3.527 \(\int \frac{1}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{16 x}{35 a^4 \sqrt{a+b x^2}}+\frac{8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{x}{7 a \left (a+b x^2\right )^{7/2}} \]

[Out]

x/(7*a*(a + b*x^2)^(7/2)) + (6*x)/(35*a^2*(a + b*x^2)^(5/2)) + (8*x)/(35*a^3*(a + b*x^2)^(3/2)) + (16*x)/(35*a
^4*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0160159, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{16 x}{35 a^4 \sqrt{a+b x^2}}+\frac{8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{x}{7 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(-9/2),x]

[Out]

x/(7*a*(a + b*x^2)^(7/2)) + (6*x)/(35*a^2*(a + b*x^2)^(5/2)) + (8*x)/(35*a^3*(a + b*x^2)^(3/2)) + (16*x)/(35*a
^4*Sqrt[a + b*x^2])

Rule 192

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

Rule 191

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{6 \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac{x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{24 \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac{16 \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac{x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac{16 x}{35 a^4 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0115378, size = 51, normalized size = 0.66 \[ \frac{x \left (70 a^2 b x^2+35 a^3+56 a b^2 x^4+16 b^3 x^6\right )}{35 a^4 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(-9/2),x]

[Out]

(x*(35*a^3 + 70*a^2*b*x^2 + 56*a*b^2*x^4 + 16*b^3*x^6))/(35*a^4*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.003, size = 48, normalized size = 0.6 \begin{align*}{\frac{x \left ( 16\,{b}^{3}{x}^{6}+56\,a{b}^{2}{x}^{4}+70\,{a}^{2}b{x}^{2}+35\,{a}^{3} \right ) }{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^(9/2),x)

[Out]

1/35*x*(16*b^3*x^6+56*a*b^2*x^4+70*a^2*b*x^2+35*a^3)/(b*x^2+a)^(7/2)/a^4

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Maxima [A]  time = 2.45681, size = 82, normalized size = 1.06 \begin{align*} \frac{16 \, x}{35 \, \sqrt{b x^{2} + a} a^{4}} + \frac{8 \, x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{6 \, x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2}} + \frac{x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

16/35*x/(sqrt(b*x^2 + a)*a^4) + 8/35*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*x/((b*x^
2 + a)^(7/2)*a)

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Fricas [A]  time = 1.31958, size = 192, normalized size = 2.49 \begin{align*} \frac{{\left (16 \, b^{3} x^{7} + 56 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 35 \, a^{3} x\right )} \sqrt{b x^{2} + a}}{35 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/35*(16*b^3*x^7 + 56*a*b^2*x^5 + 70*a^2*b*x^3 + 35*a^3*x)*sqrt(b*x^2 + a)/(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^
6*b^2*x^4 + 4*a^7*b*x^2 + a^8)

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Sympy [B]  time = 3.04892, size = 1265, normalized size = 16.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)**(9/2),x)

[Out]

35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**
4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a)
+ 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 175*a**13*b*x**
3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1
+ b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a*
*(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(35*
a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x*
*2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2
)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 429*a**11*b**3*x**7/(35*a**(37
/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a)
+ 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5
*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*a**(37/2)*sq
rt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*
a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10
*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 104*a**9*b**5*x**11/(35*a**(37/2)*sqrt(1 +
b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31
/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(
1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 16*a**8*b**6*x**13/(35*a**(37/2)*sqrt(1 + b*x**2
/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**
3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x
**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.42683, size = 74, normalized size = 0.96 \begin{align*} \frac{{\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \, b^{3} x^{2}}{a^{4}} + \frac{7 \, b^{2}}{a^{3}}\right )} + \frac{35 \, b}{a^{2}}\right )} x^{2} + \frac{35}{a}\right )} x}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

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