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

Optimal. Leaf size=44 \[ -\frac{A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0329654, antiderivative size = 44, 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} \[ -\frac{A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*b - a*B)/(3*b^2*(a + b*x^2)^(3/2)) - B/(b^2*Sqrt[a + b*x^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 )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^{5/2}}+\frac{B}{b (a+b x)^{3/2}}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0226032, size = 34, normalized size = 0.77 \[ \frac{-2 a B-A b-3 b B x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*} -{\frac{3\,bB{x}^{2}+Ab+2\,Ba}{3\,{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*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.41398, size = 68, normalized size = 1.55 \begin{align*} -\frac{B x^{2}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} - \frac{2 \, B a}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2}} - \frac{A}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5358, size = 111, normalized size = 2.52 \begin{align*} -\frac{{\left (3 \, B b x^{2} + 2 \, B a + A b\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.06531, size = 143, normalized size = 3.25 \begin{align*} \begin{cases} - \frac{A b}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{2 B a}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{3 B 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{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{a^{\frac{5}{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)**(5/2),x)

[Out]

Piecewise((-A*b/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 2*B*a/(3*a*b**2*sqrt(a + b*x**2)
+ 3*b**3*x**2*sqrt(a + b*x**2)) - 3*B*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)), ((A*x**2/2 + B*x**4/4)/a**(5/2), True))

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Giac [A]  time = 1.09832, size = 43, normalized size = 0.98 \begin{align*} -\frac{3 \,{\left (b x^{2} + a\right )} B - B a + A b}{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*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

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