3.94 \(\int \frac {1}{x^2 \sqrt {a x+b x^4}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2 \sqrt {a x+b x^4}}{3 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2014} \[ -\frac {2 \sqrt {a x+b x^4}}{3 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*Sqrt[a*x + b*x^4])/(3*a*x^2)

Rule 2014

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.00 \[ -\frac {2 \sqrt {x \left (a+b x^3\right )}}{3 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*Sqrt[x*(a + b*x^3)])/(3*a*x^2)

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fricas [A]  time = 0.80, size = 19, normalized size = 0.83 \[ -\frac {2 \, \sqrt {b x^{4} + a x}}{3 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x^4 + a*x)/(a*x^2)

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giac [A]  time = 0.22, size = 14, normalized size = 0.61 \[ -\frac {2 \, \sqrt {b + \frac {a}{x^{3}}}}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b + a/x^3)/a

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maple [A]  time = 0.04, size = 27, normalized size = 1.17 \[ -\frac {2 \left (b \,x^{3}+a \right )}{3 \sqrt {b \,x^{4}+a x}\, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.46, size = 26, normalized size = 1.13 \[ -\frac {2 \, {\left (b x^{4} + a x\right )}}{3 \, \sqrt {b x^{3} + a} a x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x^4 + a*x)/(sqrt(b*x^3 + a)*a*x^(5/2))

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mupad [B]  time = 5.13, size = 19, normalized size = 0.83 \[ -\frac {2\,\sqrt {b\,x^4+a\,x}}{3\,a\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a*x + b*x^4)^(1/2)),x)

[Out]

-(2*(a*x + b*x^4)^(1/2))/(3*a*x^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {x \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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