3.304 \(\int \frac{1}{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{5/2}} \, dx\)

Optimal. Leaf size=199 \[ \frac{5 a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 d^{7/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d^3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{2 a f^2}{d^3 e \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{\frac{a f^2}{d^2}+1}{3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}} \]

[Out]

-(1 + (a*f^2)/d^2)/(3*e*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2)) - (2*a*f^2)
/(d^3*e*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]) - (a*f^2*Sqrt[d + e*x + f*Sqr
t[a + (e^2*x^2)/f^2]])/(2*d^3*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (5*a*f^2*Ar
cTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/(2*d^(7/2)*e)

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Rubi [A]  time = 0.42237, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{5 a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 d^{7/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d^3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{2 a f^2}{d^3 e \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{\frac{a f^2}{d^2}+1}{3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-5/2),x]

[Out]

-(1 + (a*f^2)/d^2)/(3*e*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2)) - (2*a*f^2)
/(d^3*e*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]) - (a*f^2*Sqrt[d + e*x + f*Sqr
t[a + (e^2*x^2)/f^2]])/(2*d^3*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (5*a*f^2*Ar
cTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/(2*d^(7/2)*e)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{5}{2}}}\, dx \]

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))**(5/2),x)

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-5/2), x)

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Mathematica [A]  time = 0.722339, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-5/2),x]

[Out]

Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-5/2), x]

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{-{\frac{5}{2}}}\, dx \]

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))^(5/2),x)

[Out]

int(1/(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-5/2),x, algorithm="maxima")

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-5/2), x)

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Fricas [A]  time = 0.356918, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-5/2),x, algorithm="fricas")

[Out]

[-1/12*(15*((a^2*f^5 - 2*a*d*e*f^3*x - a*d^2*f^3)*sqrt(d)*sqrt((e^2*x^2 + a*f^2)
/f^2) - (2*a*d*e^2*f^2*x^2 - a^2*d*f^4 + a*d^3*f^2 - (a^2*e*f^4 - 3*a*d^2*e*f^2)
*x)*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*(15*a^2*d*f^4 + 6*d^3*e^2*x^2 - 17*a*d^3*f^2 - 2*d^5 - (35*a*d^2*e
*f^2 - d^4*e)*x + (5*a*d^2*f^3 - 6*d^3*e*f*x - d^4*f)*sqrt((e^2*x^2 + a*f^2)/f^2
))*sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/(2*d^5*e^3*x^2 - a*d^5*e*f^2 +
d^7*e - (a*d^4*e^2*f^2 - 3*d^6*e^2)*x - (a*d^4*e*f^3 - 2*d^5*e^2*f*x - d^6*e*f)
*sqrt((e^2*x^2 + a*f^2)/f^2)), -1/6*(15*((a^2*f^5 - 2*a*d*e*f^3*x - a*d^2*f^3)*s
qrt(-d)*sqrt((e^2*x^2 + a*f^2)/f^2) - (2*a*d*e^2*f^2*x^2 - a^2*d*f^4 + a*d^3*f^2
- (a^2*e*f^4 - 3*a*d^2*e*f^2)*x)*sqrt(-d))*arctan(d/(sqrt(e*x + f*sqrt((e^2*x^2
+ a*f^2)/f^2) + d)*sqrt(-d))) - (15*a^2*d*f^4 + 6*d^3*e^2*x^2 - 17*a*d^3*f^2 -
2*d^5 - (35*a*d^2*e*f^2 - d^4*e)*x + (5*a*d^2*f^3 - 6*d^3*e*f*x - d^4*f)*sqrt((e
^2*x^2 + a*f^2)/f^2))*sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/(2*d^5*e^3*
x^2 - a*d^5*e*f^2 + d^7*e - (a*d^4*e^2*f^2 - 3*d^6*e^2)*x - (a*d^4*e*f^3 - 2*d^5
*e^2*f*x - d^6*e*f)*sqrt((e^2*x^2 + a*f^2)/f^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{5}{2}}}\, dx \]

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))**(5/2),x)

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-5/2),x, algorithm="giac")

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-5/2), x)