3.1380 \(\int \sqrt{c+d x} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (c+d x)^{3/2}}{3 d} \]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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Rubi [A]  time = 0.00717338, 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 \[ \frac{2 (c+d x)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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Rubi in Sympy [A]  time = 1.37057, size = 12, normalized size = 0.75 \[ \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2),x)

[Out]

2*(c + d*x)**(3/2)/(3*d)

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Mathematica [A]  time = 0.00524516, size = 16, normalized size = 1. \[ \frac{2 (c+d x)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{2}{3\,d} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/2),x)

[Out]

2/3*(d*x+c)^(3/2)/d

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Maxima [A]  time = 1.33734, size = 16, normalized size = 1. \[ \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c),x, algorithm="maxima")

[Out]

2/3*(d*x + c)^(3/2)/d

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Fricas [A]  time = 0.208801, size = 16, normalized size = 1. \[ \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c),x, algorithm="fricas")

[Out]

2/3*(d*x + c)^(3/2)/d

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Sympy [A]  time = 0.033368, size = 12, normalized size = 0.75 \[ \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/2),x)

[Out]

2*(c + d*x)**(3/2)/(3*d)

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GIAC/XCAS [A]  time = 0.214285, size = 16, normalized size = 1. \[ \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c),x, algorithm="giac")

[Out]

2/3*(d*x + c)^(3/2)/d