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

Optimal. Leaf size=269 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{3 f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{\sqrt{2} \sqrt{e} \left (2 d e-b f^2\right )^{5/2}}-\frac{4 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}} \]

[Out]

(-4*(d^2*e - b*d*f^2 + a*e*f^2))/((2*d*e - b*f^2)^2*Sqrt[d + e*x + f*Sqrt[a + b*
x + (e^2*x^2)/f^2]]) - (f^2*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + b*x +
(e^2*x^2)/f^2]])/((2*d*e - b*f^2)^2*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e
^2*x))/f^2]))) + (3*f^2*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*
x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/Sqrt[2*d*e - b*f^2]])/(Sqrt[2]*Sqrt[e]*(2*
d*e - b*f^2)^(5/2))

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Rubi [A]  time = 0.960809, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{3 f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{\sqrt{2} \sqrt{e} \left (2 d e-b f^2\right )^{5/2}}-\frac{4 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(d^2*e - b*d*f^2 + a*e*f^2))/((2*d*e - b*f^2)^2*Sqrt[d + e*x + f*Sqrt[a + b*
x + (e^2*x^2)/f^2]]) - (f^2*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + b*x +
(e^2*x^2)/f^2]])/((2*d*e - b*f^2)^2*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e
^2*x))/f^2]))) + (3*f^2*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*
x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/Sqrt[2*d*e - b*f^2]])/(Sqrt[2]*Sqrt[e]*(2*
d*e - b*f^2)^(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.805268, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{-{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.526124, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(3*((b^2*f^5 - 4*a*e^2*f^3)*sqrt(-2*b*e*f^2 + 4*d*e^2)*sqrt((b*f^2*x + e^2*
x^2 + a*f^2)/f^2) + (b^2*d*f^4 - 4*a*d*e^2*f^2 + (b^2*e*f^4 - 4*a*e^3*f^2)*x)*sq
rt(-2*b*e*f^2 + 4*d*e^2))*log((2*sqrt(-2*b*e*f^2 + 4*d*e^2)*e*f*sqrt((b*f^2*x +
e^2*x^2 + a*f^2)/f^2) - sqrt(-2*b*e*f^2 + 4*d*e^2)*(b*f^2 - 2*e^2*x - 4*d*e) - 4
*(b*e*f^2 - 2*d*e^2)*sqrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d))/(b
*f^2 + 2*e^2*x + 2*e*f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))) + 4*(8*d^3*e^3 +
(5*b^2*d*e - 6*a*b*e^2)*f^4 - 2*(7*b*d^2*e^2 - 6*a*d*e^3)*f^2 - (b^2*e^2*f^4 - 4
*b*d*e^3*f^2 + 4*d^2*e^4)*x + (b^2*e*f^5 - 4*b*d*e^2*f^3 + 4*d^2*e^3*f)*sqrt((b*
f^2*x + e^2*x^2 + a*f^2)/f^2))*sqrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2
) + d))/(b^3*d*e*f^6 - 6*b^2*d^2*e^2*f^4 + 12*b*d^3*e^3*f^2 - 8*d^4*e^4 + (b^3*e
^2*f^6 - 6*b^2*d*e^3*f^4 + 12*b*d^2*e^4*f^2 - 8*d^3*e^5)*x + (b^3*e*f^7 - 6*b^2*
d*e^2*f^5 + 12*b*d^2*e^3*f^3 - 8*d^3*e^4*f)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2
)), -1/2*(3*((b^2*f^5 - 4*a*e^2*f^3)*sqrt(2*b*e*f^2 - 4*d*e^2)*sqrt((b*f^2*x + e
^2*x^2 + a*f^2)/f^2) + (b^2*d*f^4 - 4*a*d*e^2*f^2 + (b^2*e*f^4 - 4*a*e^3*f^2)*x)
*sqrt(2*b*e*f^2 - 4*d*e^2))*arctan((b*f^2 - 2*d*e)/(sqrt(2*b*e*f^2 - 4*d*e^2)*sq
rt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d))) - 2*(8*d^3*e^3 + (5*b^2*
d*e - 6*a*b*e^2)*f^4 - 2*(7*b*d^2*e^2 - 6*a*d*e^3)*f^2 - (b^2*e^2*f^4 - 4*b*d*e^
3*f^2 + 4*d^2*e^4)*x + (b^2*e*f^5 - 4*b*d*e^2*f^3 + 4*d^2*e^3*f)*sqrt((b*f^2*x +
e^2*x^2 + a*f^2)/f^2))*sqrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d))
/(b^3*d*e*f^6 - 6*b^2*d^2*e^2*f^4 + 12*b*d^3*e^3*f^2 - 8*d^4*e^4 + (b^3*e^2*f^6
- 6*b^2*d*e^3*f^4 + 12*b*d^2*e^4*f^2 - 8*d^3*e^5)*x + (b^3*e*f^7 - 6*b^2*d*e^2*f
^5 + 12*b*d^2*e^3*f^3 - 8*d^3*e^4*f)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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