3.298 \(\int \frac{1}{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^3} \, dx\)

Optimal. Leaf size=193 \[ -\frac{3 a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 d^4 e}+\frac{3 a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}{2 d^4 e}-\frac{a f^2}{2 d^3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{a f^2}{d^3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}-\frac{\frac{a f^2}{d^2}+1}{4 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^2} \]

[Out]

-(a*f^2)/(2*d^3*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(4*e*(d
+ e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2) - (a*f^2)/(d^3*e*(d + e*x + f*Sqrt[a + (e
^2*x^2)/f^2])) - (3*a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*d^4*e) + (3*a
*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*d^4*e)

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Rubi [A]  time = 0.287864, antiderivative size = 193, 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 \[ -\frac{3 a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 d^4 e}+\frac{3 a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}{2 d^4 e}-\frac{a f^2}{2 d^3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}-\frac{a f^2}{d^3 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )}-\frac{\frac{a f^2}{d^2}+1}{4 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-3),x]

[Out]

-(a*f^2)/(2*d^3*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(4*e*(d
+ e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2) - (a*f^2)/(d^3*e*(d + e*x + f*Sqrt[a + (e
^2*x^2)/f^2])) - (3*a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*d^4*e) + (3*a
*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*d^4*e)

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Rubi in Sympy [A]  time = 46.1686, size = 178, normalized size = 0.92 \[ - \frac{a f^{2}}{d^{3} e \left (d + f \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )\right )} - \frac{a f}{2 d^{3} e \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )} + \frac{3 a f^{2} \log{\left (d + f \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{2 d^{4} e} - \frac{3 a f^{2} \log{\left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}} \right )}}{2 d^{4} e} - \frac{a f^{2} + d^{2}}{4 d^{2} e \left (d + f \left (\frac{e x}{f} + \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )\right )^{2}} \]

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))**3,x)

[Out]

-a*f**2/(d**3*e*(d + f*(e*x/f + sqrt(a + e**2*x**2/f**2)))) - a*f/(2*d**3*e*(e*x
/f + sqrt(a + e**2*x**2/f**2))) + 3*a*f**2*log(d + f*(e*x/f + sqrt(a + e**2*x**2
/f**2)))/(2*d**4*e) - 3*a*f**2*log(e*x/f + sqrt(a + e**2*x**2/f**2))/(2*d**4*e)
- (a*f**2 + d**2)/(4*d**2*e*(d + f*(e*x/f + sqrt(a + e**2*x**2/f**2)))**2)

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Mathematica [A]  time = 1.15352, size = 309, normalized size = 1.6 \[ -\frac{\frac{4 d f \sqrt{a+\frac{e^2 x^2}{f^2}} \left (3 a^2 f^4-a d f^2 (5 d+9 e x)+d^2 e x (3 d+4 e x)\right )}{e \left (-a f^2+d^2+2 d e x\right )^2}-\frac{6 a f^2 \log \left (d^2 \left (e x-f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )-a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+2 d+e x\right )\right )}{e}+\frac{\left (a f^2+d^2\right )^3}{e \left (-a f^2+d^2+2 d e x\right )^2}+\frac{6 a f^2 \left (a f^2+d^2\right )}{e \left (-a f^2+d^2+2 d e x\right )}+\frac{6 a f^2 \log \left (a f^2-d^2-2 d e x\right )}{e}-\frac{6 a f^2 \log \left (-a f^2+d^2+2 d e x\right )}{e}+\frac{6 a f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}-4 d x}{8 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-3),x]

[Out]

-(-4*d*x + (d^2 + a*f^2)^3/(e*(d^2 - a*f^2 + 2*d*e*x)^2) + (6*a*f^2*(d^2 + a*f^2
))/(e*(d^2 - a*f^2 + 2*d*e*x)) + (4*d*f*Sqrt[a + (e^2*x^2)/f^2]*(3*a^2*f^4 + d^2
*e*x*(3*d + 4*e*x) - a*d*f^2*(5*d + 9*e*x)))/(e*(d^2 - a*f^2 + 2*d*e*x)^2) + (6*
a*f^2*Log[-d^2 + a*f^2 - 2*d*e*x])/e - (6*a*f^2*Log[d^2 - a*f^2 + 2*d*e*x])/e +
(6*a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/e - (6*a*f^2*Log[d^2*(e*x - f*Sqr
t[a + (e^2*x^2)/f^2]) - a*f^2*(2*d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])])/e)/(8*d^
4)

