3.1773 \(\int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx\)

Optimal. Leaf size=92 \[ \frac{3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac{3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac{(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac{d^3 (a+b x)^9}{9 b^4} \]

[Out]

((b*c - a*d)^3*(a + b*x)^6)/(6*b^4) + (3*d*(b*c - a*d)^2*(a + b*x)^7)/(7*b^4) +
(3*d^2*(b*c - a*d)*(a + b*x)^8)/(8*b^4) + (d^3*(a + b*x)^9)/(9*b^4)

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Rubi [A]  time = 0.40766, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac{3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac{(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac{d^3 (a+b x)^9}{9 b^4} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^3*(a + b*x)^6)/(6*b^4) + (3*d*(b*c - a*d)^2*(a + b*x)^7)/(7*b^4) +
(3*d^2*(b*c - a*d)*(a + b*x)^8)/(8*b^4) + (d^3*(a + b*x)^9)/(9*b^4)

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Rubi in Sympy [A]  time = 55.1466, size = 82, normalized size = 0.89 \[ \frac{d^{3} \left (a + b x\right )^{9}}{9 b^{4}} - \frac{3 d^{2} \left (a + b x\right )^{8} \left (a d - b c\right )}{8 b^{4}} + \frac{3 d \left (a + b x\right )^{7} \left (a d - b c\right )^{2}}{7 b^{4}} - \frac{\left (a + b x\right )^{6} \left (a d - b c\right )^{3}}{6 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*(a + b*x)**9/(9*b**4) - 3*d**2*(a + b*x)**8*(a*d - b*c)/(8*b**4) + 3*d*(a +
b*x)**7*(a*d - b*c)**2/(7*b**4) - (a + b*x)**6*(a*d - b*c)**3/(6*b**4)

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Mathematica [B]  time = 0.116356, size = 235, normalized size = 2.55 \[ \frac{1}{504} x \left (126 a^5 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+126 a^4 b x \left (10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right )+84 a^3 b^2 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )+36 a^2 b^3 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+9 a b^4 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+b^5 x^5 \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(126*a^5*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 126*a^4*b*x*(10*c^3 +
20*c^2*d*x + 15*c*d^2*x^2 + 4*d^3*x^3) + 84*a^3*b^2*x^2*(20*c^3 + 45*c^2*d*x + 3
6*c*d^2*x^2 + 10*d^3*x^3) + 36*a^2*b^3*x^3*(35*c^3 + 84*c^2*d*x + 70*c*d^2*x^2 +
20*d^3*x^3) + 9*a*b^4*x^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) +
b^5*x^5*(84*c^3 + 216*c^2*d*x + 189*c*d^2*x^2 + 56*d^3*x^3)))/504

