3.2398 \(\int \frac{1}{(d+e x) \sqrt{\frac{-c d^2+b d e}{e^2}+b x+c x^2}} \, dx\)

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))

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Rubi [A]  time = 0.12055, 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 \[ \frac{2 e \sqrt{-\frac{d (c d-b e)}{e^2}+b x+c x^2}}{(d+e x) (2 c d-b e)} \]

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))

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Rubi in Sympy [A]  time = 10.641, size = 41, normalized size = 0.85 \[ - \frac{2 e \sqrt{b x + c x^{2} + \frac{d \left (b e - c d\right )}{e^{2}}}}{\left (d + e x\right ) \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/((b*d*e-c*d**2)/e**2+b*x+c*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.149859, 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))

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Maple [A]  time = 0.016, size = 59, normalized size = 1.2 \[ -2\,{\frac{cex+be-cd}{e \left ( be-2\,cd \right ) }{\frac{1}{\sqrt{{\frac{c{e}^{2}{x}^{2}+b{e}^{2}x+bde-c{d}^{2}}{{e}^{2}}}}}}} \]

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x - (c*d^2 - b*d*e)/e^2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269745, size = 84, normalized size = 1.75 \[ \frac{2 \, e \sqrt{\frac{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}{e^{2}}}}{2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x - (c*d^2 - b*d*e)/e^2)*(e*x + d)),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)

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

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)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError