3.303 \(\int \frac{1}{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{3 a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 d^{5/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d^2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{\frac{a f^2}{d^2}+1}{e \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.341548, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{3 a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 d^{5/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d^2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{\frac{a f^2}{d^2}+1}{e \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-3/2), x)

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Mathematica [A]  time = 0.403787, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-3/2), x]

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{-{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-3/2),x, algorithm="maxima")

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-3/2), x)

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Fricas [A]  time = 0.343136, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a \sqrt{d} f^{3} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} +{\left (a e f^{2} x + a d f^{2}\right )} \sqrt{d}\right )} \log \left (\frac{\sqrt{d} f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} +{\left (e x + 2 \, d\right )} \sqrt{d} + 2 \, \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} d}{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}\right ) - 2 \,{\left (3 \, a d f^{2} - d^{2} e x + d^{2} f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{3}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{4 \,{\left (d^{3} e^{2} x + d^{3} e f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d^{4} e\right )}}, \frac{3 \,{\left (a \sqrt{-d} f^{3} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} +{\left (a e f^{2} x + a d f^{2}\right )} \sqrt{-d}\right )} \arctan \left (\frac{d}{\sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt{-d}}\right ) -{\left (3 \, a d f^{2} - d^{2} e x + d^{2} f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{3}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{2 \,{\left (d^{3} e^{2} x + d^{3} e f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d^{4} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(3*(a*sqrt(d)*f^3*sqrt((e^2*x^2 + a*f^2)/f^2) + (a*e*f^2*x + a*d*f^2)*sqrt(
d))*log((sqrt(d)*f*sqrt((e^2*x^2 + a*f^2)/f^2) + (e*x + 2*d)*sqrt(d) + 2*sqrt(e*
x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)*d)/(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2)))
- 2*(3*a*d*f^2 - d^2*e*x + d^2*f*sqrt((e^2*x^2 + a*f^2)/f^2) + 2*d^3)*sqrt(e*x
+ f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/(d^3*e^2*x + d^3*e*f*sqrt((e^2*x^2 + a*f^2
)/f^2) + d^4*e), 1/2*(3*(a*sqrt(-d)*f^3*sqrt((e^2*x^2 + a*f^2)/f^2) + (a*e*f^2*x
+ a*d*f^2)*sqrt(-d))*arctan(d/(sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)*sq
rt(-d))) - (3*a*d*f^2 - d^2*e*x + d^2*f*sqrt((e^2*x^2 + a*f^2)/f^2) + 2*d^3)*sqr
t(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/(d^3*e^2*x + d^3*e*f*sqrt((e^2*x^2 +
a*f^2)/f^2) + d^4*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-3/2), x)