3.524 \(\int \frac{x^3}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{5 b^2 \left (a+b x^2\right )^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0232085, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{5 b^2 \left (a+b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^(9/2),x]

[Out]

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{9/2}}+\frac{1}{b (a+b x)^{7/2}}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{5 b^2 \left (a+b x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0138211, size = 28, normalized size = 0.74 \[ \frac{-2 a-7 b x^2}{35 b^2 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^(9/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{7\,b{x}^{2}+2\,a}{35\,{b}^{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^3/(b*x^2+a)^(9/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.01263, size = 45, normalized size = 1.18 \begin{align*} -\frac{x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{2 \, a}{35 \,{\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^3/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.33584, size = 142, normalized size = 3.74 \begin{align*} -\frac{{\left (7 \, b x^{2} + 2 \, a\right )} \sqrt{b x^{2} + a}}{35 \,{\left (b^{6} x^{8} + 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 5.25198, size = 180, normalized size = 4.74 \begin{align*} \begin{cases} - \frac{2 a}{35 a^{3} b^{2} \sqrt{a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt{a + b x^{2}} + 105 a b^{4} x^{4} \sqrt{a + b x^{2}} + 35 b^{5} x^{6} \sqrt{a + b x^{2}}} - \frac{7 b x^{2}}{35 a^{3} b^{2} \sqrt{a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt{a + b x^{2}} + 105 a b^{4} x^{4} \sqrt{a + b x^{2}} + 35 b^{5} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-2*a/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt(a + b*x**2) + 105*a*b**4*x**4*sqrt(a
+ b*x**2) + 35*b**5*x**6*sqrt(a + b*x**2)) - 7*b*x**2/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt
(a + b*x**2) + 105*a*b**4*x**4*sqrt(a + b*x**2) + 35*b**5*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**4/(4*a**(9/2)
), True))

________________________________________________________________________________________

Giac [A]  time = 2.3914, size = 32, normalized size = 0.84 \begin{align*} -\frac{7 \, b x^{2} + 2 \, a}{35 \,{\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^3/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

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