3.1758 \(\int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx\)

Optimal. Leaf size=65 \[ \frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3} \]

[Out]

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

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Rubi [A]  time = 0.298769, antiderivative size = 65, 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 \[ \frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.3515, size = 56, normalized size = 0.86 \[ \frac{d^{2} \left (a + b x\right )^{8}}{8 b^{3}} - \frac{2 d \left (a + b x\right )^{7} \left (a d - b c\right )}{7 b^{3}} + \frac{\left (a + b x\right )^{6} \left (a d - b c\right )^{2}}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0570159, size = 189, normalized size = 2.91 \[ a^5 c^2 x+\frac{1}{2} a^4 c x^2 (2 a d+5 b c)+a b^2 x^5 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{5}{4} a^2 b x^4 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac{1}{6} b^3 x^6 \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac{1}{3} a^3 x^3 \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )+\frac{1}{7} b^4 d x^7 (5 a d+2 b c)+\frac{1}{8} b^5 d^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.001, size = 315, normalized size = 4.9 \[{\frac{{d}^{2}{b}^{5}{x}^{8}}{8}}+{\frac{ \left ( 3\,a{b}^{4}{d}^{2}+2\,{b}^{4} \left ( ad+bc \right ) d \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}{b}^{3}{d}^{2}+6\,a{b}^{3} \left ( ad+bc \right ) d+{b}^{3} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}{b}^{2}{d}^{2}+6\,{a}^{2}{b}^{2} \left ( ad+bc \right ) d+3\,a{b}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\,{b}^{3}ac \left ( ad+bc \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{3} \left ( ad+bc \right ) bd+3\,{a}^{2}b \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +6\,{a}^{2}{b}^{2}c \left ( ad+bc \right ) +{a}^{2}{b}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +6\,{a}^{3}bc \left ( ad+bc \right ) +3\,{a}^{3}{b}^{2}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{4}c \left ( ad+bc \right ) +3\,{a}^{4}b{c}^{2} \right ){x}^{2}}{2}}+{a}^{5}{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.742278, size = 266, normalized size = 4.09 \[ \frac{1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac{1}{7} \,{\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} +{\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.179742, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} d^{2} b^{5} + \frac{2}{7} x^{7} d c b^{5} + \frac{5}{7} x^{7} d^{2} b^{4} a + \frac{1}{6} x^{6} c^{2} b^{5} + \frac{5}{3} x^{6} d c b^{4} a + \frac{5}{3} x^{6} d^{2} b^{3} a^{2} + x^{5} c^{2} b^{4} a + 4 x^{5} d c b^{3} a^{2} + 2 x^{5} d^{2} b^{2} a^{3} + \frac{5}{2} x^{4} c^{2} b^{3} a^{2} + 5 x^{4} d c b^{2} a^{3} + \frac{5}{4} x^{4} d^{2} b a^{4} + \frac{10}{3} x^{3} c^{2} b^{2} a^{3} + \frac{10}{3} x^{3} d c b a^{4} + \frac{1}{3} x^{3} d^{2} a^{5} + \frac{5}{2} x^{2} c^{2} b a^{4} + x^{2} d c a^{5} + x c^{2} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.25291, size = 218, normalized size = 3.35 \[ a^{5} c^{2} x + \frac{b^{5} d^{2} x^{8}}{8} + x^{7} \left (\frac{5 a b^{4} d^{2}}{7} + \frac{2 b^{5} c d}{7}\right ) + x^{6} \left (\frac{5 a^{2} b^{3} d^{2}}{3} + \frac{5 a b^{4} c d}{3} + \frac{b^{5} c^{2}}{6}\right ) + x^{5} \left (2 a^{3} b^{2} d^{2} + 4 a^{2} b^{3} c d + a b^{4} c^{2}\right ) + x^{4} \left (\frac{5 a^{4} b d^{2}}{4} + 5 a^{3} b^{2} c d + \frac{5 a^{2} b^{3} c^{2}}{2}\right ) + x^{3} \left (\frac{a^{5} d^{2}}{3} + \frac{10 a^{4} b c d}{3} + \frac{10 a^{3} b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{5} c d + \frac{5 a^{4} b c^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207401, size = 286, normalized size = 4.4 \[ \frac{1}{8} \, b^{5} d^{2} x^{8} + \frac{2}{7} \, b^{5} c d x^{7} + \frac{5}{7} \, a b^{4} d^{2} x^{7} + \frac{1}{6} \, b^{5} c^{2} x^{6} + \frac{5}{3} \, a b^{4} c d x^{6} + \frac{5}{3} \, a^{2} b^{3} d^{2} x^{6} + a b^{4} c^{2} x^{5} + 4 \, a^{2} b^{3} c d x^{5} + 2 \, a^{3} b^{2} d^{2} x^{5} + \frac{5}{2} \, a^{2} b^{3} c^{2} x^{4} + 5 \, a^{3} b^{2} c d x^{4} + \frac{5}{4} \, a^{4} b d^{2} x^{4} + \frac{10}{3} \, a^{3} b^{2} c^{2} x^{3} + \frac{10}{3} \, a^{4} b c d x^{3} + \frac{1}{3} \, a^{5} d^{2} x^{3} + \frac{5}{2} \, a^{4} b c^{2} x^{2} + a^{5} c d x^{2} + a^{5} c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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