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

Optimal. Leaf size=129 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e^2}+\frac{\sqrt{d} \sqrt{c d-b e} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^2}+\frac{\sqrt{b x+c x^2}}{e} \]

[Out]

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

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Rubi [A]  time = 0.373672, antiderivative size = 129, 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 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e^2}+\frac{\sqrt{d} \sqrt{c d-b e} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^2}+\frac{\sqrt{b x+c x^2}}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.9377, size = 112, normalized size = 0.87 \[ - \frac{\sqrt{d} \sqrt{b e - c d} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{e^{2}} + \frac{\sqrt{b x + c x^{2}}}{e} + \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.283296, size = 168, normalized size = 1.3 \[ \frac{\sqrt{x} \sqrt{b+c x} \left (\sqrt{b e-c d} \left ((b e-2 c d) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+\sqrt{c} e \sqrt{x} \sqrt{b+c x}\right )+2 \sqrt{c} \sqrt{d} (c d-b e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{\sqrt{c} e^2 \sqrt{x (b+c x)} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 490, normalized size = 3.8 \[{\frac{1}{e}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}+{\frac{b}{2\,e}\ln \left ({1 \left ({\frac{be-2\,cd}{2\,e}}+c \left ({\frac{d}{e}}+x \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{d}{{e}^{2}}\ln \left ({1 \left ({\frac{be-2\,cd}{2\,e}}+c \left ({\frac{d}{e}}+x \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \sqrt{c}}+{\frac{bd}{{e}^{2}}\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}-{\frac{c{d}^{2}}{{e}^{3}}\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25718, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c x^{2} + b x} \sqrt{c} e -{\left (2 \, c d - b e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c} \log \left (\frac{b d +{\left (2 \, c d - b e\right )} x + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}{e x + d}\right )}{2 \, \sqrt{c} e^{2}}, \frac{2 \, \sqrt{c x^{2} + b x} \sqrt{c} e + 4 \, \sqrt{-c d^{2} + b d e} \sqrt{c} \arctan \left (\frac{\sqrt{c x^{2} + b x} d}{\sqrt{-c d^{2} + b d e} x}\right ) -{\left (2 \, c d - b e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{2 \, \sqrt{c} e^{2}}, \frac{\sqrt{c x^{2} + b x} \sqrt{-c} e -{\left (2 \, c d - b e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c d^{2} - b d e} \sqrt{-c} \log \left (\frac{b d +{\left (2 \, c d - b e\right )} x + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}{e x + d}\right )}{\sqrt{-c} e^{2}}, \frac{\sqrt{c x^{2} + b x} \sqrt{-c} e + 2 \, \sqrt{-c d^{2} + b d e} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} d}{\sqrt{-c d^{2} + b d e} x}\right ) -{\left (2 \, c d - b e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{\sqrt{-c} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError