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

Optimal. Leaf size=266 \[ \frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^3}-\frac{f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^3}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )} \]

[Out]

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

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Rubi [A]  time = 0.548343, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^3}-\frac{f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^3}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 71.4719, size = 262, normalized size = 0.98 \[ \frac{4 f^{2} \left (4 a e^{2} - b^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{b f^{2} + 2 d e + e f \left (\frac{4 e x}{f} + 4 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )}{b f^{2} - 2 d e} \right )}}{\left (b f^{2} - 2 d e\right )^{3}} - \frac{f \left (2 a b e f^{2} + 4 a d e^{2} - 3 b^{2} d f^{2} + 2 b d^{2} e\right ) - \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \left (- 8 a e^{2} f^{2} + b^{2} f^{4} + 4 b d e f^{2} - 4 d^{2} e^{2}\right )}{\left (b f^{2} - 2 d e\right )^{2} \left (b d f + 2 e f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )^{2} + \left (b f^{2} + 2 d e\right ) \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right )} \]

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))**2,x)

[Out]

4*f**2*(4*a*e**2 - b**2*f**2)*atanh((b*f**2 + 2*d*e + e*f*(4*e*x/f + 4*sqrt(a +
b*x + e**2*x**2/f**2)))/(b*f**2 - 2*d*e))/(b*f**2 - 2*d*e)**3 - (f*(2*a*b*e*f**2
+ 4*a*d*e**2 - 3*b**2*d*f**2 + 2*b*d**2*e) - (e*x/f + sqrt(a + b*x + e**2*x**2/
f**2))*(-8*a*e**2*f**2 + b**2*f**4 + 4*b*d*e*f**2 - 4*d**2*e**2))/((b*f**2 - 2*d
*e)**2*(b*d*f + 2*e*f*(e*x/f + sqrt(a + b*x + e**2*x**2/f**2))**2 + (b*f**2 + 2*
d*e)*(e*x/f + sqrt(a + b*x + e**2*x**2/f**2))))

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Mathematica [A]  time = 1.02532, size = 421, normalized size = 1.58 \[ \frac{f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f^2 (a+b x)-d^2-2 d e x\right )+\left (4 a e^2 f^2-b^2 f^4\right ) \log \left (-f^2 (a+b x)+d^2+2 d e x\right )+f^2 \left (b^2 f^2-4 a e^2\right ) \log \left (b f^2 \left (-2 d \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+a f^2+d^2\right )+2 e \left (d^2 \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+a f^2 \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}-2 d-e x\right )\right )+b^2 f^4 x\right )+f^2 \left (b^2 f^2-4 a e^2\right ) \log \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )+\frac{2 f \left (b f^2-2 d e\right ) \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )} \left (2 e \left (a f^2-d e x\right )+b f^2 (e x-d)\right )}{f^2 (a+b x)-d^2-2 d e x}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )^2}{-f^2 (a+b x)+d^2+2 d e x}+2 e^2 x \left (2 d e-b f^2\right )}{\left (2 d e-b f^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*e^2*(2*d*e - b*f^2)*x - (2*(d^2*e - b*d*f^2 + a*e*f^2)^2)/(d^2 + 2*d*e*x - f^
2*(a + b*x)) + (2*f*(-2*d*e + b*f^2)*(b*f^2*(-d + e*x) + 2*e*(a*f^2 - d*e*x))*Sq
rt[a + x*(b + (e^2*x)/f^2)])/(-d^2 - 2*d*e*x + f^2*(a + b*x)) + (4*a*e^2*f^2 - b
^2*f^4)*Log[d^2 + 2*d*e*x - f^2*(a + b*x)] + f^2*(4*a*e^2 - b^2*f^2)*Log[-d^2 -
2*d*e*x + f^2*(a + b*x)] + f^2*(-4*a*e^2 + b^2*f^2)*Log[b*f^2 + 2*e*(e*x + f*Sqr
t[a + x*(b + (e^2*x)/f^2)])] + f^2*(-4*a*e^2 + b^2*f^2)*Log[b^2*f^4*x + b*f^2*(d
^2 + a*f^2 - 2*d*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) + 2*e*(a*f^2*(-2*d - e
*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]) + d^2*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2
)]))])/(2*d*e - b*f^2)^3

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Maple [B]  time = 0.069, size = 58067, normalized size = 218.3 \[ \text{output too large to display} \]

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))^2,x)

[Out]

result too large to display

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

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)^(-2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.43574, size = 1115, normalized size = 4.19 \[ \text{result too large to display} \]

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)^(-2),x, algorithm="fricas")

[Out]

-1/2*(a*b^2*f^6 + (3*b^2*d^2 - 14*a*b*d*e + 8*a^2*e^2)*f^4 - 2*(b*d^3*e - 4*a*d^
2*e^2)*f^2 - 4*(b^2*e^2*f^4 - 4*b*d*e^3*f^2 + 4*d^2*e^4)*x^2 + (b^3*f^6 - 8*b^2*
d*e*f^4 + 20*b*d^2*e^2*f^2 - 16*d^3*e^3)*x - 2*(a*b^2*f^6 + 4*a*d^2*e^2*f^2 - (b
^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4
)*x)*log(-4*a*d*e^2*f^2 - (b^2*d - 4*a*b*e)*f^4 + 4*(b*e^3*f^2 - 2*d*e^4)*x^2 +
(3*b^2*e*f^4 - 4*(2*b*d*e^2 - a*e^3)*f^2)*x - (b^2*f^5 - 4*(b*d*e - a*e^2)*f^3 +
4*(b*e^2*f^3 - 2*d*e^3*f)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2)) - 2*(a*b^2*
f^6 + 4*a*d^2*e^2*f^2 - (b^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2
*(b^2*d*e + 2*a*b*e^2)*f^4)*x)*log(a*f^2 - d^2 + (b*f^2 - 2*d*e)*x) + 2*(a*b^2*f
^6 + 4*a*d^2*e^2*f^2 - (b^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*
(b^2*d*e + 2*a*b*e^2)*f^4)*x)*log(-e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2)
- d) - 4*((b^2*d - 2*a*b*e)*f^5 - 2*(b*d^2*e - 2*a*d*e^2)*f^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))/(a*b^3*f^8
+ 8*d^5*e^3 - (b^3*d^2 + 6*a*b^2*d*e)*f^6 + 6*(b^2*d^3*e + 2*a*b*d^2*e^2)*f^4 -
4*(3*b*d^4*e^2 + 2*a*d^3*e^3)*f^2 + (b^4*f^8 - 8*b^3*d*e*f^6 + 24*b^2*d^2*e^2*f^
4 - 32*b*d^3*e^3*f^2 + 16*d^4*e^4)*x)

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

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))**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 1.0465, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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)^(-2),x, algorithm="giac")

[Out]

sage0*x