3.4.13 \(\int \frac {x}{\sqrt {a x^3+b x^4}} \, dx\) [313]

Optimal. Leaf size=32 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2054, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 57, normalized size = 1.78 \begin {gather*} -\frac {2 x^{3/2} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b} \sqrt {x^3 (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(26)=52\).
time = 0.35, size = 56, normalized size = 1.75

method	result	size
default	\(\frac {x \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right )}{\sqrt {b \,x^{4}+a \,x^{3}}\, \sqrt {b}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(x/sqrt(b*x^4 + a*x^3), x)

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Fricas [A]
time = 2.43, size = 74, normalized size = 2.31 \begin {gather*} \left [\frac {\log \left (\frac {2 \, b x^{2} + a x + 2 \, \sqrt {b x^{4} + a x^{3}} \sqrt {b}}{x}\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{4} + a x^{3}} \sqrt {-b}}{b x^{2}}\right )}{b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[log((2*b*x^2 + a*x + 2*sqrt(b*x^4 + a*x^3)*sqrt(b))/x)/sqrt(b), -2*sqrt(-b)*arctan(sqrt(b*x^4 + a*x^3)*sqrt(-
b)/(b*x^2))/b]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(x**3*(a + b*x)), x)

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Giac [A]
time = 0.72, size = 49, normalized size = 1.53 \begin {gather*} \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {b}} - \frac {\log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{\sqrt {b} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{\sqrt {b\,x^4+a\,x^3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a*x^3 + b*x^4)^(1/2),x)

[Out]

int(x/(a*x^3 + b*x^4)^(1/2), x)

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