3.62 \(\int \frac{1}{(a+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{8 x}{15 a^3 \sqrt{a+c x^2}}+\frac{4 x}{15 a^2 \left (a+c x^2\right )^{3/2}}+\frac{x}{5 a \left (a+c x^2\right )^{5/2}} \]

[Out]

x/(5*a*(a + c*x^2)^(5/2)) + (4*x)/(15*a^2*(a + c*x^2)^(3/2)) + (8*x)/(15*a^3*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.0104298, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{8 x}{15 a^3 \sqrt{a+c x^2}}+\frac{4 x}{15 a^2 \left (a+c x^2\right )^{3/2}}+\frac{x}{5 a \left (a+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(-7/2),x]

[Out]

x/(5*a*(a + c*x^2)^(5/2)) + (4*x)/(15*a^2*(a + c*x^2)^(3/2)) + (8*x)/(15*a^3*Sqrt[a + c*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+c x^2\right )^{7/2}} \, dx &=\frac{x}{5 a \left (a+c x^2\right )^{5/2}}+\frac{4 \int \frac{1}{\left (a+c x^2\right )^{5/2}} \, dx}{5 a}\\ &=\frac{x}{5 a \left (a+c x^2\right )^{5/2}}+\frac{4 x}{15 a^2 \left (a+c x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=\frac{x}{5 a \left (a+c x^2\right )^{5/2}}+\frac{4 x}{15 a^2 \left (a+c x^2\right )^{3/2}}+\frac{8 x}{15 a^3 \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0109272, size = 40, normalized size = 0.69 \[ \frac{x \left (15 a^2+20 a c x^2+8 c^2 x^4\right )}{15 a^3 \left (a+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(-7/2),x]

[Out]

(x*(15*a^2 + 20*a*c*x^2 + 8*c^2*x^4))/(15*a^3*(a + c*x^2)^(5/2))

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Maple [A]  time = 0.048, size = 37, normalized size = 0.6 \begin{align*}{\frac{x \left ( 8\,{c}^{2}{x}^{4}+20\,a{x}^{2}c+15\,{a}^{2} \right ) }{15\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+a)^(7/2),x)

[Out]

1/15*x*(8*c^2*x^4+20*a*c*x^2+15*a^2)/(c*x^2+a)^(5/2)/a^3

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Maxima [A]  time = 1.0683, size = 62, normalized size = 1.07 \begin{align*} \frac{8 \, x}{15 \, \sqrt{c x^{2} + a} a^{3}} + \frac{4 \, x}{15 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a^{2}} + \frac{x}{5 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(7/2),x, algorithm="maxima")

[Out]

8/15*x/(sqrt(c*x^2 + a)*a^3) + 4/15*x/((c*x^2 + a)^(3/2)*a^2) + 1/5*x/((c*x^2 + a)^(5/2)*a)

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Fricas [A]  time = 2.25931, size = 146, normalized size = 2.52 \begin{align*} \frac{{\left (8 \, c^{2} x^{5} + 20 \, a c x^{3} + 15 \, a^{2} x\right )} \sqrt{c x^{2} + a}}{15 \,{\left (a^{3} c^{3} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{5} c x^{2} + a^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(7/2),x, algorithm="fricas")

[Out]

1/15*(8*c^2*x^5 + 20*a*c*x^3 + 15*a^2*x)*sqrt(c*x^2 + a)/(a^3*c^3*x^6 + 3*a^4*c^2*x^4 + 3*a^5*c*x^2 + a^6)

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Sympy [B]  time = 2.28901, size = 413, normalized size = 7.12 \begin{align*} \frac{15 a^{5} x}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{4} c x^{3}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{28 a^{3} c^{2} x^{5}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{8 a^{2} c^{3} x^{7}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+a)**(7/2),x)

[Out]

15*a**5*x/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*s
qrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a)) + 35*a**4*c*x**3/(15*a**(17/2)*sqrt(1 + c*x**2/
a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x*
*6*sqrt(1 + c*x**2/a)) + 28*a**3*c**2*x**5/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x
**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a)) + 8*a**2*c**3*
x**7/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1
+ c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.24094, size = 55, normalized size = 0.95 \begin{align*} \frac{{\left (4 \, x^{2}{\left (\frac{2 \, c^{2} x^{2}}{a^{3}} + \frac{5 \, c}{a^{2}}\right )} + \frac{15}{a}\right )} x}{15 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(7/2),x, algorithm="giac")

[Out]

1/15*(4*x^2*(2*c^2*x^2/a^3 + 5*c/a^2) + 15/a)*x/(c*x^2 + a)^(5/2)