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

Optimal. Leaf size=23 \[ -\frac{2 \left (b x+c x^2\right )^{3/2}}{3 b x^3} \]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(3*b*x^3)

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Rubi [A]  time = 0.0064643, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {650} \[ -\frac{2 \left (b x+c x^2\right )^{3/2}}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/x^3,x]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(3*b*x^3)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), 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 + 2*p + 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0094617, size = 21, normalized size = 0.91 \[ -\frac{2 (x (b+c x))^{3/2}}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x + c*x^2]/x^3,x]

[Out]

(-2*(x*(b + c*x))^(3/2))/(3*b*x^3)

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Maple [A]  time = 0.045, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,cx+2\,b}{3\,b{x}^{2}}\sqrt{c{x}^{2}+bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.00175, size = 57, normalized size = 2.48 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (c x + b\right )}}{3 \, b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(b + c*x))/x**3, x)

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Giac [B]  time = 1.40153, size = 103, normalized size = 4.48 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} c + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b \sqrt{c} + b^{2}\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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