3.2409 \(\int \frac{1}{(d+e x) \sqrt{\frac{-c d^2+b d e}{e^2}+b x+c x^2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 e \sqrt{-\frac{d (c d-b e)}{e^2}+b x+c x^2}}{(d+e x) (2 c d-b e)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0268307, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {650} \[ \frac{2 e \sqrt{-\frac{d (c d-b e)}{e^2}+b x+c x^2}}{(d+e x) (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{1}{(d+e x) \sqrt{\frac{-c d^2+b d e}{e^2}+b x+c x^2}} \, dx &=\frac{2 e \sqrt{-\frac{d (c d-b e)}{e^2}+b x+c x^2}}{(2 c d-b e) (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0566954, size = 45, normalized size = 0.94 \[ -\frac{2 e \sqrt{\frac{(d+e x) (b e-c d+c e x)}{e^2}}}{(d+e x) (b e-2 c d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.19, size = 59, normalized size = 1.2 \begin{align*} -2\,{\frac{cex+be-cd}{e \left ( be-2\,cd \right ) }{\frac{1}{\sqrt{{\frac{c{e}^{2}{x}^{2}+b{e}^{2}x+bde-c{d}^{2}}{{e}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.79547, size = 126, normalized size = 2.62 \begin{align*} \frac{2 \, e \sqrt{\frac{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}{e^{2}}}}{2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (\frac{d}{e} + x\right ) \left (b - \frac{c d}{e} + c x\right )} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError