3.39 \(\int \sqrt{c+d (a+b x)} \, dx\)

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

[Out]

(2*(c + d*(a + b*x))^(3/2))/(3*b*d)

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Rubi [A]  time = 0.0241309, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 (d (a+b x)+c)^{3/2}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*(a + b*x)],x]

[Out]

(2*(c + d*(a + b*x))^(3/2))/(3*b*d)

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Rubi in Sympy [A]  time = 2.14334, size = 17, normalized size = 0.74 \[ \frac{2 \left (c + d \left (a + b x\right )\right )^{\frac{3}{2}}}{3 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c + d*(a + b*x))**(3/2)/(3*b*d)

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Mathematica [A]  time = 0.0158129, size = 23, normalized size = 1. \[ \frac{2 (d (a+b x)+c)^{3/2}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*(a + b*x)],x]

[Out]

(2*(c + d*(a + b*x))^(3/2))/(3*b*d)

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Maple [A]  time = 0.004, size = 20, normalized size = 0.9 \[{\frac{2}{3\,db} \left ( bdx+ad+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*(b*x+a))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.33501, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{3}{2}}}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((b*x + a)*d + c)^(3/2)/(b*d)

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Fricas [A]  time = 0.197036, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}}}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*d*x + a*d + c)^(3/2)/(b*d)

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Sympy [A]  time = 0.956185, size = 82, normalized size = 3.57 \[ \begin{cases} \sqrt{c} x & \text{for}\: b = 0 \wedge d = 0 \\x \sqrt{a d + c} & \text{for}\: b = 0 \\\sqrt{c} x & \text{for}\: d = 0 \\\frac{2 a \sqrt{a d + b d x + c}}{3 b} + \frac{2 x \sqrt{a d + b d x + c}}{3} + \frac{2 c \sqrt{a d + b d x + c}}{3 b d} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d*(b*x+a))**(1/2),x)

[Out]

Piecewise((sqrt(c)*x, Eq(b, 0) & Eq(d, 0)), (x*sqrt(a*d + c), Eq(b, 0)), (sqrt(c
)*x, Eq(d, 0)), (2*a*sqrt(a*d + b*d*x + c)/(3*b) + 2*x*sqrt(a*d + b*d*x + c)/3 +
2*c*sqrt(a*d + b*d*x + c)/(3*b*d), True))

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GIAC/XCAS [A]  time = 0.209776, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}}}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*d*x + a*d + c)^(3/2)/(b*d)