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

Optimal. Leaf size=335 \[ -\frac{4 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{2 e f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{5 \sqrt{2} \sqrt{e} f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{\left (2 d e-b f^2\right )^{7/2}}-\frac{4 \left (a e f^2-b d f^2+d^2 e\right )}{3 \left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}} \]

[Out]

(-4*(d^2*e - b*d*f^2 + a*e*f^2))/(3*(2*d*e - b*f^2)^2*(d + e*x + f*Sqrt[a + b*x
+ (e^2*x^2)/f^2])^(3/2)) - (4*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*Sqrt[d
+ e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]]) - (2*e*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)^3*(b*f^2 + 2*e*(e*x
+ f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) + (5*Sqrt[2]*Sqrt[e]*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]])/(2*d*e - b*f^2)^(7/2)

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Rubi [A]  time = 1.37831, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{4 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{2 e f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{5 \sqrt{2} \sqrt{e} f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{\left (2 d e-b f^2\right )^{7/2}}-\frac{4 \left (a e f^2-b d f^2+d^2 e\right )}{3 \left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(d^2*e - b*d*f^2 + a*e*f^2))/(3*(2*d*e - b*f^2)^2*(d + e*x + f*Sqrt[a + b*x
+ (e^2*x^2)/f^2])^(3/2)) - (4*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*Sqrt[d
+ e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]]) - (2*e*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)^3*(b*f^2 + 2*e*(e*x
+ f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) + (5*Sqrt[2]*Sqrt[e]*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]])/(2*d*e - b*f^2)^(7/2)

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

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

[Out]

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

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Mathematica [A]  time = 1.13015, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{-{\frac{5}{2}}}\, dx \]

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))^(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.834223, 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)^(-5/2),x, algorithm="fricas")

[Out]

