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

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

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Rubi [A]  time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 0.68 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.69, size = 256, normalized size = 3.46 \begin {gather*} \frac {\sqrt {b^2} \left (3 a^6+a^2 b^4 x^8+4 a b^5 x^{10}+6 b^6 x^{12}\right )+\sqrt {a^2+2 a b x^2+b^2 x^4} \left (3 a^5 b-3 a^4 b^2 x^2+3 a^3 b^3 x^4-3 a^2 b^4 x^6+2 a b^5 x^8-6 b^6 x^{10}\right )}{3 b^4 \sqrt {b^2} x^8 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-8 a^3 b^3-24 a^2 b^4 x^2-24 a b^5 x^4-8 b^6 x^6\right )+3 b^4 x^8 \left (8 a^4 b^4+32 a^3 b^5 x^2+48 a^2 b^6 x^4+32 a b^7 x^6+8 b^8 x^8\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(3*a^5*b - 3*a^4*b^2*x^2 + 3*a^3*b^3*x^4 - 3*a^2*b^4*x^6 + 2*a*b^5*x^8 - 6*b^
6*x^10) + Sqrt[b^2]*(3*a^6 + a^2*b^4*x^8 + 4*a*b^5*x^10 + 6*b^6*x^12))/(3*b^4*Sqrt[b^2]*x^8*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4]*(-8*a^3*b^3 - 24*a^2*b^4*x^2 - 24*a*b^5*x^4 - 8*b^6*x^6) + 3*b^4*x^8*(8*a^4*b^4 + 32*a^3*b^5*x^2
+ 48*a^2*b^6*x^4 + 32*a*b^7*x^6 + 8*b^8*x^8))

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fricas [A]  time = 2.67, size = 69, normalized size = 0.93 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.21, size = 43, normalized size = 0.58 \begin {gather*} -\frac {6 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{24 \, {\left (b x^{2} + a\right )}^{4} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

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]

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

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maple [A]  time = 0.01, size = 43, normalized size = 0.58 \begin {gather*} -\frac {\left (b \,x^{2}+a \right ) \left (6 b^{2} x^{4}+4 a b \,x^{2}+a^{2}\right )}{24 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} b^{3}} \end {gather*}

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 = 1.79, size = 69, normalized size = 0.93 \begin {gather*} -\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 {gather*}

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]

-1/24*(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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mupad [B]  time = 4.23, size = 53, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (a^2+4\,a\,b\,x^2+6\,b^2\,x^4\right )}{24\,b^3\,{\left (b\,x^2+a\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

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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