3.1412 \(\int (b d+2 c d x)^m \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=141 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac{(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \]

[Out]

-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(1 + m))/(128*c^4*d*(1 + m)) + (3*(b^2 - 4*a*c
)^2*(b*d + 2*c*d*x)^(3 + m))/(128*c^4*d^3*(3 + m)) - (3*(b^2 - 4*a*c)*(b*d + 2*c
*d*x)^(5 + m))/(128*c^4*d^5*(5 + m)) + (b*d + 2*c*d*x)^(7 + m)/(128*c^4*d^7*(7 +
m))

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Rubi [A]  time = 0.212555, antiderivative size = 141, 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 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac{(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(1 + m))/(128*c^4*d*(1 + m)) + (3*(b^2 - 4*a*c
)^2*(b*d + 2*c*d*x)^(3 + m))/(128*c^4*d^3*(3 + m)) - (3*(b^2 - 4*a*c)*(b*d + 2*c
*d*x)^(5 + m))/(128*c^4*d^5*(5 + m)) + (b*d + 2*c*d*x)^(7 + m)/(128*c^4*d^7*(7 +
m))

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Rubi in Sympy [A]  time = 50.7838, size = 131, normalized size = 0.93 \[ - \frac{\left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{m + 1}}{128 c^{4} d \left (m + 1\right )} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{m + 3}}{128 c^{4} d^{3} \left (m + 3\right )} - \frac{3 \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{m + 5}}{128 c^{4} d^{5} \left (m + 5\right )} + \frac{\left (b d + 2 c d x\right )^{m + 7}}{128 c^{4} d^{7} \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**m*(c*x**2+b*x+a)**3,x)

[Out]

-(-4*a*c + b**2)**3*(b*d + 2*c*d*x)**(m + 1)/(128*c**4*d*(m + 1)) + 3*(-4*a*c +
b**2)**2*(b*d + 2*c*d*x)**(m + 3)/(128*c**4*d**3*(m + 3)) - 3*(-4*a*c + b**2)*(b
*d + 2*c*d*x)**(m + 5)/(128*c**4*d**5*(m + 5)) + (b*d + 2*c*d*x)**(m + 7)/(128*c
**4*d**7*(m + 7))

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Mathematica [B]  time = 0.326067, size = 321, normalized size = 2.28 \[ -\frac{(b+2 c x) \left (-6 b^2 c^2 \left (-a^2 \left (m^2+12 m+35\right )+2 a c \left (m^3+10 m^2+23 m+14\right ) x^2+c^2 \left (2 m^3+17 m^2+42 m+27\right ) x^4\right )-12 b c^3 (m+1) x \left (a^2 \left (m^2+12 m+35\right )+2 a c \left (m^2+10 m+21\right ) x^2+c^2 \left (m^2+8 m+15\right ) x^4\right )-4 c^3 \left (a^3 \left (m^3+15 m^2+71 m+105\right )+3 a^2 c \left (m^3+13 m^2+47 m+35\right ) x^2+3 a c^2 \left (m^3+11 m^2+31 m+21\right ) x^4+c^3 \left (m^3+9 m^2+23 m+15\right ) x^6\right )-6 b^4 c \left (a (m+7)-c \left (m^2+3 m+2\right ) x^2\right )-4 b^3 c^2 (m+1) x \left (c \left (m^2+5 m+6\right ) x^2-3 a (m+7)\right )+3 b^6-6 b^5 c (m+1) x\right ) (d (b+2 c x))^m}{8 c^4 (m+1) (m+3) (m+5) (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

-((b + 2*c*x)*(d*(b + 2*c*x))^m*(3*b^6 - 6*b^5*c*(1 + m)*x - 6*b^4*c*(a*(7 + m)
- c*(2 + 3*m + m^2)*x^2) - 4*b^3*c^2*(1 + m)*x*(-3*a*(7 + m) + c*(6 + 5*m + m^2)
*x^2) - 12*b*c^3*(1 + m)*x*(a^2*(35 + 12*m + m^2) + 2*a*c*(21 + 10*m + m^2)*x^2
+ c^2*(15 + 8*m + m^2)*x^4) - 6*b^2*c^2*(-(a^2*(35 + 12*m + m^2)) + 2*a*c*(14 +
23*m + 10*m^2 + m^3)*x^2 + c^2*(27 + 42*m + 17*m^2 + 2*m^3)*x^4) - 4*c^3*(a^3*(1
05 + 71*m + 15*m^2 + m^3) + 3*a^2*c*(35 + 47*m + 13*m^2 + m^3)*x^2 + 3*a*c^2*(21
+ 31*m + 11*m^2 + m^3)*x^4 + c^3*(15 + 23*m + 9*m^2 + m^3)*x^6)))/(8*c^4*(1 + m
)*(3 + m)*(5 + m)*(7 + m))

