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

Optimal. Leaf size=134 \[ \frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{3/4}}-\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{3/4}} \]

[Out]

-((Sqrt[Sqrt[c]*d - Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d -
Sqrt[a]*e]])/(Sqrt[a]*c^(3/4))) + (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*ArcTanh[(c^(1/4)*
Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(3/4))

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Rubi [A]  time = 0.315421, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{3/4}}-\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[Sqrt[c]*d - Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d -
Sqrt[a]*e]])/(Sqrt[a]*c^(3/4))) + (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*ArcTanh[(c^(1/4)*
Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(3/4))

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Rubi in Sympy [A]  time = 48.9836, size = 117, normalized size = 0.87 \[ - \frac{\sqrt{\sqrt{a} e - \sqrt{c} d} \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{3}{4}}} + \frac{\sqrt{\sqrt{a} e + \sqrt{c} d} \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(sqrt(a)*e - sqrt(c)*d)*atan(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e - sqrt(c
)*d))/(sqrt(a)*c**(3/4)) + sqrt(sqrt(a)*e + sqrt(c)*d)*atanh(c**(1/4)*sqrt(d + e
*x)/sqrt(sqrt(a)*e + sqrt(c)*d))/(sqrt(a)*c**(3/4))

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Mathematica [A]  time = 0.155931, size = 127, normalized size = 0.95 \[ \frac{\sqrt{\sqrt{a} \sqrt{c} e+c d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )-\sqrt{c d-\sqrt{a} \sqrt{c} e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt
[a]*Sqrt[c]*e]]) + Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c)

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Maple [B]  time = 0.029, size = 203, normalized size = 1.5 \[{ced{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{e{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ced\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{e\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(-c*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226173, size = 463, normalized size = 3.46 \[ \frac{1}{2} \, \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.6218, size = 76, normalized size = 0.57 \[ - 2 e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 77.037, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done