3.5.94 \(\int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx\) [494]

Optimal. Leaf size=44 \[ -\frac {\sqrt {a+b x^2}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2}}{3 a^2 x} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} \frac {2 b \sqrt {a+b x^2}}{3 a^2 x}-\frac {\sqrt {a+b x^2}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a + b*x^2]),x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx &=-\frac {\sqrt {a+b x^2}}{3 a x^3}-\frac {(2 b) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x^2}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2}}{3 a^2 x}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 31, normalized size = 0.70 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-a+2 b x^2\right )}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a + b*x^2]),x]

[Out]

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

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Maple [A]
time = 0.03, size = 37, normalized size = 0.84

method	result	size
gosper	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b \,x^{2}+a \right )}{3 a^{2} x^{3}}\)	\(26\)
trager	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b \,x^{2}+a \right )}{3 a^{2} x^{3}}\)	\(26\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b \,x^{2}+a \right )}{3 a^{2} x^{3}}\)	\(26\)
default	\(-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 36, normalized size = 0.82 \begin {gather*} \frac {2 \, \sqrt {b x^{2} + a} b}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a}}{3 \, a x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.92, size = 27, normalized size = 0.61 \begin {gather*} \frac {{\left (2 \, b x^{2} - a\right )} \sqrt {b x^{2} + a}}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.40, size = 46, normalized size = 1.05 \begin {gather*} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**(1/2),x)

[Out]

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

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Giac [A]
time = 0.65, size = 55, normalized size = 1.25 \begin {gather*} \frac {4 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.62, size = 25, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {b\,x^2+a}\,\left (a-2\,b\,x^2\right )}{3\,a^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^(1/2)),x)

[Out]

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

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