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

Optimal. Leaf size=117 \[ -\frac{a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 d^2 e}+\frac{\left (\frac{a f^2}{d^2}+1\right ) \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}{2 e}-\frac{a f^2}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )} \]

[Out]

-(a*f^2)/(2*d*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (a*f^2*Log[e*x + f*Sqrt[a +
(e^2*x^2)/f^2]])/(2*d^2*e) + ((1 + (a*f^2)/d^2)*Log[d + e*x + f*Sqrt[a + (e^2*x
^2)/f^2]])/(2*e)

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Rubi [A]  time = 0.203482, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 d^2 e}+\frac{\left (\frac{a f^2}{d^2}+1\right ) \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}{2 e}-\frac{a f^2}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a*f^2)/(2*d*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (a*f^2*Log[e*x + f*Sqrt[a +
(e^2*x^2)/f^2]])/(2*d^2*e) + ((1 + (a*f^2)/d^2)*Log[d + e*x + f*Sqrt[a + (e^2*x
^2)/f^2]])/(2*e)

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Rubi in Sympy [A]  time = 22.418, size = 102, normalized size = 0.87 \[ - \frac{a f}{2 d e \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )} - \frac{a f^{2} \log{\left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}} \right )}}{2 d^{2} e} + \frac{\left (a f^{2} + d^{2}\right ) \log{\left (d + f \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{2 d^{2} e} \]

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)),x)

[Out]

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

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Mathematica [A]  time = 0.224731, size = 141, normalized size = 1.21 \[ \frac{\left (d^2-a f^2\right ) \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )+\left (a f^2+d^2\right ) \log \left (d^2 \left (e x-f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )-a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+2 d+e x\right )\right )+2 d \left (e x-f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{4 d^2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.043, size = 1325, normalized size = 11.3 \[ \text{result too large to display} \]

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)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.302596, size = 252, normalized size = 2.15 \[ \frac{2 \, d e x - 2 \, d f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} +{\left (a f^{2} + d^{2}\right )} \log \left (a f^{2} - d e x + d f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) +{\left (a f^{2} + d^{2}\right )} \log \left (-a f^{2} + 2 \, d e x + d^{2}\right ) -{\left (a f^{2} + d^{2}\right )} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) +{\left (a f^{2} - d^{2}\right )} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right )}{4 \, d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(d + e*x + f*sqrt(a + 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(e^2*x^2/f^2 + a)*f + d),x, algorithm="giac")

[Out]

[undef, +Infinity, 1]