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 = 0.0137814, 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, Rules used = {43} \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac{2 (a+b x)^{7/2}}{7 b^3}-\frac{4 a (a+b x)^{5/2}}{5 b^3} \]

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)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \sqrt{a+b x} \, dx &=\int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx\\ &=\frac{2 a^2 (a+b x)^{3/2}}{3 b^3}-\frac{4 a (a+b x)^{5/2}}{5 b^3}+\frac{2 (a+b x)^{7/2}}{7 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0152363, size = 35, normalized size = 0.66 \[ \frac{2 (a+b x)^{3/2} \left (8 a^2-12 a b x+15 b^2 x^2\right )}{105 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{\frac{30\,{b}^{2}{x}^{2}-24\,axb+16\,{a}^{2}}{105\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

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 = 0.940389, size = 55, normalized size = 1.04 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(1/2),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 = 1.6045, size = 97, normalized size = 1.83 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(1/2),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 [B]  time = 1.63627, size = 666, normalized size = 12.57 \begin{align*} \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}} \end{align*}

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 + 315*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*sq
rt(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 [A]  time = 1.10159, size = 50, normalized size = 0.94 \begin{align*} \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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