3.648 \(\int \frac{x^5}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{x^6}{8 a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x^6}{24 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]

[Out]

x^6/(24*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + x^6/(8*a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))

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Rubi [A]  time = 0.0158215, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1109} \[ \frac{x^6}{8 a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x^6}{24 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

x^6/(24*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + x^6/(8*a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))

Rule 1109

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(2*(d*x)^(m + 1)*(a + b*x^
2 + c*x^4)^(p + 1))/(d*(m + 3)*(2*a + b*x^2)), x] - Simp[((d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*d*(m
+ 3)*(p + 1)), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && EqQ[m + 4*p + 5,
0] && LtQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{x^6}{24 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x^6}{8 a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0165833, size = 50, normalized size = 0.68 \[ \frac{-a^2-4 a b x^2-6 b^2 x^4}{24 b^3 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-a^2 - 4*a*b*x^2 - 6*b^2*x^4)/(24*b^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.17, size = 43, normalized size = 0.6 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 6\,{b}^{2}{x}^{4}+4\,ab{x}^{2}+{a}^{2} \right ) }{24\,{b}^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

-1/24*(b*x^2+a)*(6*b^2*x^4+4*a*b*x^2+a^2)/b^3/((b*x^2+a)^2)^(5/2)

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Maxima [A]  time = 0.975331, size = 85, normalized size = 1.15 \begin{align*} -\frac{1}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2}} + \frac{a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{3}} - \frac{a^{2} b^{2}}{8 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/4/((b^2)^(5/2)*(x^2 + a/b)^2) + 1/3*a*b/((b^2)^(7/2)*(x^2 + a/b)^3) - 1/8*a^2*b^2/((b^2)^(9/2)*(x^2 + a/b)^
4)

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Fricas [A]  time = 1.2508, size = 139, normalized size = 1.88 \begin{align*} -\frac{6 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{24 \,{\left (b^{7} x^{8} + 4 \, a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/24*(6*b^2*x^4 + 4*a*b*x^2 + a^2)/(b^7*x^8 + 4*a*b^6*x^6 + 6*a^2*b^5*x^4 + 4*a^3*b^4*x^2 + a^4*b^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**5/((a + b*x**2)**2)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x