3.295 \(\int (a+b x)^{3/2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (a+b x)^{5/2}}{5 b} \]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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Rubi [A]  time = 0.00691899, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 (a+b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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Rubi in Sympy [A]  time = 1.25623, size = 12, normalized size = 0.75 \[ \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a + b*x)**(5/2)/(5*b)

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Mathematica [A]  time = 0.00716154, size = 16, normalized size = 1. \[ \frac{2 (a+b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[{\frac{2}{5\,b} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2),x)

[Out]

2/5*(b*x+a)^(5/2)/b

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Maxima [A]  time = 1.3379, size = 16, normalized size = 1. \[ \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2),x, algorithm="maxima")

[Out]

2/5*(b*x + a)^(5/2)/b

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Fricas [A]  time = 0.216995, size = 38, normalized size = 2.38 \[ \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2),x, algorithm="fricas")

[Out]

2/5*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)/b

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Sympy [A]  time = 0.071451, size = 12, normalized size = 0.75 \[ \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2),x)

[Out]

2*(a + b*x)**(5/2)/(5*b)

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GIAC/XCAS [A]  time = 0.217825, size = 16, normalized size = 1. \[ \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2),x, algorithm="giac")

[Out]

2/5*(b*x + a)^(5/2)/b