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

Optimal. Leaf size=67 \[ \frac{\sqrt{a+b x^2} (A b-2 a B)}{b^3}+\frac{a (A b-a B)}{b^3 \sqrt{a+b x^2}}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0555836, antiderivative size = 67, 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} \[ \frac{\sqrt{a+b x^2} (A b-2 a B)}{b^3}+\frac{a (A b-a B)}{b^3 \sqrt{a+b x^2}}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0320527, size = 55, normalized size = 0.82 \[ \frac{-8 a^2 B+a \left (6 A b-4 b B x^2\right )+b^2 x^2 \left (3 A+B x^2\right )}{3 b^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^2*B + b^2*x^2*(3*A + B*x^2) + a*(6*A*b - 4*b*B*x^2))/(3*b^3*Sqrt[a + b*x^2])

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Maple [A]  time = 0.003, size = 52, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}B{x}^{4}+3\,A{b}^{2}{x}^{2}-4\,Bab{x}^{2}+6\,Aab-8\,{a}^{2}B}{3\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(B*b^2*x^4+3*A*b^2*x^2-4*B*a*b*x^2+6*A*a*b-8*B*a^2)/(b*x^2+a)^(1/2)/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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53889, size = 131, normalized size = 1.96 \begin{align*} \frac{{\left (B b^{2} x^{4} - 8 \, B a^{2} + 6 \, A a b -{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{2} + a 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)^(3/2),x, algorithm="fricas")

[Out]

1/3*(B*b^2*x^4 - 8*B*a^2 + 6*A*a*b - (4*B*a*b - 3*A*b^2)*x^2)*sqrt(b*x^2 + a)/(b^4*x^2 + a*b^3)

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Sympy [A]  time = 0.928273, size = 117, normalized size = 1.75 \begin{align*} \begin{cases} \frac{2 A a}{b^{2} \sqrt{a + b x^{2}}} + \frac{A x^{2}}{b \sqrt{a + b x^{2}}} - \frac{8 B a^{2}}{3 b^{3} \sqrt{a + b x^{2}}} - \frac{4 B a x^{2}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{B x^{4}}{3 b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{4}}{4} + \frac{B x^{6}}{6}}{a^{\frac{3}{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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.12725, size = 88, normalized size = 1.31 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B - 6 \, \sqrt{b x^{2} + a} B a + 3 \, \sqrt{b x^{2} + a} A b - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x^{2} + a}}}{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)^(3/2),x, algorithm="giac")

[Out]

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