3.297 \(\int \frac{1}{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^2} \, dx\)
Optimal. Leaf size=151 \[ -\frac{a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{d^3 e}+\frac{a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}{d^3 e}-\frac{a f^2}{2 d^2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{\frac{a f^2}{d^2}+1}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )} \]
[Out]
-(a*f^2)/(2*d^2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(2*e*(d
+ 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
]])/(d^3*e) + (a*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(d^3*e)
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Rubi [A] time = 0.255772, antiderivative size = 151, 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
\[ -\frac{a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{d^3 e}+\frac{a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}{d^3 e}-\frac{a f^2}{2 d^2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{\frac{a f^2}{d^2}+1}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-2),x]
[Out]
-(a*f^2)/(2*d^2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(2*e*(d
+ 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
]])/(d^3*e) + (a*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(d^3*e)
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Rubi in Sympy [A] time = 39.648, size = 136, normalized size = 0.9 \[ - \frac{a f}{2 d^{2} e \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )} + \frac{a f^{2} \log{\left (d + f \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{d^{3} e} - \frac{a f^{2} \log{\left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}} \right )}}{d^{3} e} - \frac{a f^{2} + d^{2}}{2 d^{2} e \left (d + f \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d+e*x+f*(a+e**2*x**2/f**2)**(1/2))**2,x)
[Out]
-a*f/(2*d**2*e*(e*x/f + sqrt(a + e**2*x**2/f**2))) + a*f**2*log(d + f*(e*x/f + s
qrt(a + e**2*x**2/f**2)))/(d**3*e) - a*f**2*log(e*x/f + sqrt(a + e**2*x**2/f**2)
)/(d**3*e) - (a*f**2 + d**2)/(2*d**2*e*(d + f*(e*x/f + sqrt(a + e**2*x**2/f**2))
))
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Mathematica [A] time = 0.654703, size = 248, normalized size = 1.64 \[ \frac{\frac{4 d f \sqrt{a+\frac{e^2 x^2}{f^2}} \left (a f^2-d e x\right )}{e \left (-a f^2+d^2+2 d e x\right )}+\frac{2 a f^2 \log \left (d^2 \left (e x-f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )-a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+2 d+e x\right )\right )}{e}-\frac{\left (a f^2+d^2\right )^2}{e \left (-a f^2+d^2+2 d e x\right )}-\frac{2 a f^2 \log \left (a f^2-d^2-2 d e x\right )}{e}+\frac{2 a f^2 \log \left (-a f^2+d^2+2 d e x\right )}{e}-\frac{2 a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+2 d x}{4 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-2),x]
[Out]
(2*d*x - (d^2 + a*f^2)^2/(e*(d^2 - a*f^2 + 2*d*e*x)) + (4*d*f*(a*f^2 - d*e*x)*Sq
rt[a + (e^2*x^2)/f^2])/(e*(d^2 - a*f^2 + 2*d*e*x)) - (2*a*f^2*Log[-d^2 + a*f^2 -
2*d*e*x])/e + (2*a*f^2*Log[d^2 - a*f^2 + 2*d*e*x])/e - (2*a*f^2*Log[e*x + f*Sqr
t[a + (e^2*x^2)/f^2]])/e + (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])])/e)/(4*d^3)
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Maple [B] time = 0.042, size = 4136, normalized size = 27.4 \[ \text{output too large to display} \]
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))^2,x)
[Out]
1/2/d^2*f^5/e/(a^2*f^4+2*a*d^2*f^2+d^4)/((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^3-1/2*d^2*f/e/(a^2*f^4+2*a*d^2
