3.521 \(\int \frac{x^6}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{x^7}{7 a \left (a+b x^2\right )^{7/2}} \]

[Out]

x^7/(7*a*(a + b*x^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0045666, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{x^7}{7 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^2)^(9/2),x]

[Out]

x^7/(7*a*(a + b*x^2)^(7/2))

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

Mathematica [A]  time = 0.0059893, size = 21, normalized size = 1. \[ \frac{x^7}{7 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^2)^(9/2),x]

[Out]

x^7/(7*a*(a + b*x^2)^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^2+a)^(9/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.31785, size = 139, normalized size = 6.62 \begin{align*} -\frac{x^{5}}{2 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{5 \, a x^{3}}{8 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} + \frac{x}{14 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{3}} + \frac{x}{7 \, \sqrt{b x^{2} + a} a b^{3}} + \frac{3 \, a x}{56 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{3}} - \frac{15 \, a^{2} x}{56 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/2*x^5/((b*x^2 + a)^(7/2)*b) - 5/8*a*x^3/((b*x^2 + a)^(7/2)*b^2) + 1/14*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*x/(s
qrt(b*x^2 + a)*a*b^3) + 3/56*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*a^2*x/((b*x^2 + a)^(7/2)*b^3)

________________________________________________________________________________________

Fricas [B]  time = 1.3269, size = 120, normalized size = 5.71 \begin{align*} \frac{\sqrt{b x^{2} + a} x^{7}}{7 \,{\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.00362, size = 95, normalized size = 4.52 \begin{align*} \frac{x^{7}}{7 a^{\frac{9}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 21 a^{\frac{7}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 21 a^{\frac{5}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 7 a^{\frac{3}{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**6/(b*x**2+a)**(9/2),x)

[Out]

x**7/(7*a**(9/2)*sqrt(1 + b*x**2/a) + 21*a**(7/2)*b*x**2*sqrt(1 + b*x**2/a) + 21*a**(5/2)*b**2*x**4*sqrt(1 + b
*x**2/a) + 7*a**(3/2)*b**3*x**6*sqrt(1 + b*x**2/a))

________________________________________________________________________________________

Giac [A]  time = 3.00424, size = 23, normalized size = 1.1 \begin{align*} \frac{x^{7}}{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(x^6/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/7*x^7/((b*x^2 + a)^(7/2)*a)