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

Optimal. Leaf size=330 \[ -\frac{2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}+\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^4}-\frac{2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^4}-\frac{a e f^2-b d f^2+d^2 e}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^2} \]

[Out]

-((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])^2)) - (2*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*(d + e*x + f*S
qrt[a + b*x + (e^2*x^2)/f^2])) - (2*e*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^
3*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) + (6*e*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)^4 -
(6*e*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)^4

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Rubi [A]  time = 0.70135, antiderivative size = 330, 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 \[ -\frac{2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}+\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^4}-\frac{2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^4}-\frac{a e f^2-b d f^2+d^2 e}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((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])^2)) - (2*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*(d + e*x + f*S
qrt[a + b*x + (e^2*x^2)/f^2])) - (2*e*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^
3*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) + (6*e*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)^4 -
(6*e*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)^4

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Rubi in Sympy [A]  time = 117.511, size = 306, normalized size = 0.93 \[ \frac{6 e f^{2} \left (4 a e^{2} - b^{2} f^{2}\right ) \log{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{\left (b f^{2} - 2 d e\right )^{4}} - \frac{6 e f^{2} \left (4 a e^{2} - b^{2} f^{2}\right ) \log{\left (b f + e \left (\frac{2 e x}{f} + 2 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right ) \right )}}{\left (b f^{2} - 2 d e\right )^{4}} + \frac{2 e f \left (4 a e^{2} - b^{2} f^{2}\right )}{\left (b f + e \left (\frac{2 e x}{f} + 2 \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right ) \left (b f^{2} - 2 d e\right )^{3}} + \frac{2 f^{2} \left (4 a e^{2} - b^{2} f^{2}\right )}{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right ) \left (b f^{2} - 2 d e\right )^{3}} - \frac{a e f^{2} - b d f^{2} + d^{2} e}{\left (d + f \left (\frac{e x}{f} + \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )\right )^{2} \left (b f^{2} - 2 d e\right )^{2}} \]

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))**3,x)

[Out]

6*e*f**2*(4*a*e**2 - b**2*f**2)*log(d + f*(e*x/f + sqrt(a + b*x + e**2*x**2/f**2
)))/(b*f**2 - 2*d*e)**4 - 6*e*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)))/(b*f**2 - 2*d*e)**4 + 2*e*f*(4*a*e**2 - b**2
*f**2)/((b*f + e*(2*e*x/f + 2*sqrt(a + b*x + e**2*x**2/f**2)))*(b*f**2 - 2*d*e)*
*3) + 2*f**2*(4*a*e**2 - b**2*f**2)/((d + f*(e*x/f + sqrt(a + b*x + e**2*x**2/f*
*2)))*(b*f**2 - 2*d*e)**3) - (a*e*f**2 - b*d*f**2 + d**2*e)/((d + f*(e*x/f + sqr
t(a + b*x + e**2*x**2/f**2)))**2*(b*f**2 - 2*d*e)**2)

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Mathematica [B]  time = 1.54139, size = 665, normalized size = 2.02 \[ -\frac{3 \left (4 a^2 e^3 f^4+a e f^2 \left (-b^2 f^4-4 b d e f^2+4 d^2 e^2\right )+b^2 d f^4 \left (b f^2-d e\right )\right )}{\left (b f^2-2 d e\right )^4 \left (-f^2 (a+b x)+d^2+2 d e x\right )}-\frac{2 f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )} \left (-2 e^2 \left (3 a^2 f^4-a d f^2 (5 d+9 e x)+d^2 e x (3 d+4 e x)\right )+b^2 \left (a f^6-e f^4 x (d+2 e x)\right )+b e f^2 \left (-a d f^2-9 a e f^2 x-3 d^3+d^2 e x+8 d e^2 x^2\right )+b^3 f^6 x\right )}{\left (b f^2-2 d e\right )^3 \left (-f^2 (a+b x)+d^2+2 d e x\right )^2}+\frac{3 \left (4 a e^3 f^2-b^2 e f^4\right ) \log \left (-f^2 (a+b x)+d^2+2 d e x\right )}{\left (b f^2-2 d e\right )^4}-\frac{3 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (-f^2 (a+b x)+d^2+2 d e x\right )}{\left (b f^2-2 d e\right )^4}+\frac{3 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2 \left (2 d f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+a f^2+d^2-2 d e x\right )+2 d^2 e \left (e x-f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}\right )-2 a e f^2 \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+2 d+e x\right )+b^2 f^4 x\right )}{\left (b f^2-2 d e\right )^4}-\frac{3 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )}{\left (b f^2-2 d e\right )^4}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )^3}{\left (b f^2-2 d e\right )^4 \left (-f^2 (a+b x)+d^2+2 d e x\right )^2}+\frac{4 e^3 x}{\left (2 d e-b f^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.205, size = 295147, normalized size = 894.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d+e*x+f*(a+b*x+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. \[ \int \frac{1}{{\left (e x + \sqrt{b x + \frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 10.5942, size = 2638, normalized size = 7.99 \[ \text{result too large to display} \]

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)^(-3),x, algorithm="fricas")

[Out]

