3.509 \(\int \frac {x^3}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac {a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{b^2 \sqrt {a+b x^2}} \]

[Out]

1/3*a/b^2/(b*x^2+a)^(3/2)-1/b^2/(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \[ \frac {a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{b^2 \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^(5/2),x]

[Out]

a/(3*b^2*(a + b*x^2)^(3/2)) - 1/(b^2*Sqrt[a + b*x^2])

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.78 \[ \frac {-2 a-3 b x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^(5/2),x]

[Out]

(-2*a - 3*b*x^2)/(3*b^2*(a + b*x^2)^(3/2))

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fricas [A]  time = 0.88, size = 47, normalized size = 1.31 \[ -\frac {{\left (3 \, b x^{2} + 2 \, a\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(3*b*x^2 + 2*a)*sqrt(b*x^2 + a)/(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)

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giac [A]  time = 1.10, size = 24, normalized size = 0.67 \[ -\frac {3 \, b x^{2} + 2 \, a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

-1/3*(3*b*x^2 + 2*a)/((b*x^2 + a)^(3/2)*b^2)

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maple [A]  time = 0.00, size = 25, normalized size = 0.69 \[ -\frac {3 b \,x^{2}+2 a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2+a)^(5/2),x)

[Out]

-1/3*(3*b*x^2+2*a)/(b*x^2+a)^(3/2)/b^2

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maxima [A]  time = 1.27, size = 33, normalized size = 0.92 \[ -\frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {2 \, a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

-x^2/((b*x^2 + a)^(3/2)*b) - 2/3*a/((b*x^2 + a)^(3/2)*b^2)

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mupad [B]  time = 5.17, size = 24, normalized size = 0.67 \[ -\frac {3\,b\,x^2+2\,a}{3\,b^2\,{\left (b\,x^2+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^(5/2),x)

[Out]

-(2*a + 3*b*x^2)/(3*b^2*(a + b*x^2)^(3/2))

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sympy [A]  time = 1.07, size = 92, normalized size = 2.56 \[ \begin {cases} - \frac {2 a}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} - \frac {3 b x^{2}}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**(5/2),x)

[Out]

Piecewise((-2*a/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 3*b*x**2/(3*a*b**2*sqrt(a + b*x**
2) + 3*b**3*x**2*sqrt(a + b*x**2)), Ne(b, 0)), (x**4/(4*a**(5/2)), True))

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