3.2280 \(\int \frac{\sqrt{d+e x}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=198 \[ \frac{\sqrt{2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.756636, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\sqrt{2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 85.4787, size = 185, normalized size = 0.93 \[ - \frac{\sqrt{2} \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{c} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.775799, size = 175, normalized size = 0.88 \[ \frac{\sqrt{2} \left (\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )-\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 545, normalized size = 2.8 \[{{e}^{2}\sqrt{2}b{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-2\,{\frac{ce\sqrt{2}d}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }-{e\sqrt{2}{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}+{{e}^{2}\sqrt{2}b\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-2\,{\frac{ce\sqrt{2}d}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }+{e\sqrt{2}\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + b*x + a), x)

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Fricas [A]  time = 0.23441, size = 965, normalized size = 4.87 \[ -\frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))
)/(b^2*c - 4*a*c^2))*log(sqrt(2)*(b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))
*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4
*a*c^2)) + 2*sqrt(e*x + d)*e) + 1/2*sqrt(2)*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2
)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log(-sqrt(2)*(b^2*c - 4*a*c^
2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/
(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) + 2*sqrt(e*x + d)*e) + 1/2*sqrt(2)*sqrt
((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^
2))*log(sqrt(2)*(b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))*sqrt((2*c*d - b*
e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) + 2*sqrt
(e*x + d)*e) - 1/2*sqrt(2)*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c
^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log(-sqrt(2)*(b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*
c^2 - 4*a*c^3))*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^
3)))/(b^2*c - 4*a*c^2)) + 2*sqrt(e*x + d)*e)

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Sympy [A]  time = 21.6576, size = 155, normalized size = 0.78 \[ 2 e \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} c^{3} e^{4} - 128 a b^{2} c^{2} e^{4} + 16 b^{4} c e^{4}\right ) + t^{2} \left (- 16 a b c e^{3} + 32 a c^{2} d e^{2} + 4 b^{3} e^{3} - 8 b^{2} c d e^{2}\right ) + a e^{2} - b d e + c d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} e^{2} - 16 t^{3} b^{2} c e^{2} - 2 t b e + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*RootSum(_t**4*(256*a**2*c**3*e**4 - 128*a*b**2*c**2*e**4 + 16*b**4*c*e**4) +
_t**2*(-16*a*b*c*e**3 + 32*a*c**2*d*e**2 + 4*b**3*e**3 - 8*b**2*c*d*e**2) + a*e
**2 - b*d*e + c*d**2, Lambda(_t, _t*log(64*_t**3*a*c**2*e**2 - 16*_t**3*b**2*c*e
**2 - 2*_t*b*e + 4*_t*c*d + sqrt(d + e*x))))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out