3.523 \(\int \frac{x^4}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^5}{5 a \left (a+b x^2\right )^{7/2}} \]

[Out]

x^5/(5*a*(a + b*x^2)^(7/2)) + (2*b*x^7)/(35*a^2*(a + b*x^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0106348, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^5}{5 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^2)^(9/2),x]

[Out]

x^5/(5*a*(a + b*x^2)^(7/2)) + (2*b*x^7)/(35*a^2*(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^4}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{x^5}{5 a \left (a+b x^2\right )^{7/2}}+\frac{(2 b) \int \frac{x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{5 a}\\ &=\frac{x^5}{5 a \left (a+b x^2\right )^{7/2}}+\frac{2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0115358, size = 31, normalized size = 0.7 \[ \frac{7 a x^5+2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^2)^(9/2),x]

[Out]

(7*a*x^5 + 2*b*x^7)/(35*a^2*(a + b*x^2)^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 28, normalized size = 0.6 \begin{align*}{\frac{{x}^{5} \left ( 2\,b{x}^{2}+7\,a \right ) }{35\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2+a)^(9/2),x)

[Out]

1/35*x^5*(2*b*x^2+7*a)/(b*x^2+a)^(7/2)/a^2

________________________________________________________________________________________

Maxima [B]  time = 1.85136, size = 115, normalized size = 2.61 \begin{align*} -\frac{x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{3 \, x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/4*x^3/((b*x^2 + a)^(7/2)*b) + 3/140*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*x/(sqrt(b*x^2 + a)*a^2*b^2) + 1/35*x/(
(b*x^2 + a)^(3/2)*a*b^2) - 3/28*a*x/((b*x^2 + a)^(7/2)*b^2)

________________________________________________________________________________________

Fricas [A]  time = 1.27116, size = 146, normalized size = 3.32 \begin{align*} \frac{{\left (2 \, b x^{7} + 7 \, a x^{5}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (a^{2} b^{4} x^{8} + 4 \, a^{3} b^{3} x^{6} + 6 \, a^{4} b^{2} x^{4} + 4 \, a^{5} b x^{2} + a^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.04143, size = 199, normalized size = 4.52 \begin{align*} \frac{7 a x^{5}}{35 a^{\frac{11}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{9}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{7}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 35 a^{\frac{5}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{7}}{35 a^{\frac{11}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{9}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{7}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 35 a^{\frac{5}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**2+a)**(9/2),x)

[Out]

7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sq
rt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a)) + 2*b*x**7/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105
*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1
+ b*x**2/a))

________________________________________________________________________________________

Giac [A]  time = 3.01034, size = 39, normalized size = 0.89 \begin{align*} \frac{x^{5}{\left (\frac{2 \, b x^{2}}{a^{2}} + \frac{7}{a}\right )}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/35*x^5*(2*b*x^2/a^2 + 7/a)/(b*x^2 + a)^(7/2)