3.105 \(\int \frac{1}{(\frac{b c}{d}+b x^2) \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{d x}{b c \sqrt{c+d x^2}} \]

[Out]

(d*x)/(b*c*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.0046504, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {21, 191} \[ \frac{d x}{b c \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(((b*c)/d + b*x^2)*Sqrt[c + d*x^2]),x]

[Out]

(d*x)/(b*c*Sqrt[c + d*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (\frac{b c}{d}+b x^2\right ) \sqrt{c+d x^2}} \, dx &=\frac{d \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{b}\\ &=\frac{d x}{b c \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0071051, size = 20, normalized size = 1. \[ \frac{d x}{b c \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(((b*c)/d + b*x^2)*Sqrt[c + d*x^2]),x]

[Out]

(d*x)/(b*c*Sqrt[c + d*x^2])

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Maple [A]  time = 0.003, size = 19, normalized size = 1. \begin{align*}{\frac{dx}{bc}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50166, size = 55, normalized size = 2.75 \begin{align*} \frac{\sqrt{d x^{2} + c} d x}{b c d x^{2} + b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{d \int \frac{1}{c \sqrt{c + d x^{2}} + d x^{2} \sqrt{c + d x^{2}}}\, dx}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*Integral(1/(c*sqrt(c + d*x**2) + d*x**2*sqrt(c + d*x**2)), x)/b

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Giac [A]  time = 1.11679, size = 24, normalized size = 1.2 \begin{align*} \frac{d x}{\sqrt{d x^{2} + c} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x/(sqrt(d*x^2 + c)*b*c)