3 + 2*(m^2 + m)*b*c^2*d^m*x^2 - 2*b^2*c*d^m*m*x + b^3*d^m)
*(2*c*x + b)^m*a/((m^3 + 6*m^2 + 11*m + 6)*c^2) + 1/2*(2*c*d*x + b*d)^(m + 1)*a^2/(c*d*(m + 1)) + 1/4*(8*(m^3
+ 6*m^2 + 11*m + 6)*c^4*d^m*x^4 + 4*(m^3 + 3*m^2 + 2*m)*b*c^3*d^m*x^3 - 6*(m^2 + m)*b^2*c^2*d^m*x^2 + 6*b^3*c*
d^m*m*x - 3*b^4*d^m)*(2*c*x + b)^m*b/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^3) + 1/4*(4*(m^4 + 10*m^3 + 35*m^2
+ 50*m + 24)*c^5*d^m*x^5 + 2*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b*c^4*d^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*b^2*c^3*d^m*x
^3 + 6*(m^2 + m)*b^3*c^2*d^m*x^2 - 6*b^4*c*d^m*m*x + 3*b^5*d^m)*(2*c*x + b)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^
2 + 274*m + 120)*c^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (97) = 194\).

Time = 0.28 (sec) , antiderivative size = 651, normalized size of antiderivative = 6.32 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {4 \, {\left (2 \, c d x + b d\right )}^{m} c^{5} m^{2} x^{5} + 10 \, {\left (2 \, c d x + b d\right )}^{m} b c^{4} m^{2} x^{4} + 16 \, {\left (2 \, c d x + b d\right )}^{m} c^{5} m x^{5} + 8 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{3} m^{2} x^{3} + 8 \, {\left (2 \, c d x + b d\right )}^{m} a c^{4} m^{2} x^{3} + 40 \, {\left (2 \, c d x + b d\right )}^{m} b c^{4} m x^{4} + 12 \, {\left (2 \, c d x + b d\right )}^{m} c^{5} x^{5} + 2 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{2} m^{2} x^{2} + 12 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{3} m^{2} x^{2} + 28 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{3} m x^{3} + 48 \, {\left (2 \, c d x + b d\right )}^{m} a c^{4} m x^{3} + 30 \, {\left (2 \, c d x + b d\right )}^{m} b c^{4} x^{4} + 4 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{2} m^{2} x + 4 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{3} m^{2} x + 2 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{2} m x^{2} + 72 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{3} m x^{2} + 20 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{3} x^{3} + 40 \, {\left (2 \, c d x + b d\right )}^{m} a c^{4} x^{3} + 2 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{2} m^{2} - 2 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c m x + 20 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{2} m x + 32 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{3} m x + 60 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{3} x^{2} - 2 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c m + 16 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{2} m + 60 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{3} x + {\left (2 \, c d x + b d\right )}^{m} b^{5} - 10 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c + 30 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{2}}{4 \, {\left (c^{3} m^{3} + 9 \, c^{3} m^{2} + 23 \, c^{3} m + 15 \, c^{3}\right )}} \]

[In]

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

[Out]