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Maple [B]  time = 0.018, size = 653, normalized size = 4.6 \[{\frac{ \left ( 2\,cdx+bd \right ) ^{m} \left ( 4\,{c}^{6}{m}^{3}{x}^{6}+12\,b{c}^{5}{m}^{3}{x}^{5}+36\,{c}^{6}{m}^{2}{x}^{6}+12\,a{c}^{5}{m}^{3}{x}^{4}+12\,{b}^{2}{c}^{4}{m}^{3}{x}^{4}+108\,b{c}^{5}{m}^{2}{x}^{5}+92\,{c}^{6}m{x}^{6}+24\,ab{c}^{4}{m}^{3}{x}^{3}+132\,a{c}^{5}{m}^{2}{x}^{4}+4\,{b}^{3}{c}^{3}{m}^{3}{x}^{3}+102\,{b}^{2}{c}^{4}{m}^{2}{x}^{4}+276\,b{c}^{5}m{x}^{5}+60\,{c}^{6}{x}^{6}+12\,{a}^{2}{c}^{4}{m}^{3}{x}^{2}+12\,a{b}^{2}{c}^{3}{m}^{3}{x}^{2}+264\,ab{c}^{4}{m}^{2}{x}^{3}+372\,a{c}^{5}m{x}^{4}+24\,{b}^{3}{c}^{3}{m}^{2}{x}^{3}+252\,{b}^{2}{c}^{4}m{x}^{4}+180\,b{c}^{5}{x}^{5}+12\,{a}^{2}b{c}^{3}{m}^{3}x+156\,{a}^{2}{c}^{4}{m}^{2}{x}^{2}+120\,a{b}^{2}{c}^{3}{m}^{2}{x}^{2}+744\,ab{c}^{4}m{x}^{3}+252\,{x}^{4}a{c}^{5}-6\,{b}^{4}{c}^{2}{m}^{2}{x}^{2}+44\,{b}^{3}{c}^{3}m{x}^{3}+162\,{x}^{4}{b}^{2}{c}^{4}+4\,{a}^{3}{c}^{3}{m}^{3}+156\,{a}^{2}b{c}^{3}{m}^{2}x+564\,{a}^{2}{c}^{4}m{x}^{2}-12\,a{b}^{3}{c}^{2}{m}^{2}x+276\,a{b}^{2}{c}^{3}m{x}^{2}+504\,{x}^{3}ab{c}^{4}-18\,{b}^{4}{c}^{2}m{x}^{2}+24\,{b}^{3}{c}^{3}{x}^{3}+60\,{a}^{3}{c}^{3}{m}^{2}-6\,{a}^{2}{b}^{2}{c}^{2}{m}^{2}+564\,{a}^{2}b{c}^{3}mx+420\,{x}^{2}{a}^{2}{c}^{4}-96\,a{b}^{3}{c}^{2}mx+168\,{x}^{2}a{b}^{2}{c}^{3}+6\,{b}^{5}cmx-12\,{x}^{2}{b}^{4}{c}^{2}+284\,{a}^{3}{c}^{3}m-72\,{a}^{2}{b}^{2}{c}^{2}m+420\,{a}^{2}b{c}^{3}x+6\,a{b}^{4}cm-84\,a{b}^{3}{c}^{2}x+6\,{b}^{5}cx+420\,{a}^{3}{c}^{3}-210\,{a}^{2}{b}^{2}{c}^{2}+42\,a{b}^{4}c-3\,{b}^{6} \right ) \left ( 2\,cx+b \right ) }{8\,{c}^{4} \left ({m}^{4}+16\,{m}^{3}+86\,{m}^{2}+176\,m+105 \right ) }} \]

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)^3,x)

