3.999 \(\int \frac{(a+b x)^5}{\left (\frac{a d}{b}+d x\right )^3} \, dx\)

Optimal. Leaf size=17 \[ \frac{b^2 (a+b x)^3}{3 d^3} \]

[Out]

(b^2*(a + b*x)^3)/(3*d^3)

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Rubi [A]  time = 0.0115818, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b^2 (a+b x)^3}{3 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/((a*d)/b + d*x)^3,x]

[Out]

(b^2*(a + b*x)^3)/(3*d^3)

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Rubi in Sympy [A]  time = 4.3166, size = 14, normalized size = 0.82 \[ \frac{b^{2} \left (a + b x\right )^{3}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(a*d/b+d*x)**3,x)

[Out]

b**2*(a + b*x)**3/(3*d**3)

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Mathematica [A]  time = 0.00351469, size = 17, normalized size = 1. \[ \frac{b^2 (a+b x)^3}{3 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/((a*d)/b + d*x)^3,x]

[Out]

(b^2*(a + b*x)^3)/(3*d^3)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \[{\frac{{b}^{2} \left ( bx+a \right ) ^{3}}{3\,{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(a*d/b+d*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.34232, size = 42, normalized size = 2.47 \[ \frac{b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + a*d/b)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.196277, size = 42, normalized size = 2.47 \[ \frac{b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + a*d/b)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.231112, size = 34, normalized size = 2. \[ \frac{a^{2} b^{3} x}{d^{3}} + \frac{a b^{4} x^{2}}{d^{3}} + \frac{b^{5} x^{3}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(a*d/b+d*x)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212677, size = 54, normalized size = 3.18 \[ \frac{b^{5} d^{6} x^{3} + 3 \, a b^{4} d^{6} x^{2} + 3 \, a^{2} b^{3} d^{6} x}{3 \, d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + a*d/b)^3,x, algorithm="giac")

[Out]

1/3*(b^5*d^6*x^3 + 3*a*b^4*d^6*x^2 + 3*a^2*b^3*d^6*x)/d^9