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Maple [B]  time = 0.064, size = 9721, normalized size = 50.4 \[ \text{output too large to display} \]

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))^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-3),x, algorithm="maxima")

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-3), x)

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Fricas [A]  time = 0.365868, size = 724, normalized size = 3.75 \[ \frac{5 \, a^{3} f^{6} + 8 \, d^{3} e^{3} x^{3} - 6 \, a^{2} d^{2} f^{4} - 3 \, a d^{4} f^{2} + 2 \,{\left (a d^{2} e^{2} f^{2} + 5 \, d^{4} e^{2}\right )} x^{2} - 2 \,{\left (7 \, a^{2} d e f^{4} + a d^{3} e f^{2} - 2 \, d^{5} e\right )} x + 3 \,{\left (a^{3} f^{6} + 4 \, a d^{2} e^{2} f^{2} x^{2} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} - 4 \,{\left (a^{2} d e f^{4} - a d^{3} e f^{2}\right )} x\right )} \log \left (-a e f^{2} x + 2 \, d e^{2} x^{2} + a d f^{2} +{\left (a f^{3} - 2 \, d e f x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \,{\left (a^{3} f^{6} + 4 \, a d^{2} e^{2} f^{2} x^{2} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} - 4 \,{\left (a^{2} d e f^{4} - a d^{3} e f^{2}\right )} x\right )} \log \left (-a f^{2} + 2 \, d e x + d^{2}\right ) - 3 \,{\left (a^{3} f^{6} + 4 \, a d^{2} e^{2} f^{2} x^{2} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} - 4 \,{\left (a^{2} d e f^{4} - a d^{3} e f^{2}\right )} x\right )} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) - 2 \,{\left (3 \, a^{2} d f^{5} + 4 \, d^{3} e^{2} f x^{2} - 5 \, a d^{3} f^{3} - 3 \,{\left (3 \, a d^{2} e f^{3} - d^{4} e f\right )} x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}{4 \,{\left (a^{2} d^{4} e f^{4} + 4 \, d^{6} e^{3} x^{2} - 2 \, a d^{6} e f^{2} + d^{8} e - 4 \,{\left (a d^{5} e^{2} f^{2} - d^{7} e^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(-3),x, algorithm="fricas")

[Out]

1/4*(5*a^3*f^6 + 8*d^3*e^3*x^3 - 6*a^2*d^2*f^4 - 3*a*d^4*f^2 + 2*(a*d^2*e^2*f^2
+ 5*d^4*e^2)*x^2 - 2*(7*a^2*d*e*f^4 + a*d^3*e*f^2 - 2*d^5*e)*x + 3*(a^3*f^6 + 4*
a*d^2*e^2*f^2*x^2 - 2*a^2*d^2*f^4 + a*d^4*f^2 - 4*(a^2*d*e*f^4 - a*d^3*e*f^2)*x)
*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)) + 3*(a^3*f^6 + 4*a*d^2*e^2*f^2*x^2 - 2*a^2*d^2*f^4 + a*d^4*f^2 - 4*(a
^2*d*e*f^4 - a*d^3*e*f^2)*x)*log(-a*f^2 + 2*d*e*x + d^2) - 3*(a^3*f^6 + 4*a*d^2*
e^2*f^2*x^2 - 2*a^2*d^2*f^4 + a*d^4*f^2 - 4*(a^2*d*e*f^4 - a*d^3*e*f^2)*x)*log(-
e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) - d) - 2*(3*a^2*d*f^5 + 4*d^3*e^2*f*x^2 - 5*
a*d^3*f^3 - 3*(3*a*d^2*e*f^3 - d^4*e*f)*x)*sqrt((e^2*x^2 + a*f^2)/f^2))/(a^2*d^4
*e*f^4 + 4*d^6*e^3*x^2 - 2*a*d^6*e*f^2 + d^8*e - 4*(a*d^5*e^2*f^2 - d^7*e^2)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{3}}\, dx \]

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))**3,x)

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-3), 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)^(-3),x, algorithm="giac")

[Out]

Timed out