3.119 \(\int \frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=87 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{2 a^{5/2}}-\frac{3 b \sqrt{a x+b \sqrt{x}}}{2 a^2}+\frac{\sqrt{x} \sqrt{a x+b \sqrt{x}}}{a} \]

[Out]

(-3*b*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/a + (3*b^2*ArcTanh[(Sqrt[a]*Sqrt[x])/Sq
rt[b*Sqrt[x] + a*x]])/(2*a^(5/2))

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Rubi [A]  time = 0.0788138, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2018, 670, 640, 620, 206} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{2 a^{5/2}}-\frac{3 b \sqrt{a x+b \sqrt{x}}}{2 a^2}+\frac{\sqrt{x} \sqrt{a x+b \sqrt{x}}}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

(-3*b*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/a + (3*b^2*ArcTanh[(Sqrt[a]*Sqrt[x])/Sq
rt[b*Sqrt[x] + a*x]])/(2*a^(5/2))

Rule 2018

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

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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])

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x} \sqrt{b \sqrt{x}+a x}}{a}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{2 a}\\ &=-\frac{3 b \sqrt{b \sqrt{x}+a x}}{2 a^2}+\frac{\sqrt{x} \sqrt{b \sqrt{x}+a x}}{a}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{4 a^2}\\ &=-\frac{3 b \sqrt{b \sqrt{x}+a x}}{2 a^2}+\frac{\sqrt{x} \sqrt{b \sqrt{x}+a x}}{a}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{2 a^2}\\ &=-\frac{3 b \sqrt{b \sqrt{x}+a x}}{2 a^2}+\frac{\sqrt{x} \sqrt{b \sqrt{x}+a x}}{a}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{2 a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0799227, size = 102, normalized size = 1.17 \[ \frac{\sqrt{a} \sqrt{x} \left (2 a^2 x-a b \sqrt{x}-3 b^2\right )+3 b^{5/2} \sqrt [4]{x} \sqrt{\frac{a \sqrt{x}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} \sqrt [4]{x}}{\sqrt{b}}\right )}{2 a^{5/2} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

(Sqrt[a]*Sqrt[x]*(-3*b^2 - a*b*Sqrt[x] + 2*a^2*x) + 3*b^(5/2)*Sqrt[1 + (a*Sqrt[x])/b]*x^(1/4)*ArcSinh[(Sqrt[a]
*x^(1/4))/Sqrt[b]])/(2*a^(5/2)*Sqrt[b*Sqrt[x] + a*x])

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Maple [B]  time = 0.007, size = 160, normalized size = 1.8 \begin{align*}{\frac{1}{4}\sqrt{b\sqrt{x}+ax} \left ( 4\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}b-8\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b+4\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){b}^{2}-{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{a}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*x^(1/2)+a*x)^(1/2)*(4*(b*x^(1/2)+a*x)^(1/2)*a^(5/2)*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(3/2)*b-8*a^(3/2)
*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*b+4*a*ln(1/2*(2*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2)+b)/a^(1/2))*b
^2-b^2*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*a)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/a^(7/2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{\sqrt{a x + b \sqrt{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x)/sqrt(a*x + b*sqrt(x)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{\sqrt{a x + b \sqrt{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.34643, size = 93, normalized size = 1.07 \begin{align*} \frac{1}{2} \, \sqrt{a x + b \sqrt{x}}{\left (\frac{2 \, \sqrt{x}}{a} - \frac{3 \, b}{a^{2}}\right )} - \frac{3 \, b^{2} \log \left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{4 \, a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(a*x + b*sqrt(x))*(2*sqrt(x)/a - 3*b/a^2) - 3/4*b^2*log(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b
*sqrt(x))) - b))/a^(5/2)