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

Optimal. Leaf size=215 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right )}{2 e \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{2 e \left (2 d e-b f^2\right )^2}+\frac{2 \left (a e f^2-b d f^2+d^2 e\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.449108, antiderivative size = 215, 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 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right )}{2 e \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{2 e \left (2 d e-b f^2\right )^2}+\frac{2 \left (a e f^2-b d f^2+d^2 e\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 136.281, size = 187, normalized size = 0.87 \[ \frac{2 \left (a e f^{2} - b d f^{2} + d^{2} e\right ) \log{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{\left (b f^{2} - 2 d e\right )^{2}} - \frac{f^{2} \left (4 a e^{2} - b^{2} f^{2}\right ) \log{\left (b f + e \left (\frac{2 e x}{f} + 2 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{2 e \left (b f^{2} - 2 d e\right )^{2}} + \frac{f \left (4 a e^{2} - b^{2} f^{2}\right )}{2 e \left (b f + e \left (\frac{2 e x}{f} + 2 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right ) \left (b f^{2} - 2 d e\right )} \]

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)),x)

[Out]

2*(a*e*f**2 - b*d*f**2 + d**2*e)*log(d + f*(e*x/f + sqrt(a + b*x + e**2*x**2/f**
2)))/(b*f**2 - 2*d*e)**2 - f**2*(4*a*e**2 - b**2*f**2)*log(b*f + e*(2*e*x/f + 2*
sqrt(a + b*x + e**2*x**2/f**2)))/(2*e*(b*f**2 - 2*d*e)**2) + f*(4*a*e**2 - b**2*
f**2)/(2*e*(b*f + e*(2*e*x/f + 2*sqrt(a + b*x + e**2*x**2/f**2)))*(b*f**2 - 2*d*
e))

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Mathematica [A]  time = 0.551367, size = 275, normalized size = 1.28 \[ \frac{2 e \left (a e f^2-b d f^2+d^2 e\right ) \log \left (b f^2 \left (2 d f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+a f^2+d^2-2 d e x\right )+2 d^2 e \left (e x-f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}\right )-2 a e f^2 \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+2 d+e x\right )+b^2 f^4 x\right )+\left (-2 a e^2 f^2+b^2 f^4-2 b d e f^2+2 d^2 e^2\right ) \log \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )+2 e f \left (b f^2-2 d e\right ) \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+2 e^2 x \left (2 d e-b f^2\right )}{2 e \left (b f^2-2 d e\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-1),x]

[Out]

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

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Maple [B]  time = 0.075, size = 4918, normalized size = 22.9 \[ \text{output too large to display} \]

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)),x)

[Out]

