### 3.96 $$\int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=52 $\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}}$

[Out]

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

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Rubi [A]  time = 0.0157122, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {656, 648} $\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^(3/2)/Sqrt[b*x + c*x^2],x]

[Out]

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

Rule 656

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[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

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] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{(2 b) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{3 c}\\ &=-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}\\ \end{align*}

Mathematica [A]  time = 0.0166868, size = 30, normalized size = 0.58 $\frac{2 (c x-2 b) \sqrt{x (b+c x)}}{3 c^2 \sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(3/2)/Sqrt[b*x + c*x^2],x]

[Out]

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

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Maple [A]  time = 0.055, size = 33, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -cx+2\,b \right ) }{3\,{c}^{2}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12455, size = 41, normalized size = 0.79 \begin{align*} \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )}}{3 \, \sqrt{c x + b} c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.909, size = 66, normalized size = 1.27 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (c x - 2 \, b\right )}}{3 \, c^{2} \sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25214, size = 43, normalized size = 0.83 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{3 \, c^{2}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b\right )}}{3 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*b^(3/2)/c^2 + 2/3*((c*x + b)^(3/2) - 3*sqrt(c*x + b)*b)/c^2