3.4.15 \(\int \frac {1}{x \sqrt {a x^3+b x^4}} \, dx\) [315]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2041, 2025} \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 2025

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 2041

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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.60 \begin {gather*} -\frac {2 (a-2 b x) (a+b x)}{3 a^2 \sqrt {x^3 (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.34, size = 48, normalized size = 0.92

method	result	size
trager	\(-\frac {2 \left (-2 b x +a \right ) \sqrt {b \,x^{4}+a \,x^{3}}}{3 a^{2} x^{3}}\)	\(28\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 \sqrt {x^{3} \left (b x +a \right )}\, a^{2}}\)	\(28\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 a^{2} \sqrt {b \,x^{4}+a \,x^{3}}}\)	\(30\)
default	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b \,x^{2}+a x}\, \left (-2 b x +a \right )}{3 x \sqrt {b \,x^{4}+a \,x^{3}}\, a^{2}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

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Fricas [A]
time = 1.66, 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)

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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)

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Giac [A]
time = 0.59, size = 53, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}}{3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} \mathrm {sgn}\left (x\right )} \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*(3*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a)/((sqrt(b)*x - sqrt(b*x^2 + a*x))^3*sgn(x))

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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)

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