3.2396 \(\int \frac{1}{(d+e x) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]

[Out]

(2*(b + 2*c*x)*Log[b + 2*c*x])/((2*c*d - b*e)*Sqrt[b^2/c + 4*b*x + 4*c*x^2]) - (
2*(b + 2*c*x)*Log[d + e*x])/((2*c*d - b*e)*Sqrt[b^2/c + 4*b*x + 4*c*x^2])

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Rubi [A]  time = 0.119437, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]

[Out]

(2*(b + 2*c*x)*Log[b + 2*c*x])/((2*c*d - b*e)*Sqrt[b^2/c + 4*b*x + 4*c*x^2]) - (
2*(b + 2*c*x)*Log[d + e*x])/((2*c*d - b*e)*Sqrt[b^2/c + 4*b*x + 4*c*x^2])

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Rubi in Sympy [A]  time = 22.2524, size = 95, normalized size = 0.99 \[ - \frac{4 c \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} \log{\left (b + 2 c x \right )}}{\left (2 b + 4 c x\right ) \left (b e - 2 c d\right )} + \frac{4 c \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} \log{\left (d + e x \right )}}{\left (2 b + 4 c x\right ) \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(2/(e*x+d)/(1/c*b**2+4*b*x+4*c*x**2)**(1/2),x)

[Out]

-4*c*sqrt(b**2/c + 4*b*x + 4*c*x**2)*log(b + 2*c*x)/((2*b + 4*c*x)*(b*e - 2*c*d)
) + 4*c*sqrt(b**2/c + 4*b*x + 4*c*x**2)*log(d + e*x)/((2*b + 4*c*x)*(b*e - 2*c*d
))

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Mathematica [A]  time = 0.0657165, size = 51, normalized size = 0.53 \[ \frac{2 (b+2 c x) (\log (b+2 c x)-\log (d+e x))}{\sqrt{\frac{(b+2 c x)^2}{c}} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.029, size = 58, normalized size = 0.6 \[ -2\,{\frac{ \left ( 2\,cx+b \right ) \left ( \ln \left ( 2\,cx+b \right ) -\ln \left ( ex+d \right ) \right ) }{be-2\,cd}{\frac{1}{\sqrt{{\frac{4\,{c}^{2}{x}^{2}+4\,bxc+{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(2/(e*x+d)/(b^2/c+4*b*x+4*c*x^2)^(1/2),x)

[Out]

-2*(2*c*x+b)*(ln(2*c*x+b)-ln(e*x+d))/((4*c^2*x^2+4*b*c*x+b^2)/c)^(1/2)/(b*e-2*c*
d)

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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(2/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235788, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c} \log \left (\frac{{\left (4 \, b c^{2} d^{2} - b^{3} e^{2} + 8 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (4 \, c^{3} d^{2} + 4 \, b c^{2} d e - 3 \, b^{2} c e^{2}\right )} x\right )} \sqrt{c} +{\left (8 \, c^{3} e^{2} x^{2} + 4 \, c^{3} d^{2} + b^{2} c e^{2} + 4 \,{\left (2 \, c^{3} d e + b c^{2} e^{2}\right )} x\right )} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{3} + b^{2} d + 4 \,{\left (c^{2} d + b c e\right )} x^{2} +{\left (4 \, b c d + b^{2} e\right )} x}\right )}{2 \, c d - b e}, \frac{4 \, \sqrt{-c} \arctan \left (-\frac{{\left (4 \, c^{2} e x + 2 \, c^{2} d + b c e\right )} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{{\left (2 \, b c d - b^{2} e + 2 \,{\left (2 \, c^{2} d - b c e\right )} x\right )} \sqrt{-c}}\right )}{2 \, c d - b e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(e*x + d)),x, algorithm="fricas")

[Out]

[-2*sqrt(c)*log(((4*b*c^2*d^2 - b^3*e^2 + 8*(2*c^3*d*e - b*c^2*e^2)*x^2 + 2*(4*c
^3*d^2 + 4*b*c^2*d*e - 3*b^2*c*e^2)*x)*sqrt(c) + (8*c^3*e^2*x^2 + 4*c^3*d^2 + b^
2*c*e^2 + 4*(2*c^3*d*e + b*c^2*e^2)*x)*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/c))/(4*c
^2*e*x^3 + b^2*d + 4*(c^2*d + b*c*e)*x^2 + (4*b*c*d + b^2*e)*x))/(2*c*d - b*e),
4*sqrt(-c)*arctan(-(4*c^2*e*x + 2*c^2*d + b*c*e)*sqrt((4*c^2*x^2 + 4*b*c*x + b^2
)/c)/((2*b*c*d - b^2*e + 2*(2*c^2*d - b*c*e)*x)*sqrt(-c)))/(2*c*d - b*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 \int \frac{1}{d \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} + e x \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2/(e*x+d)/(1/c*b**2+4*b*x+4*c*x**2)**(1/2),x)

[Out]

2*Integral(1/(d*sqrt(b**2/c + 4*b*x + 4*c*x**2) + e*x*sqrt(b**2/c + 4*b*x + 4*c*
x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError