3.345 \(\int \frac{x^2}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{2 a^2}{b^3 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{3/2}}{3 b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0130437, antiderivative size = 49, 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} \[ -\frac{2 a^2}{b^3 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.026362, size = 34, normalized size = 0.69 \[ \frac{2 \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 32, normalized size = 0.7 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{2}+8\,abx+16\,{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08489, size = 55, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{3}} - \frac{4 \, \sqrt{b x + a} a}{b^{3}} - \frac{2 \, a^{2}}{\sqrt{b x + a} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.55304, size = 85, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )} \sqrt{b x + a}}{3 \,{\left (b^{4} x + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.03609, size = 534, normalized size = 10.9 \begin{align*} - \frac{16 a^{\frac{19}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{19}{2}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{40 a^{\frac{17}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{17}{2}} b x}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{30 a^{\frac{15}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{15}{2}} b^{2} x^{2}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{4 a^{\frac{13}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{13}{2}} b^{3} x^{3}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{2 a^{\frac{11}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 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)**(3/2),x)

[Out]

-16*a**(19/2)*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 16*a**(19/
2)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 40*a**(17/2)*b*x*sqrt(1 + b*x/a)/(3*a
**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 48*a**(17/2)*b*x/(3*a**8*b**3 + 9*a**7*b**4*
x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 30*a**(15/2)*b**2*x**2*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x
+ 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 48*a**(15/2)*b**2*x**2/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x*
*2 + 3*a**5*b**6*x**3) - 4*a**(13/2)*b**3*x**3*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2
+ 3*a**5*b**6*x**3) + 16*a**(13/2)*b**3*x**3/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x*
*3) + 2*a**(11/2)*b**4*x**4*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3
)

________________________________________________________________________________________

Giac [A]  time = 1.13146, size = 62, normalized size = 1.27 \begin{align*} -\frac{2 \, a^{2}}{\sqrt{b x + a} b^{3}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} b^{6} - 6 \, \sqrt{b x + a} a b^{6}\right )}}{3 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2*a^2/(sqrt(b*x + a)*b^3) + 2/3*((b*x + a)^(3/2)*b^6 - 6*sqrt(b*x + a)*a*b^6)/b^9