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

Optimal. Leaf size=140 \[ -\frac{\sqrt{c} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c d-b e}}+\frac{(4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 \sqrt{d}}-\frac{2 c \sqrt{d+e x}}{b^2 (b+c x)}-\frac{\sqrt{d+e x}}{b x (b+c x)} \]

[Out]

(-2*c*Sqrt[d + e*x])/(b^2*(b + c*x)) - Sqrt[d + e*x]/(b*x*(b + c*x)) + ((4*c*d -
 b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*Sqrt[d]) - (Sqrt[c]*(4*c*d - 3*b*e)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.521599, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\sqrt{c} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c d-b e}}+\frac{(4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 \sqrt{d}}-\frac{2 c \sqrt{d+e x}}{b^2 (b+c x)}-\frac{\sqrt{d+e x}}{b x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*Sqrt[d + e*x])/(b^2*(b + c*x)) - Sqrt[d + e*x]/(b*x*(b + c*x)) + ((4*c*d -
 b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*Sqrt[d]) - (Sqrt[c]*(4*c*d - 3*b*e)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*Sqrt[c*d - b*e])

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Rubi in Sympy [A]  time = 59.5064, size = 124, normalized size = 0.89 \[ - \frac{\sqrt{d + e x}}{b x \left (b + c x\right )} - \frac{2 c \sqrt{d + e x}}{b^{2} \left (b + c x\right )} - \frac{\sqrt{c} \left (3 b e - 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \sqrt{b e - c d}} - \frac{\left (b e - 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d + e*x)/(b*x*(b + c*x)) - 2*c*sqrt(d + e*x)/(b**2*(b + c*x)) - sqrt(c)*(3
*b*e - 4*c*d)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b**3*sqrt(b*e - c*d))
 - (b*e - 4*c*d)*atanh(sqrt(d + e*x)/sqrt(d))/(b**3*sqrt(d))

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Mathematica [A]  time = 0.331311, size = 119, normalized size = 0.85 \[ \frac{-\frac{b (b+2 c x) \sqrt{d+e x}}{x (b+c x)}+\frac{(4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}+\frac{\sqrt{c} (3 b e-4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c d-b e}}}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(-((b*(b + 2*c*x)*Sqrt[d + e*x])/(x*(b + c*x))) + ((4*c*d - b*e)*ArcTanh[Sqrt[d
+ e*x]/Sqrt[d]])/Sqrt[d] + (Sqrt[c]*(-4*c*d + 3*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e
*x])/Sqrt[c*d - b*e]])/Sqrt[c*d - b*e])/b^3

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Maple [A]  time = 0.026, size = 167, normalized size = 1.2 \[ -{\frac{ce}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{ce}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}d}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}x}\sqrt{ex+d}}-{\frac{e}{{b}^{2}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{d}}}}+4\,{\frac{\sqrt{d}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e/b^2*c*(e*x+d)^(1/2)/(c*e*x+b*e)-3*e/b^2*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d
)^(1/2)/((b*e-c*d)*c)^(1/2))+4/b^3*c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2
)/((b*e-c*d)*c)^(1/2))*d-1/b^2*(e*x+d)^(1/2)/x-e/b^2/d^(1/2)*arctanh((e*x+d)^(1/
2)/d^(1/2))+4/b^3*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258633, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(((4*c^2*d - 3*b*c*e)*x^2 + (4*b*c*d - 3*b^2*e)*x)*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*(2*b*c*x + b^2)*sqrt(e*x + d)*sqrt(d) + ((4*c^2*d - b*c*e)*x^2 +
 (4*b*c*d - b^2*e)*x)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/((b^3*c*
x^2 + b^4*x)*sqrt(d)), -1/2*(2*((4*c^2*d - 3*b*c*e)*x^2 + (4*b*c*d - 3*b^2*e)*x)
*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x
 + d)*c)) + 2*(2*b*c*x + b^2)*sqrt(e*x + d)*sqrt(d) + ((4*c^2*d - b*c*e)*x^2 + (
4*b*c*d - b^2*e)*x)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/((b^3*c*x^
2 + b^4*x)*sqrt(d)), -1/2*(((4*c^2*d - 3*b*c*e)*x^2 + (4*b*c*d - 3*b^2*e)*x)*sqr
t(-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*(2*b*c*x + b^2)*sqrt(e*x + d)*sqrt(-d) + 2*
((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x)*arctan(d/(sqrt(e*x + d)*sqrt(-d)))
)/((b^3*c*x^2 + b^4*x)*sqrt(-d)), -(((4*c^2*d - 3*b*c*e)*x^2 + (4*b*c*d - 3*b^2*
e)*x)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sq
rt(e*x + d)*c)) + (2*b*c*x + b^2)*sqrt(e*x + d)*sqrt(-d) + ((4*c^2*d - b*c*e)*x^
2 + (4*b*c*d - b^2*e)*x)*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/((b^3*c*x^2 + b^4*x
)*sqrt(-d))]