[-1/6*(15*(sqrt(2)*(a*b^2*f^7 + 4*a*d^2*e^2*f^3 - (b^2*d^2 + 4*a^2*e^2)*f^5 + (b
^3*f^7 + 8*a*d*e^3*f^3 - 2*(b^2*d*e + 2*a*b*e^2)*f^5)*x)*sqrt(-e/(b*f^2 - 2*d*e)
)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + sqrt(2)*(a*b^2*d*f^6 + 4*a*d^3*e^2*f^2
- (b^2*d^3 + 4*a^2*d*e^2)*f^4 + (b^3*e*f^6 + 8*a*d*e^4*f^2 - 2*(b^2*d*e^2 + 2*a
*b*e^3)*f^4)*x^2 + (12*a*d^2*e^3*f^2 + (b^3*d + a*b^2*e)*f^6 - (3*b^2*d^2*e + 4*
a*b*d*e^2 + 4*a^2*e^3)*f^4)*x)*sqrt(-e/(b*f^2 - 2*d*e)))*log(-(b*f^2 - 2*e^2*x -
2*sqrt(2)*(b*f^2 - 2*d*e)*sqrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) +
d)*sqrt(-e/(b*f^2 - 2*d*e)) - 2*e*f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) - 4*d*
e)/(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*(6*a*b^2
*f^6 + 4*d^4*e^2 - (4*b^2*d^2 - a*b*d*e + 30*a^2*e^2)*f^4 - (9*b*d^3*e - 34*a*d^
2*e^2)*f^2 - 3*(b^2*e^2*f^4 - 4*b*d*e^3*f^2 + 4*d^2*e^4)*x^2 + (6*b^3*f^6 - 2*d^
3*e^3 - 7*(b^2*d*e + 5*a*b*e^2)*f^4 - (9*b*d^2*e^2 - 70*a*d*e^3)*f^2)*x + (2*d^3
*e^2*f - (2*b^2*d - 5*a*b*e)*f^5 + (3*b*d^2*e - 10*a*d*e^2)*f^3 + 3*(b^2*e*f^5 -
4*b*d*e^2*f^3 + 4*d^2*e^3*f)*x)*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))/(a*b^3*d*f^8 + 8*d^6*e^3 - (b^3
*d^3 + 6*a*b^2*d^2*e)*f^6 + 6*(b^2*d^4*e + 2*a*b*d^3*e^2)*f^4 - 4*(3*b*d^5*e^2 +
2*a*d^4*e^3)*f^2 + (b^4*e*f^8 - 8*b^3*d*e^2*f^6 + 24*b^2*d^2*e^3*f^4 - 32*b*d^3
*e^4*f^2 + 16*d^4*e^5)*x^2 + (24*d^5*e^4 + (b^4*d + a*b^3*e)*f^8 - 3*(3*b^3*d^2*
e + 2*a*b^2*d*e^2)*f^6 + 6*(5*b^2*d^3*e^2 + 2*a*b*d^2*e^3)*f^4 - 4*(11*b*d^4*e^3
+ 2*a*d^3*e^4)*f^2)*x + (a*b^3*f^9 + 8*d^5*e^3*f - (b^3*d^2 + 6*a*b^2*d*e)*f^7
+ 6*(b^2*d^3*e + 2*a*b*d^2*e^2)*f^5 - 4*(3*b*d^4*e^2 + 2*a*d^3*e^3)*f^3 + (b^4*f
^9 - 8*b^3*d*e*f^7 + 24*b^2*d^2*e^2*f^5 - 32*b*d^3*e^3*f^3 + 16*d^4*e^4*f)*x)*sq
rt((b*f^2*x + e^2*x^2 + a*f^2)/f^2)), 1/3*(15*(sqrt(2)*(a*b^2*f^7 + 4*a*d^2*e^2*
f^3 - (b^2*d^2 + 4*a^2*e^2)*f^5 + (b^3*f^7 + 8*a*d*e^3*f^3 - 2*(b^2*d*e + 2*a*b*
e^2)*f^5)*x)*sqrt(e/(b*f^2 - 2*d*e))*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + sqr
t(2)*(a*b^2*d*f^6 + 4*a*d^3*e^2*f^2 - (b^2*d^3 + 4*a^2*d*e^2)*f^4 + (b^3*e*f^6 +
8*a*d*e^4*f^2 - 2*(b^2*d*e^2 + 2*a*b*e^3)*f^4)*x^2 + (12*a*d^2*e^3*f^2 + (b^3*d
+ a*b^2*e)*f^6 - (3*b^2*d^2*e + 4*a*b*d*e^2 + 4*a^2*e^3)*f^4)*x)*sqrt(e/(b*f^2
- 2*d*e)))*arctan(1/2*sqrt(2)*(b*f^2 - 2*d*e)*sqrt(e/(b*f^2 - 2*d*e))/(sqrt(e*x
+ f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d)*e)) - 2*(6*a*b^2*f^6 + 4*d^4*e^2
- (4*b^2*d^2 - a*b*d*e + 30*a^2*e^2)*f^4 - (9*b*d^3*e - 34*a*d^2*e^2)*f^2 - 3*(b
^2*e^2*f^4 - 4*b*d*e^3*f^2 + 4*d^2*e^4)*x^2 + (6*b^3*f^6 - 2*d^3*e^3 - 7*(b^2*d*
e + 5*a*b*e^2)*f^4 - (9*b*d^2*e^2 - 70*a*d*e^3)*f^2)*x + (2*d^3*e^2*f - (2*b^2*d
- 5*a*b*e)*f^5 + (3*b*d^2*e - 10*a*d*e^2)*f^3 + 3*(b^2*e*f^5 - 4*b*d*e^2*f^3 +
4*d^2*e^3*f)*x)*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))/(a*b^3*d*f^8 + 8*d^6*e^3 - (b^3*d^3 + 6*a*b^2*d^
2*e)*f^6 + 6*(b^2*d^4*e + 2*a*b*d^3*e^2)*f^4 - 4*(3*b*d^5*e^2 + 2*a*d^4*e^3)*f^2
+ (b^4*e*f^8 - 8*b^3*d*e^2*f^6 + 24*b^2*d^2*e^3*f^4 - 32*b*d^3*e^4*f^2 + 16*d^4
*e^5)*x^2 + (24*d^5*e^4 + (b^4*d + a*b^3*e)*f^8 - 3*(3*b^3*d^2*e + 2*a*b^2*d*e^2
)*f^6 + 6*(5*b^2*d^3*e^2 + 2*a*b*d^2*e^3)*f^4 - 4*(11*b*d^4*e^3 + 2*a*d^3*e^4)*f
^2)*x + (a*b^3*f^9 + 8*d^5*e^3*f - (b^3*d^2 + 6*a*b^2*d*e)*f^7 + 6*(b^2*d^3*e +
2*a*b*d^2*e^2)*f^5 - 4*(3*b*d^4*e^2 + 2*a*d^3*e^3)*f^3 + (b^4*f^9 - 8*b^3*d*e*f^
7 + 24*b^2*d^2*e^2*f^5 - 32*b*d^3*e^3*f^3 + 16*d^4*e^4*f)*x)*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. \[ \int \frac{1}{\left (d + e x + f \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{5}{2}}}\, dx \]

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

[Out]

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

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GIAC/XCAS [A]  time = 1.41446, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^(-5/2),x, algorithm="giac")

[Out]

sage0*x