3.2.93 \(\int \frac {\sqrt {x}}{\sqrt {a x^2+b x^3}} \, dx\)

Optimal. Leaf size=34 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{\sqrt {b}} \]

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Rubi [A]  time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2029, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[a*x^2 + b*x^3],x]

[Out]

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

Rule 206

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

Rule 2029

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

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Mathematica [A]  time = 0.02, size = 55, normalized size = 1.62 \begin {gather*} \frac {2 \sqrt {a} x \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {x^2 (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[a]*x*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x^2*(a + b*x)])

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IntegrateAlgebraic [A]  time = 0.09, size = 50, normalized size = 1.47 \begin {gather*} \frac {4 \log \left (\sqrt {x}\right )}{\sqrt {b}}-\frac {2 \log \left (\sqrt {a x^2+b x^3}-\sqrt {b} x^{3/2}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x]/Sqrt[a*x^2 + b*x^3],x]

[Out]

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

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fricas [A]  time = 0.42, size = 77, normalized size = 2.26 \begin {gather*} \left [\frac {\log \left (\frac {2 \, b x^{2} + a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {b} \sqrt {x}}{x}\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-b}}{b x^{\frac {3}{2}}}\right )}{b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 23, normalized size = 0.68 \begin {gather*} -\frac {2 \, \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 58, normalized size = 1.71 \begin {gather*} \frac {\sqrt {\left (b x +a \right ) x}\, \sqrt {x}\, \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )}{\sqrt {b \,x^{3}+a \,x^{2}}\, \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {b x^{3} + a x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a*x^2 + b*x^3)^(1/2),x)

[Out]

int(x^(1/2)/(a*x^2 + b*x^3)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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