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Sympy [A]  time = 95.5427, size = 1114, normalized size = 7.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**2*d*e*sqrt(d + e*x)/(2*b**4*e**2 - 2*b**3*c*d*e + 2*b**3*c*e**2*x - 2*b**2*
c**2*d*e*x) - 2*c*e**2*sqrt(d + e*x)/(2*b**3*e**2 - 2*b**2*c*d*e + 2*b**2*c*e**2
*x - 2*b*c**2*d*e*x) + c*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1
/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1
/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - c*e**2*sqrt(-1/(c*(b*e - c*d)**3))
*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3
)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - c**2*d*e*sqr
t(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*
sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e
*x))/(2*b**2) + c**2*d*e*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sqrt(-1/(c*(b
*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b
*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) - 2*c*e*Piecewise((atan(sqrt(d + e*x)/s
qrt(b*e/c - d))/(c*sqrt(b*e/c - d)), b*e/c - d > 0), (-acoth(sqrt(d + e*x)/sqrt(
-b*e/c + d))/(c*sqrt(-b*e/c + d)), (b*e/c - d < 0) & (d + e*x > -b*e/c + d)), (-
atanh(sqrt(d + e*x)/sqrt(-b*e/c + d))/(c*sqrt(-b*e/c + d)), (b*e/c - d < 0) & (d
 + e*x < -b*e/c + d)))/b**2 - d*e*sqrt(d**(-3))*log(-d**2*sqrt(d**(-3)) + sqrt(d
 + e*x))/(2*b**2) + d*e*sqrt(d**(-3))*log(d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2
*b**2) - 2*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**2 - sqrt(d + e*x)/(b**2*x) + 4*c
**2*d*Piecewise((atan(sqrt(d + e*x)/sqrt(b*e/c - d))/(c*sqrt(b*e/c - d)), b*e/c
- d > 0), (-acoth(sqrt(d + e*x)/sqrt(-b*e/c + d))/(c*sqrt(-b*e/c + d)), (b*e/c -
 d < 0) & (d + e*x > -b*e/c + d)), (-atanh(sqrt(d + e*x)/sqrt(-b*e/c + d))/(c*sq
rt(-b*e/c + d)), (b*e/c - d < 0) & (d + e*x < -b*e/c + d)))/b**3 + 4*c*d*Piecewi
se((-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**3

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GIAC/XCAS [A]  time = 0.215298, size = 244, normalized size = 1.74 \[ \frac{{\left (4 \, c^{2} d - 3 \, b c e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3}} - \frac{{\left (4 \, c d - b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c e - 2 \, \sqrt{x e + d} c d e + \sqrt{x e + d} b e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*c^2*d - 3*b*c*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d +
b*c*e)*b^3) - (4*c*d - b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*(
x*e + d)^(3/2)*c*e - 2*sqrt(x*e + d)*c*d*e + sqrt(x*e + d)*b*e^2)/(((x*e + d)^2*
c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2)

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