((3*a*b^3*d - 4*a^2*b^2*e)*f^8 - (b^3*d^3 + 4*a*b^2*d^2*e + 10*a^2*b*d*e^2 - 20*
a^3*e^3)*f^6 - 4*(b^2*d^4*e - 8*a*b*d^3*e^2 + 6*a^2*d^2*e^3)*f^4 - 4*(b^3*e^3*f^
6 - 6*b^2*d*e^4*f^4 + 12*b*d^2*e^5*f^2 - 8*d^3*e^6)*x^3 + 2*(b*d^5*e^2 - 6*a*d^4
*e^3)*f^2 - (b^4*e*f^8 - 2*a*b^2*e^3*f^6 - 40*d^4*e^5 - 2*(11*b^2*d^2*e^3 - 4*a*
b*d*e^4)*f^4 + 8*(7*b*d^3*e^4 - a*d^2*e^5)*f^2)*x^2 + (16*d^5*e^4 + (3*b^4*d - 5
*a*b^3*e)*f^8 - (7*b^3*d^2*e + 10*a*b^2*d*e^2 - 28*a^2*b*e^3)*f^6 + 2*(5*b^2*d^3
*e^2 + 22*a*b*d^2*e^3 - 28*a^2*d*e^4)*f^4 - 8*(3*b*d^4*e^3 + a*d^3*e^4)*f^2)*x -
3*(a^2*b^2*e*f^8 - 4*a*d^4*e^3*f^2 - 2*(a*b^2*d^2*e + 2*a^3*e^3)*f^6 + (b^2*d^4
*e + 8*a^2*d^2*e^3)*f^4 + (b^4*e*f^8 - 16*a*d^2*e^5*f^2 - 4*(b^3*d*e^2 + a*b^2*e
^3)*f^6 + 4*(b^2*d^2*e^3 + 4*a*b*d*e^4)*f^4)*x^2 + 2*(a*b^3*e*f^8 - 8*a*d^3*e^4*
f^2 - (b^3*d^2*e + 2*a*b^2*d*e^2 + 4*a^2*b*e^3)*f^6 + 2*(b^2*d^3*e^2 + 2*a*b*d^2
*e^3 + 4*a^2*d*e^4)*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)) - 3*(a^2*b^2*e*f^8 - 4*a*d^4*e^3*f^2 - 2*(a*b^2*d^2*e + 2*a^3*e^3)*f
^6 + (b^2*d^4*e + 8*a^2*d^2*e^3)*f^4 + (b^4*e*f^8 - 16*a*d^2*e^5*f^2 - 4*(b^3*d*
e^2 + a*b^2*e^3)*f^6 + 4*(b^2*d^2*e^3 + 4*a*b*d*e^4)*f^4)*x^2 + 2*(a*b^3*e*f^8 -
8*a*d^3*e^4*f^2 - (b^3*d^2*e + 2*a*b^2*d*e^2 + 4*a^2*b*e^3)*f^6 + 2*(b^2*d^3*e^
2 + 2*a*b*d^2*e^3 + 4*a^2*d*e^4)*f^4)*x)*log(a*f^2 - d^2 + (b*f^2 - 2*d*e)*x) +
3*(a^2*b^2*e*f^8 - 4*a*d^4*e^3*f^2 - 2*(a*b^2*d^2*e + 2*a^3*e^3)*f^6 + (b^2*d^4*
e + 8*a^2*d^2*e^3)*f^4 + (b^4*e*f^8 - 16*a*d^2*e^5*f^2 - 4*(b^3*d*e^2 + a*b^2*e^
3)*f^6 + 4*(b^2*d^2*e^3 + 4*a*b*d*e^4)*f^4)*x^2 + 2*(a*b^3*e*f^8 - 8*a*d^3*e^4*f
^2 - (b^3*d^2*e + 2*a*b^2*d*e^2 + 4*a^2*b*e^3)*f^6 + 2*(b^2*d^3*e^2 + 2*a*b*d^2*
e^3 + 4*a^2*d*e^4)*f^4)*x)*log(-e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) -
d) - 2*(a*b^3*f^9 - 3*(a*b^2*d*e + 2*a^2*b*e^2)*f^7 - 3*(b^2*d^3*e - 4*a*b*d^2*e
^2 - 4*a^2*d*e^3)*f^5 + 2*(3*b*d^4*e^2 - 10*a*d^3*e^3)*f^3 - 2*(b^3*e^2*f^7 - 6*
b^2*d*e^3*f^5 + 12*b*d^2*e^4*f^3 - 8*d^3*e^5*f)*x^2 + (b^4*f^9 + 12*d^4*e^4*f -
3*(b^3*d*e + 3*a*b^2*e^2)*f^7 + 3*(b^2*d^2*e^2 + 12*a*b*d*e^3)*f^5 - 4*(2*b*d^3*
e^3 + 9*a*d^2*e^4)*f^3)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))/(a^2*b^4*f^12
+ 16*d^8*e^4 - 2*(a*b^4*d^2 + 4*a^2*b^3*d*e)*f^10 + (b^4*d^4 + 16*a*b^3*d^3*e +
24*a^2*b^2*d^2*e^2)*f^8 - 8*(b^3*d^5*e + 6*a*b^2*d^4*e^2 + 4*a^2*b*d^3*e^3)*f^6
+ 8*(3*b^2*d^6*e^2 + 8*a*b*d^5*e^3 + 2*a^2*d^4*e^4)*f^4 - 32*(b*d^7*e^3 + a*d^6*
e^4)*f^2 + (b^6*f^12 - 12*b^5*d*e*f^10 + 60*b^4*d^2*e^2*f^8 - 160*b^3*d^3*e^3*f^
6 + 240*b^2*d^4*e^4*f^4 - 192*b*d^5*e^5*f^2 + 64*d^6*e^6)*x^2 + 2*(a*b^5*f^12 +
32*d^7*e^5 - (b^5*d^2 + 10*a*b^4*d*e)*f^10 + 10*(b^4*d^3*e + 4*a*b^3*d^2*e^2)*f^
8 - 40*(b^3*d^4*e^2 + 2*a*b^2*d^3*e^3)*f^6 + 80*(b^2*d^5*e^3 + a*b*d^4*e^4)*f^4
- 16*(5*b*d^6*e^4 + 2*a*d^5*e^5)*f^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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))**3,x)

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out