3.460 \(\int \left (a+b \sqrt{c+d x}\right )^2 \, dx\)

Optimal. Leaf size=41 \[ a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0629925, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[c + d*x])^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a b \left (c + d x\right )^{\frac{3}{2}}}{3 d} + \frac{b^{2} \int ^{c + d x} x\, dx}{d} + \frac{\int ^{c + d x} a^{2}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a*b*(c + d*x)**(3/2)/(3*d) + b**2*Integral(x, (x, c + d*x))/d + Integral(a**2,
(x, c + d*x))/d

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Mathematica [A]  time = 0.0246211, size = 41, normalized size = 1. \[ \frac{(c+d x) \left (6 a^2+8 a b \sqrt{c+d x}+3 b^2 (c+d x)\right )}{6 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[c + d*x])^2,x]

[Out]

((c + d*x)*(6*a^2 + 8*a*b*Sqrt[c + d*x] + 3*b^2*(c + d*x)))/(6*d)

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Maple [A]  time = 0.002, size = 35, normalized size = 0.9 \[{b}^{2} \left ({\frac{d{x}^{2}}{2}}+cx \right ) +{\frac{4\,ab}{3\,d} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(d*x+c)^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 0.726655, size = 47, normalized size = 1.15 \[ \frac{1}{2} \,{\left (d x^{2} + 2 \, c x\right )} b^{2} + a^{2} x + \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a b}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.295526, size = 66, normalized size = 1.61 \[ \frac{3 \, b^{2} d^{2} x^{2} + 6 \,{\left (b^{2} c + a^{2}\right )} d x + 8 \,{\left (a b d x + a b c\right )} \sqrt{d x + c}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2,x, algorithm="fricas")

[Out]

1/6*(3*b^2*d^2*x^2 + 6*(b^2*c + a^2)*d*x + 8*(a*b*d*x + a*b*c)*sqrt(d*x + c))/d

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Sympy [A]  time = 0.538076, size = 68, normalized size = 1.66 \[ \begin{cases} a^{2} x + \frac{4 a b c \sqrt{c + d x}}{3 d} + \frac{4 a b x \sqrt{c + d x}}{3} + b^{2} c x + \frac{b^{2} d x^{2}}{2} & \text{for}\: d \neq 0 \\x \left (a + b \sqrt{c}\right )^{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**2*x + 4*a*b*c*sqrt(c + d*x)/(3*d) + 4*a*b*x*sqrt(c + d*x)/3 + b**2
*c*x + b**2*d*x**2/2, Ne(d, 0)), (x*(a + b*sqrt(c))**2, True))

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GIAC/XCAS [A]  time = 0.273487, size = 53, normalized size = 1.29 \[ \frac{3 \,{\left (d x + c\right )}^{2} b^{2} + 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b + 6 \,{\left (d x + c\right )} a^{2}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2,x, algorithm="giac")

[Out]

1/6*(3*(d*x + c)^2*b^2 + 8*(d*x + c)^(3/2)*a*b + 6*(d*x + c)*a^2)/d