3.1778 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx\)

Optimal. Leaf size=14 \[ \frac{(c+d x)^4}{4 d} \]

[Out]

(c + d*x)^4/(4*d)

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Rubi [A]  time = 0.0246838, 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 \[ \frac{(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^4/(4*d)

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Rubi in Sympy [A]  time = 7.61968, size = 8, normalized size = 0.57 \[ \frac{\left (c + d x\right )^{4}}{4 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c + d*x)**4/(4*d)

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Mathematica [A]  time = 0.00288753, size = 14, normalized size = 1. \[ \frac{(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^4/(4*d)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \[{\frac{ \left ( dx+c \right ) ^{4}}{4\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(d*x+c)^4/d

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Maxima [A]  time = 0.741105, size = 42, normalized size = 3. \[ \frac{1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac{3}{2} \, c^{2} d x^{2} + c^{3} x \]

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.197349, size = 42, normalized size = 3. \[ \frac{1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac{3}{2} \, c^{2} d x^{2} + c^{3} x \]

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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.321443, size = 32, normalized size = 2.29 \[ c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4

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GIAC/XCAS [A]  time = 0.208905, size = 42, normalized size = 3. \[ \frac{1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac{3}{2} \, c^{2} d x^{2} + c^{3} x \]

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)^3,x, algorithm="giac")

[Out]

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