3.1763 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx\)

Optimal. Leaf size=14 \[ \frac{(c+d x)^3}{3 d} \]

[Out]

(c + d*x)^3/(3*d)

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Rubi [A]  time = 0.0203599, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(c+d x)^3}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^3/(3*d)

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Rubi in Sympy [A]  time = 7.64738, size = 8, normalized size = 0.57 \[ \frac{\left (c + d x\right )^{3}}{3 d} \]

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)**2,x)

[Out]

(c + d*x)**3/(3*d)

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Mathematica [A]  time = 0.00259954, size = 14, normalized size = 1. \[ \frac{(c+d x)^3}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^3/(3*d)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[{\frac{ \left ( dx+c \right ) ^{3}}{3\,d}} \]

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)^2,x)

[Out]

1/3*(d*x+c)^3/d

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Maxima [A]  time = 0.724111, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^2*x^3 + c*d*x^2 + c^2*x

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Fricas [A]  time = 0.199503, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^2*x^3 + c*d*x^2 + c^2*x

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Sympy [A]  time = 0.21576, size = 19, normalized size = 1.36 \[ c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3} \]

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)**2,x)

[Out]

c**2*x + c*d*x**2 + d**2*x**3/3

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GIAC/XCAS [A]  time = 0.210549, size = 113, normalized size = 8.07 \[ \frac{{\left (\frac{3 \, b^{2} c^{2}}{{\left (b x + a\right )}^{2}} + \frac{3 \, b c d}{b x + a} - \frac{6 \, a b c d}{{\left (b x + a\right )}^{2}} - \frac{3 \, a d^{2}}{b x + a} + \frac{3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + d^{2}\right )}{\left (b x + a\right )}^{3}}{3 \, b^{3}} \]

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)^2,x, algorithm="giac")

[Out]

1/3*(3*b^2*c^2/(b*x + a)^2 + 3*b*c*d/(b*x + a) - 6*a*b*c*d/(b*x + a)^2 - 3*a*d^2
/(b*x + a) + 3*a^2*d^2/(b*x + a)^2 + d^2)*(b*x + a)^3/b^3