3.880 \(\int \frac{1}{\sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} c^{3/2} d^{5/2} e}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \]

[Out]

-1/(2*c*d*e*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]) + (3*Sqrt[d + e*x])/(4*c*d^2*
e*Sqrt[c*d^2 - c*e^2*x^2]) - (3*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]
*Sqrt[d]*Sqrt[d + e*x])])/(4*Sqrt[2]*c^(3/2)*d^(5/2)*e)

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Rubi [A]  time = 0.218672, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} c^{3/2} d^{5/2} e}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2)),x]

[Out]

-1/(2*c*d*e*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]) + (3*Sqrt[d + e*x])/(4*c*d^2*
e*Sqrt[c*d^2 - c*e^2*x^2]) - (3*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]
*Sqrt[d]*Sqrt[d + e*x])])/(4*Sqrt[2]*c^(3/2)*d^(5/2)*e)

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Rubi in Sympy [A]  time = 23.1806, size = 133, normalized size = 0.89 \[ - \frac{1}{2 c d e \sqrt{d + e x} \sqrt{c d^{2} - c e^{2} x^{2}}} + \frac{3 \sqrt{d + e x}}{4 c d^{2} e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{3 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{8 c^{\frac{3}{2}} d^{\frac{5}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

-1/(2*c*d*e*sqrt(d + e*x)*sqrt(c*d**2 - c*e**2*x**2)) + 3*sqrt(d + e*x)/(4*c*d**
2*e*sqrt(c*d**2 - c*e**2*x**2)) - 3*sqrt(2)*atanh(sqrt(2)*sqrt(c*d**2 - c*e**2*x
**2)/(2*sqrt(c)*sqrt(d)*sqrt(d + e*x)))/(8*c**(3/2)*d**(5/2)*e)

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Mathematica [A]  time = 0.126654, size = 128, normalized size = 0.85 \[ \frac{2 \sqrt{d} \sqrt{d+e x} (d+3 e x)-3 \sqrt{2} (d+e x) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{8 c d^{5/2} e (d+e x) \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2)),x]

[Out]

(2*Sqrt[d]*Sqrt[d + e*x]*(d + 3*e*x) - 3*Sqrt[2]*(d + e*x)*Sqrt[d^2 - e^2*x^2]*A
rcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(8*c*d^(5/2)*e*(d +
e*x)*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.033, size = 152, normalized size = 1. \[{\frac{1}{8\,{c}^{2} \left ( ex-d \right ) e{d}^{2}}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xe+3\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) d-6\,\sqrt{cd}xe-2\,\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

1/8/(e*x+d)^(3/2)*(-c*(e^2*x^2-d^2))^(1/2)/c^2*(3*(-(e*x-d)*c)^(1/2)*2^(1/2)*arc
tanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*x*e+3*(-(e*x-d)*c)^(1/2)*2^(1/2
)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*d-6*(c*d)^(1/2)*x*e-2*(c*d
)^(1/2)*d)/(e*x-d)/e/d^2/(c*d)^(1/2)

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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(1/((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237058, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d}{\left (3 \, e x + d\right )} \sqrt{e x + d} - 3 \,{\left (c e^{3} x^{3} + c d e^{2} x^{2} - c d^{2} e x - c d^{3}\right )} \log \left (-\frac{\sqrt{2}{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )} \sqrt{c d} + 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{16 \,{\left (c^{2} d^{2} e^{4} x^{3} + c^{2} d^{3} e^{3} x^{2} - c^{2} d^{4} e^{2} x - c^{2} d^{5} e\right )} \sqrt{c d}}, -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d}{\left (3 \, e x + d\right )} \sqrt{e x + d} - 3 \,{\left (c e^{3} x^{3} + c d e^{2} x^{2} - c d^{2} e x - c d^{3}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right )\right )}}{8 \,{\left (c^{2} d^{2} e^{4} x^{3} + c^{2} d^{3} e^{3} x^{2} - c^{2} d^{4} e^{2} x - c^{2} d^{5} e\right )} \sqrt{-c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[-1/16*sqrt(2)*(2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(c*d)*(3*e*x + d)*sqrt(e*
x + d) - 3*(c*e^3*x^3 + c*d*e^2*x^2 - c*d^2*e*x - c*d^3)*log(-(sqrt(2)*(e^2*x^2
- 2*d*e*x - 3*d^2)*sqrt(c*d) + 4*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*d)/(e^2*
x^2 + 2*d*e*x + d^2)))/((c^2*d^2*e^4*x^3 + c^2*d^3*e^3*x^2 - c^2*d^4*e^2*x - c^2
*d^5*e)*sqrt(c*d)), -1/8*sqrt(2)*(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)*(3
*e*x + d)*sqrt(e*x + d) - 3*(c*e^3*x^3 + c*d*e^2*x^2 - c*d^2*e*x - c*d^3)*arctan
(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)*sqrt(e*x + d)/(c*e^2*x^2 - c*d^2)))
/((c^2*d^2*e^4*x^3 + c^2*d^3*e^3*x^2 - c^2*d^4*e^2*x - c^2*d^5*e)*sqrt(-c*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral(1/((-c*(-d + e*x)*(d + e*x))**(3/2)*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.565378, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

sage0*x