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Maple [B]  time = 0.003, size = 601, normalized size = 6.5 \[{\frac{{d}^{3}{b}^{5}{x}^{9}}{9}}+{\frac{ \left ( 2\,a{b}^{4}{d}^{3}+3\,{b}^{4} \left ( ad+bc \right ){d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}{b}^{3}{d}^{3}+6\,a{b}^{3} \left ( ad+bc \right ){d}^{2}+{b}^{2} \left ( a{b}^{2}c{d}^{2}+2\, \left ( ad+bc \right ) ^{2}bd+bd \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2} \left ( ad+bc \right ){b}^{2}{d}^{2}+2\,ab \left ( a{b}^{2}c{d}^{2}+2\, \left ( ad+bc \right ) ^{2}bd+bd \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) +{b}^{2} \left ( 4\,ac \left ( ad+bc \right ) bd+ \left ( ad+bc \right ) \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2} \left ( a{b}^{2}c{d}^{2}+2\, \left ( ad+bc \right ) ^{2}bd+bd \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) +2\,ab \left ( 4\,ac \left ( ad+bc \right ) bd+ \left ( ad+bc \right ) \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) +{b}^{2} \left ( ac \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\, \left ( ad+bc \right ) ^{2}ac+{a}^{2}b{c}^{2}d \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2} \left ( 4\,ac \left ( ad+bc \right ) bd+ \left ( ad+bc \right ) \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) +2\,ab \left ( ac \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\, \left ( ad+bc \right ) ^{2}ac+{a}^{2}b{c}^{2}d \right ) +3\,{b}^{2}{a}^{2}{c}^{2} \left ( ad+bc \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2} \left ( ac \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\, \left ( ad+bc \right ) ^{2}ac+{a}^{2}b{c}^{2}d \right ) +6\,{a}^{3}b{c}^{2} \left ( ad+bc \right ) +{a}^{3}{b}^{2}{c}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{4}{c}^{2} \left ( ad+bc \right ) +2\,{a}^{4}b{c}^{3} \right ){x}^{2}}{2}}+{a}^{5}{c}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*d^3*b^5*x^9+1/8*(2*a*b^4*d^3+3*b^4*(a*d+b*c)*d^2)*x^8+1/7*(a^2*b^3*d^3+6*a*b
^3*(a*d+b*c)*d^2+b^2*(a*b^2*c*d^2+2*(a*d+b*c)^2*b*d+b*d*(2*c*a*b*d+(a*d+b*c)^2))
)*x^7+1/6*(3*a^2*(a*d+b*c)*b^2*d^2+2*a*b*(a*b^2*c*d^2+2*(a*d+b*c)^2*b*d+b*d*(2*c
*a*b*d+(a*d+b*c)^2))+b^2*(4*a*c*(a*d+b*c)*b*d+(a*d+b*c)*(2*c*a*b*d+(a*d+b*c)^2))
)*x^6+1/5*(a^2*(a*b^2*c*d^2+2*(a*d+b*c)^2*b*d+b*d*(2*c*a*b*d+(a*d+b*c)^2))+2*a*b
*(4*a*c*(a*d+b*c)*b*d+(a*d+b*c)*(2*c*a*b*d+(a*d+b*c)^2))+b^2*(a*c*(2*c*a*b*d+(a*
d+b*c)^2)+2*(a*d+b*c)^2*a*c+a^2*b*c^2*d))*x^5+1/4*(a^2*(4*a*c*(a*d+b*c)*b*d+(a*d
+b*c)*(2*c*a*b*d+(a*d+b*c)^2))+2*a*b*(a*c*(2*c*a*b*d+(a*d+b*c)^2)+2*(a*d+b*c)^2*
a*c+a^2*b*c^2*d)+3*b^2*a^2*c^2*(a*d+b*c))*x^4+1/3*(a^2*(a*c*(2*c*a*b*d+(a*d+b*c)
^2)+2*(a*d+b*c)^2*a*c+a^2*b*c^2*d)+6*a^3*b*c^2*(a*d+b*c)+a^3*b^2*c^3)*x^3+1/2*(3
*a^4*c^2*(a*d+b*c)+2*a^4*b*c^3)*x^2+a^5*c^3*x

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Maxima [A]  time = 0.740692, size = 374, normalized size = 4.07 \[ \frac{1}{9} \, b^{5} d^{3} x^{9} + a^{5} c^{3} x + \frac{1}{8} \,{\left (3 \, b^{5} c d^{2} + 5 \, a b^{4} d^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, b^{5} c^{2} d + 15 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} c^{3} + 15 \, a b^{4} c^{2} d + 30 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{6} +{\left (a b^{4} c^{3} + 6 \, a^{2} b^{3} c^{2} d + 6 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, a^{2} b^{3} c^{3} + 30 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} c^{3} + 15 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b c^{3} + 3 \, a^{5} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*b^5*d^3*x^9 + a^5*c^3*x + 1/8*(3*b^5*c*d^2 + 5*a*b^4*d^3)*x^8 + 1/7*(3*b^5*c
^2*d + 15*a*b^4*c*d^2 + 10*a^2*b^3*d^3)*x^7 + 1/6*(b^5*c^3 + 15*a*b^4*c^2*d + 30
*a^2*b^3*c*d^2 + 10*a^3*b^2*d^3)*x^6 + (a*b^4*c^3 + 6*a^2*b^3*c^2*d + 6*a^3*b^2*
c*d^2 + a^4*b*d^3)*x^5 + 1/4*(10*a^2*b^3*c^3 + 30*a^3*b^2*c^2*d + 15*a^4*b*c*d^2
+ a^5*d^3)*x^4 + 1/3*(10*a^3*b^2*c^3 + 15*a^4*b*c^2*d + 3*a^5*c*d^2)*x^3 + 1/2*
(5*a^4*b*c^3 + 3*a^5*c^2*d)*x^2

