3.179 \(\int x^2 \sqrt{a+b x} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac{2 (a+b x)^{7/2}}{7 b^3}-\frac{4 a (a+b x)^{5/2}}{5 b^3} \]

[Out]

(2*a^2*(a + b*x)^(3/2))/(3*b^3) - (4*a*(a + b*x)^(5/2))/(5*b^3) + (2*(a + b*x)^(
7/2))/(7*b^3)

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Rubi [A]  time = 0.0381906, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac{2 (a+b x)^{7/2}}{7 b^3}-\frac{4 a (a+b x)^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a + b*x],x]

[Out]

(2*a^2*(a + b*x)^(3/2))/(3*b^3) - (4*a*(a + b*x)^(5/2))/(5*b^3) + (2*(a + b*x)^(
7/2))/(7*b^3)

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Rubi in Sympy [A]  time = 4.03145, size = 49, normalized size = 0.92 \[ \frac{2 a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} - \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*(a + b*x)**(3/2)/(3*b**3) - 4*a*(a + b*x)**(5/2)/(5*b**3) + 2*(a + b*x)**
(7/2)/(7*b**3)

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Mathematica [A]  time = 0.0152769, size = 46, normalized size = 0.87 \[ \frac{2 \sqrt{a+b x} \left (8 a^3-4 a^2 b x+3 a b^2 x^2+15 b^3 x^3\right )}{105 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(8*a^3 - 4*a^2*b*x + 3*a*b^2*x^2 + 15*b^3*x^3))/(105*b^3)

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Maple [A]  time = 0.004, size = 32, normalized size = 0.6 \[{\frac{30\,{b}^{2}{x}^{2}-24\,axb+16\,{a}^{2}}{105\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(1/2),x)

[Out]

2/105*(b*x+a)^(3/2)*(15*b^2*x^2-12*a*b*x+8*a^2)/b^3

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Maxima [A]  time = 1.36013, size = 55, normalized size = 1.04 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{5}{2}} a}{5 \, b^{3}} + \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(b*x + a)^(7/2)/b^3 - 4/5*(b*x + a)^(5/2)*a/b^3 + 2/3*(b*x + a)^(3/2)*a^2/b^
3

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Fricas [A]  time = 0.20006, size = 57, normalized size = 1.08 \[ \frac{2 \,{\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*b^3*x^3 + 3*a*b^2*x^2 - 4*a^2*b*x + 8*a^3)*sqrt(b*x + a)/b^3

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Sympy [A]  time = 2.70939, size = 666, normalized size = 12.57 \[ \frac{16 a^{\frac{23}{2}} \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{23}{2}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac{40 a^{\frac{21}{2}} b x \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{21}{2}} b x}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac{30 a^{\frac{19}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{19}{2}} b^{2} x^{2}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac{40 a^{\frac{17}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{17}{2}} b^{3} x^{3}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac{100 a^{\frac{15}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac{96 a^{\frac{13}{2}} b^{5} x^{5} \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac{30 a^{\frac{11}{2}} b^{6} x^{6} \sqrt{1 + \frac{b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*a**(23/2)*sqrt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x*
*2 + 105*a**5*b**6*x**3) - 16*a**(23/2)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a
**6*b**5*x**2 + 105*a**5*b**6*x**3) + 40*a**(21/2)*b*x*sqrt(1 + b*x/a)/(105*a**8
*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 48*a**(21/2
)*b*x/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3
) + 30*a**(19/2)*b**2*x**2*sqrt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 31
5*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 48*a**(19/2)*b**2*x**2/(105*a**8*b**3 +
315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 40*a**(17/2)*b**3*
x**3*sqrt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105
*a**5*b**6*x**3) - 16*a**(17/2)*b**3*x**3/(105*a**8*b**3 + 315*a**7*b**4*x + 315
*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 100*a**(15/2)*b**4*x**4*sqrt(1 + b*x/a)/
(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 96
*a**(13/2)*b**5*x**5*sqrt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6
*b**5*x**2 + 105*a**5*b**6*x**3) + 30*a**(11/2)*b**6*x**6*sqrt(1 + b*x/a)/(105*a
**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3)

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GIAC/XCAS [A]  time = 0.198964, size = 62, normalized size = 1.17 \[ \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )}}{105 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*
a^2*b^12)/b^15