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

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

[Out]

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

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Rubi [A]  time = 0.321445, antiderivative size = 140, 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 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 \sqrt{d} e^2 \sqrt{c d-b e}}-\frac{\sqrt{b x+c x^2}}{e (d+e x)}+\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.8666, size = 121, normalized size = 0.86 \[ \frac{2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{e^{2}} - \frac{\sqrt{b x + c x^{2}}}{e \left (d + e x\right )} + \frac{\left (b e - 2 c d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 \sqrt{d} e^{2} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.330972, size = 144, normalized size = 1.03 \[ \frac{\sqrt{x (b+c x)} \left (\frac{(b e-2 c d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{b e-c d}}+\frac{2 \sqrt{c} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}-\frac{e}{d+e x}\right )}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 885, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d/(b*e-c*d)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-
1/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b+1/e/(b
*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c+1/e/(b*e-c*d
)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b-1/e^2*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x
))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)-1/
2/e/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e
+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2))/(d/e+x))*b^2+3/2/e^2*d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*
e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c-1/e^3*d^2/(b*e-c*d)/(-d*(b*e
-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^
2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^2
-c/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264052, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} - b d e}{\left (e x + d\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x} e -{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{2 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{c d^{2} - b d e}}, \frac{\sqrt{-c d^{2} + b d e}{\left (e x + d\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - \sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x} e +{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{{\left (e^{3} x + d e^{2}\right )} \sqrt{-c d^{2} + b d e}}, \frac{4 \, \sqrt{c d^{2} - b d e}{\left (e x + d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) - 2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x} e -{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{2 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{c d^{2} - b d e}}, \frac{2 \, \sqrt{-c d^{2} + b d e}{\left (e x + d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) - \sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x} e +{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{{\left (e^{3} x + d e^{2}\right )} \sqrt{-c d^{2} + b d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError