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

Optimal. Leaf size=52 \[ \frac {4 b \sqrt {a x^3+b x^4}}{3 a^2 x^2}-\frac {2 \sqrt {a x^3+b x^4}}{3 a x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2000} \begin {gather*} \frac {4 b \sqrt {a x^3+b x^4}}{3 a^2 x^2}-\frac {2 \sqrt {a x^3+b x^4}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a*x^3 + b*x^4]),x]

[Out]

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

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 29, normalized size = 0.56 \begin {gather*} -\frac {2 (a-2 b x) \sqrt {x^3 (a+b x)}}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a*x^3 + b*x^4]),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.17, size = 33, normalized size = 0.63 \begin {gather*} \frac {2 (2 b x-a) \sqrt {a x^3+b x^4}}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*Sqrt[a*x^3 + b*x^4]),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 29, normalized size = 0.56 \begin {gather*} \frac {2 \, \sqrt {b x^{4} + a x^{3}} {\left (2 \, b x - a\right )}}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.23, size = 27, normalized size = 0.52 \begin {gather*} -\frac {2 \, {\left ({\left (b + \frac {a}{x}\right )}^{\frac {3}{2}} - 3 \, \sqrt {b + \frac {a}{x}} b\right )}}{3 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 30, normalized size = 0.58 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 \sqrt {b \,x^{4}+a \,x^{3}}\, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.06, size = 42, normalized size = 0.81 \begin {gather*} -\frac {2\,a\,\sqrt {b\,x^4+a\,x^3}-4\,b\,x\,\sqrt {b\,x^4+a\,x^3}}{3\,a^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a*x^3 + b*x^4)^(1/2)),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________