3.604 \(\int \frac{1}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.329794, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.5996, size = 117, normalized size = 0.87 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e + \sqrt{c} d}} + \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e - \sqrt{c} d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.156811, size = 128, normalized size = 0.96 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \sqrt{c d-\sqrt{a} \sqrt{c} e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 110, normalized size = 0.8 \[{ce{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (c x^{2} - a\right )} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 - a)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 - a)*sqrt(e*x + d)), x)

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Fricas [A]  time = 0.228358, size = 1281, normalized size = 9.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 - a)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- a \sqrt{d + e x} + c x^{2} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(1/(-a*sqrt(d + e*x) + c*x**2*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 15.6814, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 - a)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Done