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Fricas [A]  time = 0.186881, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} d^{3} b^{5} + \frac{3}{8} x^{8} d^{2} c b^{5} + \frac{5}{8} x^{8} d^{3} b^{4} a + \frac{3}{7} x^{7} d c^{2} b^{5} + \frac{15}{7} x^{7} d^{2} c b^{4} a + \frac{10}{7} x^{7} d^{3} b^{3} a^{2} + \frac{1}{6} x^{6} c^{3} b^{5} + \frac{5}{2} x^{6} d c^{2} b^{4} a + 5 x^{6} d^{2} c b^{3} a^{2} + \frac{5}{3} x^{6} d^{3} b^{2} a^{3} + x^{5} c^{3} b^{4} a + 6 x^{5} d c^{2} b^{3} a^{2} + 6 x^{5} d^{2} c b^{2} a^{3} + x^{5} d^{3} b a^{4} + \frac{5}{2} x^{4} c^{3} b^{3} a^{2} + \frac{15}{2} x^{4} d c^{2} b^{2} a^{3} + \frac{15}{4} x^{4} d^{2} c b a^{4} + \frac{1}{4} x^{4} d^{3} a^{5} + \frac{10}{3} x^{3} c^{3} b^{2} a^{3} + 5 x^{3} d c^{2} b a^{4} + x^{3} d^{2} c a^{5} + \frac{5}{2} x^{2} c^{3} b a^{4} + \frac{3}{2} x^{2} d c^{2} a^{5} + x c^{3} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9*d^3*b^5 + 3/8*x^8*d^2*c*b^5 + 5/8*x^8*d^3*b^4*a + 3/7*x^7*d*c^2*b^5 + 15
/7*x^7*d^2*c*b^4*a + 10/7*x^7*d^3*b^3*a^2 + 1/6*x^6*c^3*b^5 + 5/2*x^6*d*c^2*b^4*
a + 5*x^6*d^2*c*b^3*a^2 + 5/3*x^6*d^3*b^2*a^3 + x^5*c^3*b^4*a + 6*x^5*d*c^2*b^3*
a^2 + 6*x^5*d^2*c*b^2*a^3 + x^5*d^3*b*a^4 + 5/2*x^4*c^3*b^3*a^2 + 15/2*x^4*d*c^2
*b^2*a^3 + 15/4*x^4*d^2*c*b*a^4 + 1/4*x^4*d^3*a^5 + 10/3*x^3*c^3*b^2*a^3 + 5*x^3
*d*c^2*b*a^4 + x^3*d^2*c*a^5 + 5/2*x^2*c^3*b*a^4 + 3/2*x^2*d*c^2*a^5 + x*c^3*a^5

