3.37 \(\int \frac{x^2 (A+B x)}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{x^2 (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0217641, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {805, 261} \[ -\frac{x^2 (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x))/(a + b*x^2)^(5/2),x]

[Out]

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

Rule 805

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[(m*(c*d*f + a*e*g))/(2*a*c*(p + 1)), Int[(d + e*
x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplif
y[m + 2*p + 3], 0] && LtQ[p, -1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0169559, size = 44, normalized size = 0.83 \[ \frac{-2 a^2 B-3 a b B x^2+A b^2 x^3}{3 a b^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x))/(a + b*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 41, normalized size = 0.8 \begin{align*}{\frac{A{x}^{3}{b}^{2}-3\,B{x}^{2}ab-2\,B{a}^{2}}{3\,a{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x+A)/(b*x^2+a)^(5/2),x)

[Out]

1/3*(A*b^2*x^3-3*B*a*b*x^2-2*B*a^2)/(b*x^2+a)^(3/2)/a/b^2

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Maxima [A]  time = 1.01277, size = 95, normalized size = 1.79 \begin{align*} -\frac{B x^{2}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} - \frac{A x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} + \frac{A x}{3 \, \sqrt{b x^{2} + a} a b} - \frac{2 \, B a}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

-B*x^2/((b*x^2 + a)^(3/2)*b) - 1/3*A*x/((b*x^2 + a)^(3/2)*b) + 1/3*A*x/(sqrt(b*x^2 + a)*a*b) - 2/3*B*a/((b*x^2
+ a)^(3/2)*b^2)

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Fricas [A]  time = 1.54911, size = 128, normalized size = 2.42 \begin{align*} \frac{{\left (A b^{2} x^{3} - 3 \, B a b x^{2} - 2 \, B a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/3*(A*b^2*x^3 - 3*B*a*b*x^2 - 2*B*a^2)*sqrt(b*x^2 + a)/(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)

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Sympy [B]  time = 10.6081, size = 141, normalized size = 2.66 \begin{align*} \frac{A x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + B \left (\begin{cases} - \frac{2 a}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{3 b x^{2}}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(B*x+A)/(b*x**2+a)**(5/2),x)

[Out]

A*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b*x**2*sqrt(1 + b*x**2/a)) + B*Piecewise((-2*a/(3*a*b**2*sq
rt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 3*b*x**2/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*
x**2)), Ne(b, 0)), (x**4/(4*a**(5/2)), True))

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Giac [A]  time = 1.18286, size = 49, normalized size = 0.92 \begin{align*} \frac{{\left (\frac{A x}{a} - \frac{3 \, B}{b}\right )} x^{2} - \frac{2 \, B a}{b^{2}}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/3*((A*x/a - 3*B/b)*x^2 - 2*B*a/b^2)/(b*x^2 + a)^(3/2)