3.54 \(\int \frac{x^2}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 x}{c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0175548, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {652, 620, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 x}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(b*x + c*x^2)^(3/2),x]

[Out]

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

Rule 652

Int[((d_.) + (e_.)*(x_))^2*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + b*x +
c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fr
eeQ[{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] &&  !IntegerQ[p] && LtQ[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^2}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 x}{c \sqrt{b x+c x^2}}+\frac{\int \frac{1}{\sqrt{b x+c x^2}} \, dx}{c}\\ &=-\frac{2 x}{c \sqrt{b x+c x^2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{c}\\ &=-\frac{2 x}{c \sqrt{b x+c x^2}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0812652, size = 67, normalized size = 1.4 \[ \frac{2 \sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )-2 \sqrt{c} x}{c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*Sqrt[c]*x + 2*Sqrt[b]*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 = 0.048, size = 47, normalized size = 1. \begin{align*} -2\,{\frac{x}{c\sqrt{c{x}^{2}+bx}}}+{\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^2/(c*x^2+b*x)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90494, size = 289, normalized size = 6.02 \begin{align*} \left [\frac{{\left (c x + b\right )} \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}{c^{3} x + b c^{2}}, -\frac{2 \,{\left ({\left (c x + b\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x} c\right )}}{c^{3} x + b c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(x**2/(x*(b + c*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError