3.365 \(\int \frac{1}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.144979, 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 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3935, size = 68, normalized size = 0.88 \[ - \frac{2 \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \sqrt{b e - c d}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0834119, size = 77, normalized size = 1. \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 62, normalized size = 0.8 \[ -2\,{\frac{c}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{1}{b\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244325, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d} \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right )}{b \sqrt{d}}, \frac{2 \, \sqrt{d} \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{-\frac{c}{c d - b e}}}{\sqrt{e x + d} c}\right ) + \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right )}{b \sqrt{d}}, \frac{\sqrt{-d} \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + 2 \, \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right )}{b \sqrt{-d}}, \frac{2 \,{\left (\sqrt{-d} \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{-\frac{c}{c d - b e}}}{\sqrt{e x + d} c}\right ) + \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right )\right )}}{b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.99083, size = 308, normalized size = 4. \[ \frac{2 c \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{c}{b e - c d}} \left (b e - c d\right )} & \text{for}\: \frac{c}{b e - c d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{c}{b e - c d}} \left (b e - c d\right )} & \text{for}\: \frac{1}{d + e x} > - \frac{c}{b e - c d} \wedge \frac{c}{b e - c d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{c}{b e - c d}} \left (b e - c d\right )} & \text{for}\: \frac{c}{b e - c d} < 0 \wedge \frac{1}{d + e x} < - \frac{c}{b e - c d} \end{cases}\right )}{b} + \frac{2 \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{d \sqrt{- \frac{1}{d}}} & \text{for}\: - \frac{1}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{d + e x} \sqrt{\frac{1}{d}}} \right )}}{d \sqrt{\frac{1}{d}}} & \text{for}\: - \frac{1}{d} < 0 \wedge \frac{1}{d} < \frac{1}{d + e x} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{d + e x} \sqrt{\frac{1}{d}}} \right )}}{d \sqrt{\frac{1}{d}}} & \text{for}\: \frac{1}{d} > \frac{1}{d + e x} \wedge - \frac{1}{d} < 0 \end{cases}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*Piecewise((atan(1/(sqrt(c/(b*e - c*d))*sqrt(d + e*x)))/(sqrt(c/(b*e - c*d))*
(b*e - c*d)), c/(b*e - c*d) > 0), (-acoth(1/(sqrt(-c/(b*e - c*d))*sqrt(d + e*x))
)/(sqrt(-c/(b*e - c*d))*(b*e - c*d)), (c/(b*e - c*d) < 0) & (1/(d + e*x) > -c/(b
*e - c*d))), (-atanh(1/(sqrt(-c/(b*e - c*d))*sqrt(d + e*x)))/(sqrt(-c/(b*e - c*d
))*(b*e - c*d)), (c/(b*e - c*d) < 0) & (1/(d + e*x) < -c/(b*e - c*d))))/b + 2*Pi
ecewise((atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/(d*sqrt(-1/d)), -1/d > 0), (-acoth(1
/(sqrt(d + e*x)*sqrt(1/d)))/(d*sqrt(1/d)), (-1/d < 0) & (1/d < 1/(d + e*x))), (-
atanh(1/(sqrt(d + e*x)*sqrt(1/d)))/(d*sqrt(1/d)), (-1/d < 0) & (1/d > 1/(d + e*x
))))/b

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GIAC/XCAS [A]  time = 0.20694, size = 96, normalized size = 1.25 \[ -\frac{2 \, c \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b} + \frac{2 \, \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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