3.1424 \(\int (b d+2 c d x)^m (a+b x+c x^2)^2 \, dx\)
Optimal. Leaf size=103 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{m+3}}{16 c^3 d^3 (m+3)}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+1}}{32 c^3 d (m+1)}+\frac{(b d+2 c d x)^{m+5}}{32 c^3 d^5 (m+5)} \]
[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))
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Rubi [A] time = 0.0505594, antiderivative size = 103, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used =
{683} \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{m+3}}{16 c^3 d^3 (m+3)}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+1}}{32 c^3 d (m+1)}+\frac{(b d+2 c d x)^{m+5}}{32 c^3 d^5 (m+5)} \]
Antiderivative was successfully verified.
[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 683
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*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^2 \, dx &=\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] time = 0.0470792, size = 77, normalized size = 0.75 \[ \frac{(b+2 c x) \left (-\frac{2 \left (b^2-4 a c\right ) (b+2 c x)^2}{m+3}+\frac{\left (b^2-4 a c\right )^2}{m+1}+\frac{(b+2 c x)^4}{m+5}\right ) (d (b+2 c x))^m}{32 c^3} \]
Antiderivative was successfully verified.
[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)
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Maple [B] time = 0.046, size = 255, normalized size = 2.5 \begin{align*}{\frac{ \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\,ab{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\,ab{c}^{2}mx+20\,{x}^{2}a{c}^{3}-2\,{b}^{3}cmx+4\,{x}^{2}{b}^{2}{c}^{2}+16\,{a}^{2}{c}^{2}m-2\,a{b}^{2}cm+20\,ba{c}^{2}x-2\,{b}^{3}cx+30\,{a}^{2}{c}^{2}-10\,ac{b}^{2}+{b}^{4} \right ) \left ( 2\,cx+b \right ) \left ( 2\,cdx+bd \right ) ^{m}}{4\,{c}^{3} \left ({m}^{3}+9\,{m}^{2}+23\,m+15 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^2,x)
[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)
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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((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.21289, size = 641, normalized size = 6.22 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [A] time = 6.70394, size = 3434, normalized size = 33.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)), (-48*a**2*b**2*c**2/(384*b**6*c**3*d**5 + 30
72*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) - 16*a*
b**4*c/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 61
44*b**2*c**7*d**5*x**4) - 128*a*b**3*c**2*x/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*
x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) + 256*a*b*c**4*x**3/(384*b**6*c**3*d**5 + 3072*b*
*5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) + 128*a*c**5
*x**4/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 614
4*b**2*c**7*d**5*x**4) + 12*b**6*log(b/(2*c) + x)/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5
*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) + 7*b**6/(384*b**6*c**3*d**5 + 3072*b**5*c*
*4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) + 96*b**5*c*x*log
(b/(2*c) + x)/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x*
*3 + 6144*b**2*c**7*d**5*x**4) + 32*b**5*c*x/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5
*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) + 288*b**4*c**2*x**2*log(b/(2*c) + x)/(384*b**6*
c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5
*x**4) + 384*b**3*c**3*x**3*log(b/(2*c) + x)/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5
*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) - 64*b**3*c**3*x**3/(384*b**6*c**3*d**5 + 3072*b
**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) + 192*b**2*
c**4*x**4*log(b/(2*c) + x)/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x + 9216*b**4*c**5*d**5*x**2 + 12288*b**3
*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4) - 32*b**2*c**4*x**4/(384*b**6*c**3*d**5 + 3072*b**5*c**4*d**5*x +
9216*b**4*c**5*d**5*x**2 + 12288*b**3*c**6*d**5*x**3 + 6144*b**2*c**7*d**5*x**4), Eq(m, -5)), (-8*a**2*c**2/(3
2*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*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) - 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 + 3
6*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))
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Giac [B] time = 1.20724, size = 879, normalized size = 8.53 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)