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

Optimal. Leaf size=51 \[ \frac{2 a^2 \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{5/2}}{5 b^3}-\frac{4 a (a+b x)^{3/2}}{3 b^3} \]

[Out]

(2*a^2*Sqrt[a + b*x])/b^3 - (4*a*(a + b*x)^(3/2))/(3*b^3) + (2*(a + b*x)^(5/2))/(5*b^3)

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Rubi [A]  time = 0.012585, antiderivative size = 51, 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} \[ \frac{2 a^2 \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{5/2}}{5 b^3}-\frac{4 a (a+b x)^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a + b*x],x]

[Out]

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

Mathematica [A]  time = 0.0140645, size = 35, normalized size = 0.69 \[ \frac{2 \sqrt{a+b x} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 0.6 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}-8\,axb+16\,{a}^{2}}{15\,{b}^{3}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^(1/2),x)

[Out]

2/15*(b*x+a)^(1/2)*(3*b^2*x^2-4*a*b*x+8*a^2)/b^3

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Maxima [A]  time = 0.965589, size = 55, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} a}{3 \, b^{3}} + \frac{2 \, \sqrt{b x + a} a^{2}}{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]

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

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Fricas [A]  time = 1.61231, size = 73, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a}}{15 \, 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]

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

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Sympy [B]  time = 1.56892, size = 600, normalized size = 11.76 \begin{align*} \frac{16 a^{\frac{21}{2}} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{21}{2}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{40 a^{\frac{19}{2}} b x \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{19}{2}} b x}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{30 a^{\frac{17}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{17}{2}} b^{2} x^{2}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{10 a^{\frac{15}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{15}{2}} b^{3} x^{3}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{10 a^{\frac{13}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{6 a^{\frac{11}{2}} b^{5} x^{5} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 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**(21/2)*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 16*a**(
21/2)/(15*a**8*b**3 + 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 40*a**(19/2)*b*x*sqrt(1 + b*x/
a)/(15*a**8*b**3 + 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 48*a**(19/2)*b*x/(15*a**8*b**3 +
45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 30*a**(17/2)*b**2*x**2*sqrt(1 + b*x/a)/(15*a**8*b**3
+ 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 48*a**(17/2)*b**2*x**2/(15*a**8*b**3 + 45*a**7*b*
*4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 10*a**(15/2)*b**3*x**3*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7
*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 16*a**(15/2)*b**3*x**3/(15*a**8*b**3 + 45*a**7*b**4*x + 45*
a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 10*a**(13/2)*b**4*x**4*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7*b**4*x +
45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 6*a**(11/2)*b**5*x**5*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7*b**4*x
+ 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3)

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Giac [A]  time = 1.06844, size = 50, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )}}{15 \, 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]

2/15*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)/b^3