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

Optimal. Leaf size=41 \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0321411, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^{3/2}}+\frac{B}{b \sqrt{a+b x}}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b-a B}{b^2 \sqrt{a+b x^2}}+\frac{B \sqrt{a+b x^2}}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.0207583, size = 30, normalized size = 0.73 \[ \frac{2 a B-A b+b B x^2}{b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*} -{\frac{-bB{x}^{2}+Ab-2\,Ba}{{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.54058, size = 80, normalized size = 1.95 \begin{align*} \frac{{\left (B b x^{2} + 2 \, B a - A b\right )} \sqrt{b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.588677, size = 66, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{A}{b \sqrt{a + b x^{2}}} + \frac{2 B a}{b^{2} \sqrt{a + b x^{2}}} + \frac{B x^{2}}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.10764, size = 46, normalized size = 1.12 \begin{align*} \frac{\sqrt{b x^{2} + a} B + \frac{B a - A b}{\sqrt{b x^{2} + a}}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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