3.1762 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{a+b x} \, dx\)

Optimal. Leaf size=38 \[ \frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.0843894, antiderivative size = 38, 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 \[ \frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5365, size = 31, normalized size = 0.82 \[ \frac{b \left (c + d x\right )^{4}}{4 d^{2}} + \frac{\left (c + d x\right )^{3} \left (a d - b c\right )}{3 d^{2}} \]

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/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0185805, size = 47, normalized size = 1.24 \[ \frac{1}{12} x \left (4 d x^2 (a d+2 b c)+6 c x (2 a d+b c)+12 a c^2+3 b d^2 x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(12*a*c^2 + 6*c*(b*c + 2*a*d)*x + 4*d*(2*b*c + a*d)*x^2 + 3*b*d^2*x^3))/12

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Maple [A]  time = 0.001, size = 55, normalized size = 1.5 \[{\frac{b{d}^{2}{x}^{4}}{4}}+{\frac{ \left ( bcd+d \left ( ad+bc \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( c \left ( ad+bc \right ) +acd \right ){x}^{2}}{2}}+a{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.718251, size = 65, normalized size = 1.71 \[ \frac{1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac{1}{3} \,{\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{2} + 2 \, a c d\right )} x^{2} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.193703, size = 65, normalized size = 1.71 \[ \frac{1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac{1}{3} \,{\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{2} + 2 \, a c d\right )} x^{2} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.222585, size = 49, normalized size = 1.29 \[ a c^{2} x + \frac{b d^{2} x^{4}}{4} + x^{3} \left (\frac{a d^{2}}{3} + \frac{2 b c d}{3}\right ) + x^{2} \left (a c d + \frac{b c^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208536, size = 66, normalized size = 1.74 \[ \frac{1}{4} \, b d^{2} x^{4} + \frac{2}{3} \, b c d x^{3} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{2} \, b c^{2} x^{2} + a c d x^{2} + a c^{2} x \]

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),x, algorithm="giac")

[Out]

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