3.1761 \(\int \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx\)

Optimal. Leaf size=65 \[ -\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3} \]

[Out]

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

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Rubi [A]  time = 0.167913, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a c \left (a d + b c\right ) \int x\, dx + \frac{b^{2} d^{2} x^{5}}{5} + \frac{b d x^{4} \left (a d + b c\right )}{2} + c^{2} \int a^{2}\, dx + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*c*(a*d + b*c)*Integral(x, x) + b**2*d**2*x**5/5 + b*d*x**4*(a*d + b*c)/2 + c
**2*Integral(a**2, x) + x**3*(a**2*d**2/3 + 4*a*b*c*d/3 + b**2*c**2/3)

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Mathematica [A]  time = 0.0210594, size = 79, normalized size = 1.22 \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{1}{2} b d x^4 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{5} b^2 d^2 x^5 \]

Antiderivative was successfully verified.

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

[Out]

a^2*c^2*x + a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3)/3 + (b*d
*(b*c + a*d)*x^4)/2 + (b^2*d^2*x^5)/5

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Maple [A]  time = 0.001, size = 69, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{ \left ( ad+bc \right ) bd{x}^{4}}{2}}+{\frac{ \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ){x}^{3}}{3}}+ac \left ( ad+bc \right ){x}^{2}+{a}^{2}{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.741145, size = 97, normalized size = 1.49 \[ \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{1}{2} \,{\left (b c + a d\right )} b d x^{4} + a^{2} c^{2} x + \frac{1}{3} \,{\left (b c + a d\right )}^{2} x^{3} + \frac{1}{3} \,{\left (2 \, b d x^{3} + 3 \,{\left (b c + a d\right )} x^{2}\right )} a c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.179922, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} d^{2} b^{2} + \frac{1}{2} x^{4} d c b^{2} + \frac{1}{2} x^{4} d^{2} b a + \frac{1}{3} x^{3} c^{2} b^{2} + \frac{4}{3} x^{3} d c b a + \frac{1}{3} x^{3} d^{2} a^{2} + x^{2} c^{2} b a + x^{2} d c a^{2} + x c^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5*d^2*b^2 + 1/2*x^4*d*c*b^2 + 1/2*x^4*d^2*b*a + 1/3*x^3*c^2*b^2 + 4/3*x^3*
d*c*b*a + 1/3*x^3*d^2*a^2 + x^2*c^2*b*a + x^2*d*c*a^2 + x*c^2*a^2

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Sympy [A]  time = 0.154192, size = 87, normalized size = 1.34 \[ a^{2} c^{2} x + \frac{b^{2} d^{2} x^{5}}{5} + x^{4} \left (\frac{a b d^{2}}{2} + \frac{b^{2} c d}{2}\right ) + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**2*x + b**2*d**2*x**5/5 + x**4*(a*b*d**2/2 + b**2*c*d/2) + x**3*(a**2*d**
2/3 + 4*a*b*c*d/3 + b**2*c**2/3) + x**2*(a**2*c*d + a*b*c**2)

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GIAC/XCAS [A]  time = 0.207673, size = 120, normalized size = 1.85 \[ \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{1}{2} \, b^{2} c d x^{4} + \frac{1}{2} \, a b d^{2} x^{4} + \frac{1}{3} \, b^{2} c^{2} x^{3} + \frac{4}{3} \, a b c d x^{3} + \frac{1}{3} \, a^{2} d^{2} x^{3} + a b c^{2} x^{2} + a^{2} c d x^{2} + a^{2} c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*d^2*x^5 + 1/2*b^2*c*d*x^4 + 1/2*a*b*d^2*x^4 + 1/3*b^2*c^2*x^3 + 4/3*a*b*
c*d*x^3 + 1/3*a^2*d^2*x^3 + a*b*c^2*x^2 + a^2*c*d*x^2 + a^2*c^2*x