3.525 \(\int \frac{x^2}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{8 b^2 x^7}{105 a^3 \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^3}{3 a \left (a+b x^2\right )^{7/2}} \]

[Out]

x^3/(3*a*(a + b*x^2)^(7/2)) + (4*b*x^5)/(15*a^2*(a + b*x^2)^(7/2)) + (8*b^2*x^7)/(105*a^3*(a + b*x^2)^(7/2))

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Rubi [A]  time = 0.0205032, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{8 b^2 x^7}{105 a^3 \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^3}{3 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^2)^(9/2),x]

[Out]

x^3/(3*a*(a + b*x^2)^(7/2)) + (4*b*x^5)/(15*a^2*(a + b*x^2)^(7/2)) + (8*b^2*x^7)/(105*a^3*(a + b*x^2)^(7/2))

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

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

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{x^3}{3 a \left (a+b x^2\right )^{7/2}}+\frac{(4 b) \int \frac{x^4}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a}\\ &=\frac{x^3}{3 a \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac{\left (8 b^2\right ) \int \frac{x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^2}\\ &=\frac{x^3}{3 a \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac{8 b^2 x^7}{105 a^3 \left (a+b x^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0131602, size = 42, normalized size = 0.62 \[ \frac{x^3 \left (35 a^2+28 a b x^2+8 b^2 x^4\right )}{105 a^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^2)^(9/2),x]

[Out]

(x^3*(35*a^2 + 28*a*b*x^2 + 8*b^2*x^4))/(105*a^3*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.005, size = 39, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( 8\,{b}^{2}{x}^{4}+28\,ab{x}^{2}+35\,{a}^{2} \right ) }{105\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2+a)^(9/2),x)

[Out]

1/105*x^3*(8*b^2*x^4+28*a*b*x^2+35*a^2)/(b*x^2+a)^(7/2)/a^3

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Maxima [A]  time = 2.70413, size = 95, normalized size = 1.4 \begin{align*} -\frac{x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{8 \, x}{105 \, \sqrt{b x^{2} + a} a^{3} b} + \frac{4 \, x}{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b} + \frac{x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/7*x/((b*x^2 + a)^(7/2)*b) + 8/105*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*x/((
b*x^2 + a)^(5/2)*a*b)

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Fricas [A]  time = 1.3595, size = 171, normalized size = 2.51 \begin{align*} \frac{{\left (8 \, b^{2} x^{7} + 28 \, a b x^{5} + 35 \, a^{2} x^{3}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{3} b^{4} x^{8} + 4 \, a^{4} b^{3} x^{6} + 6 \, a^{5} b^{2} x^{4} + 4 \, a^{6} b x^{2} + a^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(8*b^2*x^7 + 28*a*b*x^5 + 35*a^2*x^3)*sqrt(b*x^2 + a)/(a^3*b^4*x^8 + 4*a^4*b^3*x^6 + 6*a^5*b^2*x^4 + 4*a
^6*b*x^2 + a^7)

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Sympy [B]  time = 2.31883, size = 517, normalized size = 7.6 \begin{align*} \frac{35 a^{5} x^{3}}{105 a^{\frac{19}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{17}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 630 a^{\frac{15}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{13}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{11}{2}} b^{4} x^{8} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{63 a^{4} b x^{5}}{105 a^{\frac{19}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{17}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 630 a^{\frac{15}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{13}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{11}{2}} b^{4} x^{8} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{36 a^{3} b^{2} x^{7}}{105 a^{\frac{19}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{17}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 630 a^{\frac{15}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{13}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{11}{2}} b^{4} x^{8} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{8 a^{2} b^{3} x^{9}}{105 a^{\frac{19}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{17}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 630 a^{\frac{15}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 420 a^{\frac{13}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{11}{2}} b^{4} x^{8} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**2+a)**(9/2),x)

[Out]

35*a**5*x**3/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*
x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2
/a)) + 63*a**4*b*x**5/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15
/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1
+ b*x**2/a)) + 36*a**3*b**2*x**7/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a)
+ 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4
*x**8*sqrt(1 + b*x**2/a)) + 8*a**2*b**3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 +
b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**
(11/2)*b**4*x**8*sqrt(1 + b*x**2/a))

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Giac [A]  time = 2.5777, size = 58, normalized size = 0.85 \begin{align*} \frac{{\left (4 \, x^{2}{\left (\frac{2 \, b^{2} x^{2}}{a^{3}} + \frac{7 \, b}{a^{2}}\right )} + \frac{35}{a}\right )} x^{3}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*(4*x^2*(2*b^2*x^2/a^3 + 7*b/a^2) + 35/a)*x^3/(b*x^2 + a)^(7/2)