3.555 \(\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b*x^2)) - (b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x
^2))

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Rubi [A]  time = 0.0205168, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1112, 14} \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^6,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b*x^2)) - (b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x
^2))

Rule 1112

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0072181, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a+5 b x^2\right )}{15 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^6,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(3*a + 5*b*x^2))/(15*x^5*(a + b*x^2))

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Maple [A]  time = 0.042, size = 36, normalized size = 0.5 \begin{align*} -{\frac{5\,b{x}^{2}+3\,a}{15\,{x}^{5} \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^2+a)^2)^(1/2)/x^6,x)

[Out]

-1/15*(5*b*x^2+3*a)*((b*x^2+a)^2)^(1/2)/x^5/(b*x^2+a)

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Maxima [A]  time = 1.01464, size = 20, normalized size = 0.25 \begin{align*} -\frac{5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/x^6,x, algorithm="maxima")

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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Fricas [A]  time = 1.47216, size = 36, normalized size = 0.46 \begin{align*} -\frac{5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/x^6,x, algorithm="fricas")

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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Sympy [A]  time = 0.306549, size = 15, normalized size = 0.19 \begin{align*} - \frac{3 a + 5 b x^{2}}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**2+a)**2)**(1/2)/x**6,x)

[Out]

-(3*a + 5*b*x**2)/(15*x**5)

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Giac [A]  time = 1.16003, size = 42, normalized size = 0.53 \begin{align*} -\frac{5 \, b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 3 \, a \mathrm{sgn}\left (b x^{2} + a\right )}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/x^6,x, algorithm="giac")

[Out]

-1/15*(5*b*x^2*sgn(b*x^2 + a) + 3*a*sgn(b*x^2 + a))/x^5