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

Optimal. Leaf size=97 \[ -\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} (A b-3 a B)}{b^4}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0759132, antiderivative size = 97, 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} \[ -\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} (A b-3 a B)}{b^4}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (A+B x)}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^2 (-A b+a B)}{b^3 (a+b x)^{5/2}}+\frac{a (-2 A b+3 a B)}{b^3 (a+b x)^{3/2}}+\frac{A b-3 a B}{b^3 \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a+b x^2}}+\frac{(A b-3 a B) \sqrt{a+b x^2}}{b^4}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0498619, size = 73, normalized size = 0.75 \[ \frac{8 a^2 b \left (A-3 B x^2\right )-16 a^3 B-6 a b^2 x^2 \left (B x^2-2 A\right )+b^3 x^4 \left (3 A+B x^2\right )}{3 b^4 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*a^3*B + 8*a^2*b*(A - 3*B*x^2) - 6*a*b^2*x^2*(-2*A + B*x^2) + b^3*x^4*(3*A + B*x^2))/(3*b^4*(a + b*x^2)^(3
/2))

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Maple [A]  time = 0.004, size = 76, normalized size = 0.8 \begin{align*}{\frac{{x}^{6}B{b}^{3}+3\,A{b}^{3}{x}^{4}-6\,Ba{b}^{2}{x}^{4}+12\,Aa{b}^{2}{x}^{2}-24\,B{a}^{2}b{x}^{2}+8\,A{a}^{2}b-16\,B{a}^{3}}{3\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(B*b^3*x^6+3*A*b^3*x^4-6*B*a*b^2*x^4+12*A*a*b^2*x^2-24*B*a^2*b*x^2+8*A*a^2*b-16*B*a^3)/(b*x^2+a)^(3/2)/b^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56345, size = 201, normalized size = 2.07 \begin{align*} \frac{{\left (B b^{3} x^{6} - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{4} - 16 \, B a^{3} + 8 \, A a^{2} b - 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(B*b^3*x^6 - 3*(2*B*a*b^2 - A*b^3)*x^4 - 16*B*a^3 + 8*A*a^2*b - 12*(2*B*a^2*b - A*a*b^2)*x^2)*sqrt(b*x^2 +
a)/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

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Sympy [A]  time = 1.82665, size = 337, normalized size = 3.47 \begin{align*} \begin{cases} \frac{8 A a^{2} b}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} + \frac{12 A a b^{2} x^{2}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} + \frac{3 A b^{3} x^{4}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} - \frac{16 B a^{3}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} - \frac{24 B a^{2} b x^{2}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} - \frac{6 B a b^{2} x^{4}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} + \frac{B b^{3} x^{6}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{8}}{8}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*A*a**2*b/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) + 12*A*a*b**2*x**2/(3*a*b**4*
sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) + 3*A*b**3*x**4/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt
(a + b*x**2)) - 16*B*a**3/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) - 24*B*a**2*b*x**2/(3*a*b
**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) - 6*B*a*b**2*x**4/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**
2*sqrt(a + b*x**2)) + B*b**3*x**6/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)), Ne(b, 0)), ((A*x
**6/6 + B*x**8/8)/a**(5/2), True))

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Giac [A]  time = 1.12941, size = 124, normalized size = 1.28 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B - 9 \, \sqrt{b x^{2} + a} B a + 3 \, \sqrt{b x^{2} + a} A b - \frac{9 \,{\left (b x^{2} + a\right )} B a^{2} - B a^{3} - 6 \,{\left (b x^{2} + a\right )} A a b + A a^{2} b}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{3 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((b*x^2 + a)^(3/2)*B - 9*sqrt(b*x^2 + a)*B*a + 3*sqrt(b*x^2 + a)*A*b - (9*(b*x^2 + a)*B*a^2 - B*a^3 - 6*(b
*x^2 + a)*A*a*b + A*a^2*b)/(b*x^2 + a)^(3/2))/b^4