1/4*(4*(2*c*d*x + b*d)^m*c^5*m^2*x^5 + 10*(2*c*d*x + b*d)^m*b*c^4*m^2*x^4 + 16*(2*c*d*x + b*d)^m*c^5*m*x^5 + 8
*(2*c*d*x + b*d)^m*b^2*c^3*m^2*x^3 + 8*(2*c*d*x + b*d)^m*a*c^4*m^2*x^3 + 40*(2*c*d*x + b*d)^m*b*c^4*m*x^4 + 12
*(2*c*d*x + b*d)^m*c^5*x^5 + 2*(2*c*d*x + b*d)^m*b^3*c^2*m^2*x^2 + 12*(2*c*d*x + b*d)^m*a*b*c^3*m^2*x^2 + 28*(
2*c*d*x + b*d)^m*b^2*c^3*m*x^3 + 48*(2*c*d*x + b*d)^m*a*c^4*m*x^3 + 30*(2*c*d*x + b*d)^m*b*c^4*x^4 + 4*(2*c*d*
x + b*d)^m*a*b^2*c^2*m^2*x + 4*(2*c*d*x + b*d)^m*a^2*c^3*m^2*x + 2*(2*c*d*x + b*d)^m*b^3*c^2*m*x^2 + 72*(2*c*d
*x + b*d)^m*a*b*c^3*m*x^2 + 20*(2*c*d*x + b*d)^m*b^2*c^3*x^3 + 40*(2*c*d*x + b*d)^m*a*c^4*x^3 + 2*(2*c*d*x + b
*d)^m*a^2*b*c^2*m^2 - 2*(2*c*d*x + b*d)^m*b^4*c*m*x + 20*(2*c*d*x + b*d)^m*a*b^2*c^2*m*x + 32*(2*c*d*x + b*d)^
m*a^2*c^3*m*x + 60*(2*c*d*x + b*d)^m*a*b*c^3*x^2 - 2*(2*c*d*x + b*d)^m*a*b^3*c*m + 16*(2*c*d*x + b*d)^m*a^2*b*
c^2*m + 60*(2*c*d*x + b*d)^m*a^2*c^3*x + (2*c*d*x + b*d)^m*b^5 - 10*(2*c*d*x + b*d)^m*a*b^3*c + 30*(2*c*d*x +
b*d)^m*a^2*b*c^2)/(c^3*m^3 + 9*c^3*m^2 + 23*c^3*m + 15*c^3)

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.98 \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx={\left (b\,d+2\,c\,d\,x\right )}^m\,\left (\frac {x\,\left (4\,a^2\,c^3\,m^2+32\,a^2\,c^3\,m+60\,a^2\,c^3+4\,a\,b^2\,c^2\,m^2+20\,a\,b^2\,c^2\,m-2\,b^4\,c\,m\right )}{4\,c^3\,\left (m^3+9\,m^2+23\,m+15\right )}+\frac {b\,\left (2\,a^2\,c^2\,m^2+16\,a^2\,c^2\,m+30\,a^2\,c^2-2\,a\,b^2\,c\,m-10\,a\,b^2\,c+b^4\right )}{4\,c^3\,\left (m^3+9\,m^2+23\,m+15\right )}+\frac {x^3\,\left (m+1\right )\,\left (10\,a\,c+2\,b^2\,m+5\,b^2+2\,a\,c\,m\right )}{m^3+9\,m^2+23\,m+15}+\frac {c^2\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {5\,b\,c\,x^4\,\left (m^2+4\,m+3\right )}{2\,\left (m^3+9\,m^2+23\,m+15\right )}+\frac {b\,x^2\,\left (m+1\right )\,\left (m\,b^2+30\,a\,c+6\,a\,c\,m\right )}{2\,c\,\left (m^3+9\,m^2+23\,m+15\right )}\right ) \]

[In]

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

[Out]

(b*d + 2*c*d*x)^m*((x*(60*a^2*c^3 + 32*a^2*c^3*m + 4*a^2*c^3*m^2 - 2*b^4*c*m + 4*a*b^2*c^2*m^2 + 20*a*b^2*c^2*
m))/(4*c^3*(23*m + 9*m^2 + m^3 + 15)) + (b*(b^4 + 30*a^2*c^2 + 16*a^2*c^2*m + 2*a^2*c^2*m^2 - 10*a*b^2*c - 2*a
*b^2*c*m))/(4*c^3*(23*m + 9*m^2 + m^3 + 15)) + (x^3*(m + 1)*(10*a*c + 2*b^2*m + 5*b^2 + 2*a*c*m))/(23*m + 9*m^
2 + m^3 + 15) + (c^2*x^5*(4*m + m^2 + 3))/(23*m + 9*m^2 + m^3 + 15) + (5*b*c*x^4*(4*m + m^2 + 3))/(2*(23*m + 9
*m^2 + m^3 + 15)) + (b*x^2*(m + 1)*(30*a*c + b^2*m + 6*a*c*m))/(2*c*(23*m + 9*m^2 + m^3 + 15)))