[Out]

1/8*(2*c*d*x+b*d)^m*(4*c^6*m^3*x^6+12*b*c^5*m^3*x^5+36*c^6*m^2*x^6+12*a*c^5*m^3*
x^4+12*b^2*c^4*m^3*x^4+108*b*c^5*m^2*x^5+92*c^6*m*x^6+24*a*b*c^4*m^3*x^3+132*a*c
^5*m^2*x^4+4*b^3*c^3*m^3*x^3+102*b^2*c^4*m^2*x^4+276*b*c^5*m*x^5+60*c^6*x^6+12*a
^2*c^4*m^3*x^2+12*a*b^2*c^3*m^3*x^2+264*a*b*c^4*m^2*x^3+372*a*c^5*m*x^4+24*b^3*c
^3*m^2*x^3+252*b^2*c^4*m*x^4+180*b*c^5*x^5+12*a^2*b*c^3*m^3*x+156*a^2*c^4*m^2*x^
2+120*a*b^2*c^3*m^2*x^2+744*a*b*c^4*m*x^3+252*a*c^5*x^4-6*b^4*c^2*m^2*x^2+44*b^3
*c^3*m*x^3+162*b^2*c^4*x^4+4*a^3*c^3*m^3+156*a^2*b*c^3*m^2*x+564*a^2*c^4*m*x^2-1
2*a*b^3*c^2*m^2*x+276*a*b^2*c^3*m*x^2+504*a*b*c^4*x^3-18*b^4*c^2*m*x^2+24*b^3*c^
3*x^3+60*a^3*c^3*m^2-6*a^2*b^2*c^2*m^2+564*a^2*b*c^3*m*x+420*a^2*c^4*x^2-96*a*b^
3*c^2*m*x+168*a*b^2*c^3*x^2+6*b^5*c*m*x-12*b^4*c^2*x^2+284*a^3*c^3*m-72*a^2*b^2*
c^2*m+420*a^2*b*c^3*x+6*a*b^4*c*m-84*a*b^3*c^2*x+6*b^5*c*x+420*a^3*c^3-210*a^2*b
^2*c^2+42*a*b^4*c-3*b^6)*(2*c*x+b)/c^4/(m^4+16*m^3+86*m^2+176*m+105)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(2*c*d*x + b*d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235793, size = 933, normalized size = 6.62 \[ \frac{{\left (4 \, a^{3} b c^{3} m^{3} + 8 \,{\left (c^{7} m^{3} + 9 \, c^{7} m^{2} + 23 \, c^{7} m + 15 \, c^{7}\right )} x^{7} - 3 \, b^{7} + 42 \, a b^{5} c - 210 \, a^{2} b^{3} c^{2} + 420 \, a^{3} b c^{3} + 28 \,{\left (b c^{6} m^{3} + 9 \, b c^{6} m^{2} + 23 \, b c^{6} m + 15 \, b c^{6}\right )} x^{6} + 12 \,{\left (42 \, b^{2} c^{5} + 42 \, a c^{6} +{\left (3 \, b^{2} c^{5} + 2 \, a c^{6}\right )} m^{3} + 2 \,{\left (13 \, b^{2} c^{5} + 11 \, a c^{6}\right )} m^{2} +{\left (65 \, b^{2} c^{5} + 62 \, a c^{6}\right )} m\right )} x^{5} + 10 \,{\left (21 \, b^{3} c^{4} + 126 \, a b c^{5} + 2 \,{\left (b^{3} c^{4} + 3 \, a b c^{5}\right )} m^{3} + 3 \,{\left (5 \, b^{3} c^{4} + 22 \, a b c^{5}\right )} m^{2} + 2 \,{\left (17 \, b^{3} c^{4} + 93 \, a b c^{5}\right )} m\right )} x^{4} + 4 \,{\left (210 \, a b^{2} c^{4} + 210 \, a^{2} c^{5} +{\left (b^{4} c^{3} + 12 \, a b^{2} c^{4} + 6 \, a^{2} c^{5}\right )} m^{3} + 3 \,{\left (b^{4} c^{3} + 42 \, a b^{2} c^{4} + 26 \, a^{2} c^{5}\right )} m^{2} + 2 \,{\left (b^{4} c^{3} + 162 \, a b^{2} c^{4} + 141 \, a^{2} c^{5}\right )} m\right )} x^{3} - 6 \,{\left (a^{2} b^{3} c^{2} - 10 \, a^{3} b c^{3}\right )} m^{2} + 6 \,{\left (210 \, a^{2} b c^{4} + 2 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b c^{4}\right )} m^{3} -{\left (b^{5} c^{2} - 16 \, a b^{3} c^{3} - 78 \, a^{2} b c^{4}\right )} m^{2} -{\left (b^{5} c^{2} - 14 \, a b^{3} c^{3} - 282 \, a^{2} b c^{4}\right )} m\right )} x^{2} + 2 \,{\left (3 \, a b^{5} c - 36 \, a^{2} b^{3} c^{2} + 142 \, a^{3} b c^{3}\right )} m + 2 \,{\left (420 \, a^{3} c^{4} + 2 \,{\left (3 \, a^{2} b^{2} c^{3} + 2 \, a^{3} c^{4}\right )} m^{3} - 6 \,{\left (a b^{4} c^{2} - 12 \, a^{2} b^{2} c^{3} - 10 \, a^{3} c^{4}\right )} m^{2} +{\left (3 \, b^{6} c - 42 \, a b^{4} c^{2} + 210 \, a^{2} b^{2} c^{3} + 284 \, a^{3} c^{4}\right )} m\right )} x\right )}{\left (2 \, c d x + b d\right )}^{m}}{8 \,{\left (c^{4} m^{4} + 16 \, c^{4} m^{3} + 86 \, c^{4} m^{2} + 176 \, c^{4} m + 105 \, c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(4*a^3*b*c^3*m^3 + 8*(c^7*m^3 + 9*c^7*m^2 + 23*c^7*m + 15*c^7)*x^7 - 3*b^7 +
42*a*b^5*c - 210*a^2*b^3*c^2 + 420*a^3*b*c^3 + 28*(b*c^6*m^3 + 9*b*c^6*m^2 + 23
*b*c^6*m + 15*b*c^6)*x^6 + 12*(42*b^2*c^5 + 42*a*c^6 + (3*b^2*c^5 + 2*a*c^6)*m^3
+ 2*(13*b^2*c^5 + 11*a*c^6)*m^2 + (65*b^2*c^5 + 62*a*c^6)*m)*x^5 + 10*(21*b^3*c
^4 + 126*a*b*c^5 + 2*(b^3*c^4 + 3*a*b*c^5)*m^3 + 3*(5*b^3*c^4 + 22*a*b*c^5)*m^2
+ 2*(17*b^3*c^4 + 93*a*b*c^5)*m)*x^4 + 4*(210*a*b^2*c^4 + 210*a^2*c^5 + (b^4*c^3
+ 12*a*b^2*c^4 + 6*a^2*c^5)*m^3 + 3*(b^4*c^3 + 42*a*b^2*c^4 + 26*a^2*c^5)*m^2 +
2*(b^4*c^3 + 162*a*b^2*c^4 + 141*a^2*c^5)*m)*x^3 - 6*(a^2*b^3*c^2 - 10*a^3*b*c^
3)*m^2 + 6*(210*a^2*b*c^4 + 2*(a*b^3*c^3 + 3*a^2*b*c^4)*m^3 - (b^5*c^2 - 16*a*b^
3*c^3 - 78*a^2*b*c^4)*m^2 - (b^5*c^2 - 14*a*b^3*c^3 - 282*a^2*b*c^4)*m)*x^2 + 2*
(3*a*b^5*c - 36*a^2*b^3*c^2 + 142*a^3*b*c^3)*m + 2*(420*a^3*c^4 + 2*(3*a^2*b^2*c
^3 + 2*a^3*c^4)*m^3 - 6*(a*b^4*c^2 - 12*a^2*b^2*c^3 - 10*a^3*c^4)*m^2 + (3*b^6*c
- 42*a*b^4*c^2 + 210*a^2*b^2*c^3 + 284*a^3*c^4)*m)*x)*(2*c*d*x + b*d)^m/(c^4*m^
4 + 16*c^4*m^3 + 86*c^4*m^2 + 176*c^4*m + 105*c^4)

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

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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(2*c*d*x + b*d)^m,x, algorithm="giac")

[Out]

Done