### 3.107 $$\int \frac{x^{3/2}}{(b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=23 $-\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}}$

[Out]

(-2*Sqrt[x])/(c*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0070982, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {648} $-\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^(3/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*Sqrt[x])/(c*Sqrt[b*x + c*x^2])

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

Mathematica [A]  time = 0.0067679, size = 21, normalized size = 0.91 $-\frac{2 \sqrt{x}}{c \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*Sqrt[x])/(c*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.049, size = 25, normalized size = 1.1 \begin{align*} -2\,{\frac{ \left ( cx+b \right ){x}^{3/2}}{c \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08504, size = 65, normalized size = 2.83 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x} \sqrt{x}}{c^{2} x^{2} + b c x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.21502, size = 28, normalized size = 1.22 \begin{align*} -\frac{2}{\sqrt{c x + b} c} + \frac{2}{\sqrt{b} c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(c*x + b)*c) + 2/(sqrt(b)*c)