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

Optimal. Leaf size=47 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}} \]

[Out]

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

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Rubi [A]  time = 0.128169, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.43813, size = 54, normalized size = 1.15 \[ - \frac{2 \operatorname{atanh}{\left (\frac{b + 2 c d e + 2 c e^{2} x}{\sqrt{b} \sqrt{b + 4 c d e}} \right )}}{\sqrt{b} \sqrt{b + 4 c d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0403079, size = 47, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{b} \sqrt{b+4 c d e}}\right )}{\sqrt{b} \sqrt{b+4 c d e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 43, normalized size = 0.9 \[ -2\,{\frac{1}{\sqrt{4\,bcde+{b}^{2}}}{\it Artanh} \left ({\frac{2\,c{e}^{2}x+2\,cde+b}{\sqrt{4\,bcde+{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+c*(e*x+d)^2),x)

[Out]

-2/(4*b*c*d*e+b^2)^(1/2)*arctanh((2*c*e^2*x+2*c*d*e+b)/(4*b*c*d*e+b^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/((e*x + d)^2*c + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283352, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{8 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e + b^{3} + 2 \,{\left (4 \, b c^{2} d e^{3} + b^{2} c e^{2}\right )} x -{\left (2 \, c^{2} e^{4} x^{2} + 2 \, c^{2} d^{2} e^{2} + 4 \, b c d e + b^{2} + 2 \,{\left (2 \, c^{2} d e^{3} + b c e^{2}\right )} x\right )} \sqrt{4 \, b c d e + b^{2}}}{c e^{2} x^{2} + c d^{2} +{\left (2 \, c d e + b\right )} x}\right )}{\sqrt{4 \, b c d e + b^{2}}}, -\frac{2 \, \arctan \left (\frac{\sqrt{-4 \, b c d e - b^{2}}{\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{4 \, b c d e + b^{2}}\right )}{\sqrt{-4 \, b c d e - b^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[log(-(8*b*c^2*d^2*e^2 + 6*b^2*c*d*e + b^3 + 2*(4*b*c^2*d*e^3 + b^2*c*e^2)*x - (
2*c^2*e^4*x^2 + 2*c^2*d^2*e^2 + 4*b*c*d*e + b^2 + 2*(2*c^2*d*e^3 + b*c*e^2)*x)*s
qrt(4*b*c*d*e + b^2))/(c*e^2*x^2 + c*d^2 + (2*c*d*e + b)*x))/sqrt(4*b*c*d*e + b^
2), -2*arctan(sqrt(-4*b*c*d*e - b^2)*(2*c*e^2*x + 2*c*d*e + b)/(4*b*c*d*e + b^2)
)/sqrt(-4*b*c*d*e - b^2)]

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Sympy [A]  time = 0.830143, size = 151, normalized size = 3.21 \[ \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} \log{\left (x + \frac{- b^{2} \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} - 4 b c d e \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} + b + 2 c d e}{2 c e^{2}} \right )} - \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} \log{\left (x + \frac{b^{2} \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} + 4 b c d e \sqrt{\frac{1}{b \left (b + 4 c d e\right )}} + b + 2 c d e}{2 c e^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(1/(b*(b + 4*c*d*e)))*log(x + (-b**2*sqrt(1/(b*(b + 4*c*d*e))) - 4*b*c*d*e*s
qrt(1/(b*(b + 4*c*d*e))) + b + 2*c*d*e)/(2*c*e**2)) - sqrt(1/(b*(b + 4*c*d*e)))*
log(x + (b**2*sqrt(1/(b*(b + 4*c*d*e))) + 4*b*c*d*e*sqrt(1/(b*(b + 4*c*d*e))) +
b + 2*c*d*e)/(2*c*e**2))

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GIAC/XCAS [A]  time = 0.261169, size = 65, normalized size = 1.38 \[ \frac{2 \, \arctan \left (\frac{2 \, c x e^{2} + 2 \, c d e + b}{\sqrt{-4 \, b c d e - b^{2}}}\right )}{\sqrt{-4 \, b c d e - b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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