3.63 \(\int \frac{1}{\left (a+c x^2\right )^{9/2}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0425395, 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 \[ \frac{16 x}{35 a^4 \sqrt{a+c x^2}}+\frac{8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac{6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac{x}{7 a \left (a+c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(-9/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.01404, size = 70, normalized size = 0.91 \[ \frac{x}{7 a \left (a + c x^{2}\right )^{\frac{7}{2}}} + \frac{6 x}{35 a^{2} \left (a + c x^{2}\right )^{\frac{5}{2}}} + \frac{8 x}{35 a^{3} \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{16 x}{35 a^{4} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+a)**(9/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0316585, size = 51, normalized size = 0.66 \[ \frac{x \left (35 a^3+70 a^2 c x^2+56 a c^2 x^4+16 c^3 x^6\right )}{35 a^4 \left (a+c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(-9/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 48, normalized size = 0.6 \[{\frac{x \left ( 16\,{c}^{3}{x}^{6}+56\,a{c}^{2}{x}^{4}+70\,{a}^{2}c{x}^{2}+35\,{a}^{3} \right ) }{35\,{a}^{4}} \left ( c{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+a)^(9/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.698505, size = 82, normalized size = 1.06 \[ \frac{16 \, x}{35 \, \sqrt{c x^{2} + a} a^{4}} + \frac{8 \, x}{35 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{6 \, x}{35 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} a^{2}} + \frac{x}{7 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(-9/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.251975, size = 123, normalized size = 1.6 \[ \frac{{\left (16 \, c^{3} x^{7} + 56 \, a c^{2} x^{5} + 70 \, a^{2} c x^{3} + 35 \, a^{3} x\right )} \sqrt{c x^{2} + a}}{35 \,{\left (a^{4} c^{4} x^{8} + 4 \, a^{5} c^{3} x^{6} + 6 \, a^{6} c^{2} x^{4} + 4 \, a^{7} c x^{2} + a^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(-9/2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 9.96288, size = 1265, normalized size = 16.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+a)**(9/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216599, size = 74, normalized size = 0.96 \[ \frac{{\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \, c^{3} x^{2}}{a^{4}} + \frac{7 \, c^{2}}{a^{3}}\right )} + \frac{35 \, c}{a^{2}}\right )} x^{2} + \frac{35}{a}\right )} x}{35 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(-9/2),x, algorithm="giac")

[Out]

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