3.1000 \(\int \frac{(a+b x)^4}{\left (\frac{a d}{b}+d x\right )^3} \, dx\)

Optimal. Leaf size=23 \[ \frac{a b^3 x}{d^3}+\frac{b^4 x^2}{2 d^3} \]

[Out]

(a*b^3*x)/d^3 + (b^4*x^2)/(2*d^3)

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Rubi [A]  time = 0.0162692, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{a b^3 x}{d^3}+\frac{b^4 x^2}{2 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4/((a*d)/b + d*x)^3,x]

[Out]

(a*b^3*x)/d^3 + (b^4*x^2)/(2*d^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{4} \int x\, dx}{d^{3}} + \frac{b^{3} \int a\, dx}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4/(a*d/b+d*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.00148056, size = 19, normalized size = 0.83 \[ \frac{b^3 \left (a x+\frac{b x^2}{2}\right )}{d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4/((a*d)/b + d*x)^3,x]

[Out]

(b^3*(a*x + (b*x^2)/2))/d^3

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Maple [A]  time = 0.001, size = 18, normalized size = 0.8 \[{\frac{{b}^{3}}{{d}^{3}} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4/(a*d/b+d*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.34012, size = 27, normalized size = 1.17 \[ \frac{b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(d*x + a*d/b)^3,x, algorithm="maxima")

[Out]

1/2*(b^4*x^2 + 2*a*b^3*x)/d^3

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Fricas [A]  time = 0.191669, size = 27, normalized size = 1.17 \[ \frac{b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(d*x + a*d/b)^3,x, algorithm="fricas")

[Out]

1/2*(b^4*x^2 + 2*a*b^3*x)/d^3

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Sympy [A]  time = 0.219927, size = 20, normalized size = 0.87 \[ \frac{a b^{3} x}{d^{3}} + \frac{b^{4} x^{2}}{2 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4/(a*d/b+d*x)**3,x)

[Out]

a*b**3*x/d**3 + b**4*x**2/(2*d**3)

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GIAC/XCAS [A]  time = 0.213342, size = 35, normalized size = 1.52 \[ \frac{b^{4} d^{3} x^{2} + 2 \, a b^{3} d^{3} x}{2 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(d*x + a*d/b)^3,x, algorithm="giac")

[Out]

1/2*(b^4*d^3*x^2 + 2*a*b^3*d^3*x)/d^6