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

Optimal. Leaf size=77 \[ \frac{2 \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.147165, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2 \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8563, size = 66, normalized size = 0.86 \[ - \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b} + \frac{2 \sqrt{b e - c d} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.129217, size = 75, normalized size = 0.97 \[ \frac{2 \left (\frac{\sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c}}-\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 100, normalized size = 1.3 \[ 2\,{\frac{e}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{cd}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{\sqrt{d}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249281, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b}, \frac{2 \, \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b}, -\frac{2 \, \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b}, -\frac{2 \,{\left (\sqrt{-d} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right )\right )}}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b
*e)/c))/(c*x + b)) + sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/b, (2
*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + sqrt(d)*log((
e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/b, -(2*sqrt(-d)*arctan(sqrt(e*x + d)/sq
rt(-d)) - sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt(
(c*d - b*e)/c))/(c*x + b)))/b, -2*(sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - sqr
t(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)))/b]

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Sympy [A]  time = 5.80949, size = 252, normalized size = 3.27 \[ \frac{2 \left (- \frac{d e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} & \text{for}\: - d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: - d < 0 \wedge d < d + e x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: d > d + e x \wedge - d < 0 \end{cases}\right )}{b} + \frac{e \left (b e - c d\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{c \sqrt{\frac{b e - c d}{c}}} & \text{for}\: \frac{b e - c d}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: d + e x > \frac{- b e + c d}{c} \wedge \frac{b e - c d}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: \frac{b e - c d}{c} < 0 \wedge d + e x < \frac{- b e + c d}{c} \end{cases}\right )}{b}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-d*e*Piecewise((-atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d), -d > 0), (acoth(sqrt(
d + e*x)/sqrt(d))/sqrt(d), (-d < 0) & (d < d + e*x)), (atanh(sqrt(d + e*x)/sqrt(
d))/sqrt(d), (-d < 0) & (d > d + e*x)))/b + e*(b*e - c*d)*Piecewise((atan(sqrt(d
+ e*x)/sqrt((b*e - c*d)/c))/(c*sqrt((b*e - c*d)/c)), (b*e - c*d)/c > 0), (-acot
h(sqrt(d + e*x)/sqrt((-b*e + c*d)/c))/(c*sqrt((-b*e + c*d)/c)), ((b*e - c*d)/c <
0) & (d + e*x > (-b*e + c*d)/c)), (-atanh(sqrt(d + e*x)/sqrt((-b*e + c*d)/c))/(
c*sqrt((-b*e + c*d)/c)), ((b*e - c*d)/c < 0) & (d + e*x < (-b*e + c*d)/c)))/b)/e

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GIAC/XCAS [A]  time = 0.211709, size = 108, normalized size = 1.4 \[ -\frac{2 \,{\left (c d - b e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b} + \frac{2 \, d \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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