f/(b*f^2-2*d*e)*(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2
*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f
^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2
*d*e)^2)^(1/2)+1/2*f^3/(b*f^2-2*d*e)^2*ln((-1/2*(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^
2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))/f^2)/(1/f^2*e^2
)^(1/2)+(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f
^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b
*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)
^(1/2))/(1/f^2*e^2)^(1/2)*b^2-f/(b*f^2-2*d*e)^2*ln((-1/2*(-b^2*f^4+2*a*e^2*f^2+2
*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))/f^2)/(
1/f^2*e^2)^(1/2)+(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+
2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*
f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-
2*d*e)^2)^(1/2))/(1/f^2*e^2)^(1/2)*a*e^2-f/(b*f^2-2*d*e)^2*ln((-1/2*(-b^2*f^4+2*
a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+e^2*(x+(a*f^2-d^2)/(b*f^2-2*d
*e))/f^2)/(1/f^2*e^2)^(1/2)+(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2
*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e)
)+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/
f^2/(b*f^2-2*d*e)^2)^(1/2))/(1/f^2*e^2)^(1/2)*b*d*e+1/f/(b*f^2-2*d*e)^2*ln((-1/2
*(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+e^2*(x+(a*f^2-d^
2)/(b*f^2-2*d*e))/f^2)/(1/f^2*e^2)^(1/2)+(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^
2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/
(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*
f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(1/f^2*e^2)^(1/2)*d^2*e^2-f^3/(b*f^2-2*
d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4
*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*
a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2
*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^
2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^
2-2*d*e)^2)^(1/2)*(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2
+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2
*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2
-2*d*e)^2)^(1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*a^2*e^2+2*f^3/(b*f^2-2*d*e)^3/(
(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^
2/(b*f^2-2*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^
2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f
^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*
a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)
^2)^(1/2)*(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e
*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a
*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^
2)^(1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*a*b*d*e-f^3/(b*f^2-2*d*e)^3/((a^2*e^2*f
^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2
*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*
d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e
^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*a*b*d*e*f^
4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*
(e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2
*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4
+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/
(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*b^2*d^2-2*f/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d
*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(
1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+
d^4*e^2)/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b
*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*
f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(e^2*(x+(a
*f^2-d^2)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/
(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f
^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(x+(a*f^2-
d^2)/(b*f^2-2*d*e)))*a*d^2*e^2+2*f/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b
^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*ln(
(2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)
/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d
*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*
d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(e^2*(x+(a*f^2-d^2
)/(b*f^2-2*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2
*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d
^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(x+(a*f^2-d^2)/(b*
f^2-2*d*e)))*b*d^3*e-1/f/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4
+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*ln((2*(a^2*e^
2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^
2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*
f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^
2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(e^2*(x+(a*f^2-d^2)/(b*f^2-2
*d*e))^2/f^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(
a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2
-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)
))*d^4*e^2-d*ln((b*f^2-2*d*e)*x+a*f^2-d^2)/(b*f^2-2*d*e)-e/(b*f^2-2*d*e)*x+e/(b*
f^2-2*d*e)^2*ln(b*f^2*x+a*f^2-2*d*e*x-d^2)*a*f^2-e/(b*f^2-2*d*e)^2*ln(b*f^2*x+a*
f^2-2*d*e*x-d^2)*d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d),x, algorithm="maxima")

[Out]

integrate(1/(e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d), x)

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Fricas [A]  time = 4.61315, size = 501, normalized size = 2.33 \[ -\frac{2 \,{\left (b e^{2} f^{2} - 2 \, d e^{3}\right )} x - 2 \,{\left (d^{2} e^{2} -{\left (b d e - a e^{2}\right )} f^{2}\right )} \log \left ({\left (b d - 2 \, a e\right )} f^{2} -{\left (b e f^{2} - 2 \, d e^{2}\right )} x +{\left (b f^{3} - 2 \, d e f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) - 2 \,{\left (d^{2} e^{2} -{\left (b d e - a e^{2}\right )} f^{2}\right )} \log \left (a f^{2} - d^{2} +{\left (b f^{2} - 2 \, d e\right )} x\right ) +{\left (b^{2} f^{4} + 2 \, d^{2} e^{2} - 2 \,{\left (b d e + a e^{2}\right )} f^{2}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 2 \,{\left (d^{2} e^{2} -{\left (b d e - a e^{2}\right )} f^{2}\right )} \log \left (-e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) - 2 \,{\left (b e f^{3} - 2 \, d e^{2} f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \,{\left (b^{2} e f^{4} - 4 \, b d e^{2} f^{2} + 4 \, d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d),x, algorithm="fricas")

[Out]

-1/2*(2*(b*e^2*f^2 - 2*d*e^3)*x - 2*(d^2*e^2 - (b*d*e - a*e^2)*f^2)*log((b*d - 2
*a*e)*f^2 - (b*e*f^2 - 2*d*e^2)*x + (b*f^3 - 2*d*e*f)*sqrt((b*f^2*x + e^2*x^2 +
a*f^2)/f^2)) - 2*(d^2*e^2 - (b*d*e - a*e^2)*f^2)*log(a*f^2 - d^2 + (b*f^2 - 2*d*
e)*x) + (b^2*f^4 + 2*d^2*e^2 - 2*(b*d*e + a*e^2)*f^2)*log(-b*f^2 - 2*e^2*x + 2*e
*f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2)) + 2*(d^2*e^2 - (b*d*e - a*e^2)*f^2)*lo
g(-e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) - d) - 2*(b*e*f^3 - 2*d*e^2*f)*
sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))/(b^2*e*f^4 - 4*b*d*e^2*f^2 + 4*d^2*e^3)

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

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)),x)

[Out]

Integral(1/(d + e*x + f*sqrt(a + b*x + e**2*x**2/f**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, +\infty , 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d),x, algorithm="giac")

[Out]

[undef, +Infinity, 1]