### 3.45 $$\int \frac{x}{\sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=47 $\frac{\sqrt{b x+c x^2}}{c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0151187, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {640, 620, 206} $\frac{\sqrt{b x+c x^2}}{c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[b*x + c*x^2],x]

[Out]

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

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

Mathematica [A]  time = 0.0414106, size = 71, normalized size = 1.51 $\frac{\sqrt{c} x (b+c x)-b^{3/2} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{3/2} \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[b*x + c*x^2],x]

[Out]

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

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Maple [A]  time = 0.046, size = 47, normalized size = 1. \begin{align*}{\frac{1}{c}\sqrt{c{x}^{2}+bx}}-{\frac{b}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03164, size = 236, normalized size = 5.02 \begin{align*} \left [\frac{b \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \, \sqrt{c x^{2} + b x} c}{2 \, c^{2}}, \frac{b \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x} c}{c^{2}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(b*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*sqrt(c*x^2 + b*x)*c)/c^2, (b*sqrt(-c)*arctan(
sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + sqrt(c*x^2 + b*x)*c)/c^2]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(x*(b + c*x)), x)

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Giac [A]  time = 1.31484, size = 70, normalized size = 1.49 \begin{align*} \frac{b \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{2} + b x}}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(3/2) + sqrt(c*x^2 + b*x)/c