*f^2+d^4)/((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+1/2/d^2*x-1/4/e*f/d^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)+1/4/f/d*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/e/f/((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))-1/4*f/d^3*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-d*f/(a^2*f^4+2*a*d^2*f^2+d^4)*(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)*x-1/2*a*f^2/(-a*f^2+2*d*e*x+d^2)/d/e-1/4/e/d^3/(-a*f^2+2*d*e*x+d^2)*a^
2*f^4+1/4*d^2*f/e/(a^2*f^4+2*a*d^2*f^2+d^4)*(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)+1/2/e/d^3*ln(-a*f^2+2*d*e*x+d^2)*a*f^2-1/4*d^3/f/(a^2*f^4+2*a*d^2*f^2+
d^4)*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/e^2*f^5/
d/(a^2*f^4+2*a*d^2*f^2+d^4)/(x-1/2/d/e*a*f^2+1/2*d/e)*(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)^(3/2)*a+1/4/e*f^7/d^4/(a^2*f^4+2*a*d^2*f^2+d^4)/((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^4+d*f^3/
e^2/(a^2*f^4+2*a*d^2*f^2+d^4)/(x-1/2/d/e*a*f^2+1/2*d/e)*(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)^(3/2)-3/4/d*f^3/(a^2*f^4+2*a*d^2*f^2+d^4)*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^2-3/4*d*f/(a^2*f^4+2*a*d^2*f^2+d^4)
*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*d^4/f/e/
(a^2*f^4+2*a*d^2*f^2+d^4)/((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))-f^3/d/(a^2*f^4+2*a*d^2*f^2+d^4)*(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)*x*a+1/4/e*f^3/d^4/((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/e*f/d^2/((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/e*f^5/d^2/(a^2*f^4+2*a*d^2*f^2+d^4)*(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)*a^2-1/4*f^5/d^3/(a^2*f^4+2*a*d^2*f^2+d^4)*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^3-1/4*d/(-a*f^2+2*d*e*x+d^2)/e
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-2),x, algorithm="maxima")
[Out]
integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-2), x)
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Fricas [A] time = 0.30386, size = 383, normalized size = 2.54 \[ \frac{a^{2} f^{4} - 2 \, d^{2} e^{2} x^{2} + a d^{2} f^{2} - 2 \, d^{3} e x +{\left (a^{2} f^{4} - 2 \, a d e f^{2} x - a d^{2} f^{2}\right )} \log \left (-a e f^{2} x + 2 \, d e^{2} x^{2} + a d f^{2} +{\left (a f^{3} - 2 \, d e f x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) +{\left (a^{2} f^{4} - 2 \, a d e f^{2} x - a d^{2} f^{2}\right )} \log \left (-a f^{2} + 2 \, d e x + d^{2}\right ) -{\left (a^{2} f^{4} - 2 \, a d e f^{2} x - a d^{2} f^{2}\right )} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) - 2 \,{\left (a d f^{3} - d^{2} e f x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \,{\left (a d^{3} e f^{2} - 2 \, d^{4} e^{2} x - d^{5} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-2),x, algorithm="fricas")
[Out]
1/2*(a^2*f^4 - 2*d^2*e^2*x^2 + a*d^2*f^2 - 2*d^3*e*x + (a^2*f^4 - 2*a*d*e*f^2*x
- a*d^2*f^2)*log(-a*e*f^2*x + 2*d*e^2*x^2 + a*d*f^2 + (a*f^3 - 2*d*e*f*x)*sqrt((
e^2*x^2 + a*f^2)/f^2)) + (a^2*f^4 - 2*a*d*e*f^2*x - a*d^2*f^2)*log(-a*f^2 + 2*d*
e*x + d^2) - (a^2*f^4 - 2*a*d*e*f^2*x - a*d^2*f^2)*log(-e*x + f*sqrt((e^2*x^2 +
a*f^2)/f^2) - d) - 2*(a*d*f^3 - d^2*e*f*x)*sqrt((e^2*x^2 + a*f^2)/f^2))/(a*d^3*e
*f^2 - 2*d^4*e^2*x - d^5*e)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d+e*x+f*(a+e**2*x**2/f**2)**(1/2))**2,x)
[Out]
Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-2), x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-2),x, algorithm="giac")
[Out]
Timed out