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Sympy [A]  time = 0.31998, size = 308, normalized size = 3.35 \[ a^{5} c^{3} x + \frac{b^{5} d^{3} x^{9}}{9} + x^{8} \left (\frac{5 a b^{4} d^{3}}{8} + \frac{3 b^{5} c d^{2}}{8}\right ) + x^{7} \left (\frac{10 a^{2} b^{3} d^{3}}{7} + \frac{15 a b^{4} c d^{2}}{7} + \frac{3 b^{5} c^{2} d}{7}\right ) + x^{6} \left (\frac{5 a^{3} b^{2} d^{3}}{3} + 5 a^{2} b^{3} c d^{2} + \frac{5 a b^{4} c^{2} d}{2} + \frac{b^{5} c^{3}}{6}\right ) + x^{5} \left (a^{4} b d^{3} + 6 a^{3} b^{2} c d^{2} + 6 a^{2} b^{3} c^{2} d + a b^{4} c^{3}\right ) + x^{4} \left (\frac{a^{5} d^{3}}{4} + \frac{15 a^{4} b c d^{2}}{4} + \frac{15 a^{3} b^{2} c^{2} d}{2} + \frac{5 a^{2} b^{3} c^{3}}{2}\right ) + x^{3} \left (a^{5} c d^{2} + 5 a^{4} b c^{2} d + \frac{10 a^{3} b^{2} c^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{5} c^{2} d}{2} + \frac{5 a^{4} b c^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*c**3*x + b**5*d**3*x**9/9 + x**8*(5*a*b**4*d**3/8 + 3*b**5*c*d**2/8) + x**7
*(10*a**2*b**3*d**3/7 + 15*a*b**4*c*d**2/7 + 3*b**5*c**2*d/7) + x**6*(5*a**3*b**
2*d**3/3 + 5*a**2*b**3*c*d**2 + 5*a*b**4*c**2*d/2 + b**5*c**3/6) + x**5*(a**4*b*
d**3 + 6*a**3*b**2*c*d**2 + 6*a**2*b**3*c**2*d + a*b**4*c**3) + x**4*(a**5*d**3/
4 + 15*a**4*b*c*d**2/4 + 15*a**3*b**2*c**2*d/2 + 5*a**2*b**3*c**3/2) + x**3*(a**
5*c*d**2 + 5*a**4*b*c**2*d + 10*a**3*b**2*c**3/3) + x**2*(3*a**5*c**2*d/2 + 5*a*
*4*b*c**3/2)

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GIAC/XCAS [A]  time = 0.207992, size = 409, normalized size = 4.45 \[ \frac{1}{9} \, b^{5} d^{3} x^{9} + \frac{3}{8} \, b^{5} c d^{2} x^{8} + \frac{5}{8} \, a b^{4} d^{3} x^{8} + \frac{3}{7} \, b^{5} c^{2} d x^{7} + \frac{15}{7} \, a b^{4} c d^{2} x^{7} + \frac{10}{7} \, a^{2} b^{3} d^{3} x^{7} + \frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{5}{2} \, a b^{4} c^{2} d x^{6} + 5 \, a^{2} b^{3} c d^{2} x^{6} + \frac{5}{3} \, a^{3} b^{2} d^{3} x^{6} + a b^{4} c^{3} x^{5} + 6 \, a^{2} b^{3} c^{2} d x^{5} + 6 \, a^{3} b^{2} c d^{2} x^{5} + a^{4} b d^{3} x^{5} + \frac{5}{2} \, a^{2} b^{3} c^{3} x^{4} + \frac{15}{2} \, a^{3} b^{2} c^{2} d x^{4} + \frac{15}{4} \, a^{4} b c d^{2} x^{4} + \frac{1}{4} \, a^{5} d^{3} x^{4} + \frac{10}{3} \, a^{3} b^{2} c^{3} x^{3} + 5 \, a^{4} b c^{2} d x^{3} + a^{5} c d^{2} x^{3} + \frac{5}{2} \, a^{4} b c^{3} x^{2} + \frac{3}{2} \, a^{5} c^{2} d x^{2} + a^{5} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*b^5*d^3*x^9 + 3/8*b^5*c*d^2*x^8 + 5/8*a*b^4*d^3*x^8 + 3/7*b^5*c^2*d*x^7 + 15
/7*a*b^4*c*d^2*x^7 + 10/7*a^2*b^3*d^3*x^7 + 1/6*b^5*c^3*x^6 + 5/2*a*b^4*c^2*d*x^
6 + 5*a^2*b^3*c*d^2*x^6 + 5/3*a^3*b^2*d^3*x^6 + a*b^4*c^3*x^5 + 6*a^2*b^3*c^2*d*
x^5 + 6*a^3*b^2*c*d^2*x^5 + a^4*b*d^3*x^5 + 5/2*a^2*b^3*c^3*x^4 + 15/2*a^3*b^2*c
^2*d*x^4 + 15/4*a^4*b*c*d^2*x^4 + 1/4*a^5*d^3*x^4 + 10/3*a^3*b^2*c^3*x^3 + 5*a^4
*b*c^2*d*x^3 + a^5*c*d^2*x^3 + 5/2*a^4*b*c^3*x^2 + 3/2*a^5*c^2*d*x^2 + a^5*c^3*x