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

Optimal. Leaf size=69 \[ \frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}-\frac{2 x^2}{c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0266901, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {668, 640, 620, 206} \[ \frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}-\frac{2 x^2}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x + c*x^2)^(3/2),x]

[Out]

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

Rule 668

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*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 1] && 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{x^3}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 x^2}{c \sqrt{b x+c x^2}}+\frac{3 \int \frac{x}{\sqrt{b x+c x^2}} \, dx}{c}\\ &=-\frac{2 x^2}{c \sqrt{b x+c x^2}}+\frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{(3 b) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2 c^2}\\ &=-\frac{2 x^2}{c \sqrt{b x+c x^2}}+\frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{c^2}\\ &=-\frac{2 x^2}{c \sqrt{b x+c x^2}}+\frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0126394, size = 50, normalized size = 0.72 \[ \frac{2 x^3 \sqrt{\frac{c x}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{c x}{b}\right )}{5 b \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(b*x + c*x^2)^(3/2),x]

[Out]

(2*x^3*Sqrt[1 + (c*x)/b]*Hypergeometric2F1[3/2, 5/2, 7/2, -((c*x)/b)])/(5*b*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.05, size = 68, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+3\,{\frac{bx}{{c}^{2}\sqrt{c{x}^{2}+bx}}}-{\frac{3\,b}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2/c/(c*x^2+b*x)^(1/2)+3*b/c^2/(c*x^2+b*x)^(1/2)*x-3/2*b/c^(5/2)*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^3/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.94351, size = 344, normalized size = 4.99 \begin{align*} \left [\frac{3 \,{\left (b c x + b^{2}\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (c^{2} x + 3 \, b c\right )} \sqrt{c x^{2} + b x}}{2 \,{\left (c^{4} x + b c^{3}\right )}}, \frac{3 \,{\left (b c x + b^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (c^{2} x + 3 \, b c\right )} \sqrt{c x^{2} + b x}}{c^{4} x + b c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*(b*c*x + b^2)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(c^2*x + 3*b*c)*sqrt(c*x^2 + b*
x))/(c^4*x + b*c^3), (3*(b*c*x + b^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (c^2*x + 3*b*c)*sqrt
(c*x^2 + b*x))/(c^4*x + b*c^3)]

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

[Out]

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

[Out]

Exception raised: TypeError