3.178 \(\int x \sqrt{a+b x} \, dx\)

Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{5/2}}{5 b^2}-\frac{2 a (a+b x)^{3/2}}{3 b^2} \]

[Out]

(-2*a*(a + b*x)^(3/2))/(3*b^2) + (2*(a + b*x)^(5/2))/(5*b^2)

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Rubi [A]  time = 0.0238743, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (a+b x)^{5/2}}{5 b^2}-\frac{2 a (a+b x)^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a + b*x],x]

[Out]

(-2*a*(a + b*x)^(3/2))/(3*b^2) + (2*(a + b*x)^(5/2))/(5*b^2)

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Rubi in Sympy [A]  time = 2.43236, size = 31, normalized size = 0.91 \[ - \frac{2 a \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(a + b*x)**(3/2)/(3*b**2) + 2*(a + b*x)**(5/2)/(5*b**2)

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Mathematica [A]  time = 0.0123065, size = 34, normalized size = 1. \[ \frac{2 \sqrt{a+b x} \left (-2 a^2+a b x+3 b^2 x^2\right )}{15 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(-2*a^2 + a*b*x + 3*b^2*x^2))/(15*b^2)

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Maple [A]  time = 0.002, size = 21, normalized size = 0.6 \[ -{\frac{-6\,bx+4\,a}{15\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(1/2),x)

[Out]

-2/15*(b*x+a)^(3/2)*(-3*b*x+2*a)/b^2

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Maxima [A]  time = 1.34248, size = 35, normalized size = 1.03 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(b*x + a)^(5/2)/b^2 - 2/3*(b*x + a)^(3/2)*a/b^2

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Fricas [A]  time = 0.202072, size = 41, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*b^2*x^2 + a*b*x - 2*a^2)*sqrt(b*x + a)/b^2

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Sympy [A]  time = 1.69484, size = 202, normalized size = 5.94 \[ - \frac{4 a^{\frac{9}{2}} \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{4 a^{\frac{9}{2}}}{15 a^{2} b^{2} + 15 a b^{3} x} - \frac{2 a^{\frac{7}{2}} b x \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{4 a^{\frac{7}{2}} b x}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{8 a^{\frac{5}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{6 a^{\frac{3}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**(1/2),x)

[Out]

-4*a**(9/2)*sqrt(1 + b*x/a)/(15*a**2*b**2 + 15*a*b**3*x) + 4*a**(9/2)/(15*a**2*b
**2 + 15*a*b**3*x) - 2*a**(7/2)*b*x*sqrt(1 + b*x/a)/(15*a**2*b**2 + 15*a*b**3*x)
+ 4*a**(7/2)*b*x/(15*a**2*b**2 + 15*a*b**3*x) + 8*a**(5/2)*b**2*x**2*sqrt(1 + b
*x/a)/(15*a**2*b**2 + 15*a*b**3*x) + 6*a**(3/2)*b**3*x**3*sqrt(1 + b*x/a)/(15*a*
*2*b**2 + 15*a*b**3*x)

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GIAC/XCAS [A]  time = 0.198204, size = 34, normalized size = 1. \[ \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)/b^2