3.314 \(\int \frac {1}{\sqrt {a x^3+b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2000} \[ -\frac {2 \sqrt {a x^3+b x^4}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(a*x^2)

Rule 2000

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \[ -\frac {2 \sqrt {x^3 (a+b x)}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-2*Sqrt[x^3*(a + b*x)])/(a*x^2)

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fricas [A]  time = 0.38, size = 21, normalized size = 0.91 \[ -\frac {2 \, \sqrt {b x^{4} + a x^{3}}}{a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a*x^3)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x^4 + a*x^3)/(a*x^2)

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giac [A]  time = 0.21, size = 27, normalized size = 1.17 \[ \frac {2}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a*x^3)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 25, normalized size = 1.09 \[ -\frac {2 \left (b x +a \right ) x}{\sqrt {b \,x^{4}+a \,x^{3}}\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a*x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(b*x^4 + a*x^3), x)

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mupad [B]  time = 5.14, size = 21, normalized size = 0.91 \[ -\frac {2\,\sqrt {b\,x^4+a\,x^3}}{a\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^3 + b*x^4)^(1/2),x)

[Out]

-(2*(a*x^3 + b*x^4)^(1/2))/(a*x^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a x^{3} + b x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a*x**3)**(1/2),x)

[Out]

Integral(1/sqrt(a*x**3 + b*x**4), x)

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