3.588 \(\int \frac{x^3 (A+B x^2)}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{A b-2 a B}{b^3 \sqrt{a+b x^2}}+\frac{a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{B \sqrt{a+b x^2}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0535126, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ -\frac{A b-2 a B}{b^3 \sqrt{a+b x^2}}+\frac{a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{B \sqrt{a+b x^2}}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0346629, size = 54, normalized size = 0.79 \[ \frac{8 a^2 B-2 a b \left (A-6 B x^2\right )+3 b^2 x^2 \left (B x^2-A\right )}{3 b^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(8*a^2*B - 2*a*b*(A - 6*B*x^2) + 3*b^2*x^2*(-A + B*x^2))/(3*b^3*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.006, size = 53, normalized size = 0.8 \begin{align*} -{\frac{-3\,{b}^{2}B{x}^{4}+3\,A{b}^{2}{x}^{2}-12\,Bab{x}^{2}+2\,Aab-8\,{a}^{2}B}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

-1/3*(-3*B*b^2*x^4+3*A*b^2*x^2-12*B*a*b*x^2+2*A*a*b-8*B*a^2)/(b*x^2+a)^(3/2)/b^3

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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^3*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76132, size = 155, normalized size = 2.28 \begin{align*} \frac{{\left (3 \, B b^{2} x^{4} + 8 \, B a^{2} - 2 \, A a b + 3 \,{\left (4 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/3*(3*B*b^2*x^4 + 8*B*a^2 - 2*A*a*b + 3*(4*B*a*b - A*b^2)*x^2)*sqrt(b*x^2 + a)/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b
^3)

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Sympy [A]  time = 1.15936, size = 240, normalized size = 3.53 \begin{align*} \begin{cases} - \frac{2 A a b}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} - \frac{3 A b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{8 B a^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{12 B a b x^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{3 B b^{2} x^{4}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{4}}{4} + \frac{B x^{6}}{6}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

Piecewise((-2*A*a*b/(3*a*b**3*sqrt(a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2)) - 3*A*b**2*x**2/(3*a*b**3*sqrt(
a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2)) + 8*B*a**2/(3*a*b**3*sqrt(a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**
2)) + 12*B*a*b*x**2/(3*a*b**3*sqrt(a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2)) + 3*B*b**2*x**4/(3*a*b**3*sqrt(
a + b*x**2) + 3*b**4*x**2*sqrt(a + b*x**2)), Ne(b, 0)), ((A*x**4/4 + B*x**6/6)/a**(5/2), True))

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Giac [A]  time = 1.11266, size = 82, normalized size = 1.21 \begin{align*} \frac{3 \, \sqrt{b x^{2} + a} B + \frac{6 \,{\left (b x^{2} + a\right )} B a - B a^{2} - 3 \,{\left (b x^{2} + a\right )} A b + A a b}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/3*(3*sqrt(b*x^2 + a)*B + (6*(b*x^2 + a)*B*a - B*a^2 - 3*(b*x^2 + a)*A*b + A*a*b)/(b*x^2 + a)^(3/2))/b^3