3.444 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx\)

Optimal. Leaf size=286 \[ \frac{\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*d*x^3) - ((c/(a*e) - (5*e)/d^2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*
d^2 + 3*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*a^2*d^3*e^2*x) -
((c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*ArcTanh[(2*a*d*e + (c*d^
2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2])])/(16*a^(5/2)*d^(7/2)*e^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.18343, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*d*x^3) - ((c/(a*e) - (5*e)/d^2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*
d^2 + 3*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*a^2*d^3*e^2*x) -
((c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*ArcTanh[(2*a*d*e + (c*d^
2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2])])/(16*a^(5/2)*d^(7/2)*e^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 147.703, size = 270, normalized size = 0.94 \[ \frac{\left (\frac{5 e}{12 d^{2}} - \frac{c}{12 a e}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{x^{2}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 d x^{3}} - \frac{\left (3 a e^{2} + c d^{2}\right ) \left (5 a e^{2} - 3 c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{24 a^{2} d^{3} e^{2} x} + \frac{\left (a e^{2} - c d^{2}\right ) \left (5 a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{16 a^{\frac{5}{2}} d^{\frac{7}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**4/(e*x+d),x)

[Out]

(5*e/(12*d**2) - c/(12*a*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/x**2
- sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3*d*x**3) - (3*a*e**2 + c*d**
2)*(5*a*e**2 - 3*c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(24*a**2
*d**3*e**2*x) + (a*e**2 - c*d**2)*(5*a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)*at
anh((2*a*d*e + x*(a*e**2 + c*d**2))/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*
e*x**2 + x*(a*e**2 + c*d**2))))/(16*a**(5/2)*d**(7/2)*e**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.573724, size = 299, normalized size = 1.05 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^2 e^2 \left (-8 d^2+10 d e x-15 e^2 x^2\right )-2 a c d^2 e x (d-2 e x)+3 c^2 d^4 x^2\right )+3 x^3 \log (x) \left (-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right )-3 x^3 \left (-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )\right )}{48 a^{5/2} d^{7/2} e^{5/2} x^3 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sq
rt[d + e*x]*(3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(d - 2*e*x) + a^2*e^2*(-8*d^2 + 10*d*
e*x - 15*e^2*x^2)) + 3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x
^3*Log[x] - 3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x^3*Log[c*
d^2*x + 2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + a*e*(2*d + e
*x)]))/(48*a^(5/2)*d^(7/2)*e^(5/2)*x^3*Sqrt[(a*e + c*d*x)*(d + e*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.024, size = 1165, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x)

[Out]

3/4/d^3/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/8/a^2/e*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*c^2+1/4/d/a^2/e^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)*c-1/2/d^2/a^2/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-1/16*d^3/a^2/e
^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2))/x)*c^3+11/8/d^3/a*e^2*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x+1/8*d/a^3/e^2*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/2/d^2*e^3*
ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2))/(c*d*e)^(1/2)*c-1/2/d^4*e^5*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/
2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)*a-1/3/d^2/a/e/x^3*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-11/8/d^4/a*e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)-1/8/a^3/e^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2+9/8/d^2/a*e
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+1/8*d^2/a^3/e^3*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*c^3+5/16/d^3*a*e^4/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+
2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)-3/16/d*e^2/(a*d*e)^(
1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2))/x)*c-1/16*d/a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2
)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2+1/2/d/a^2*c^2*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*x+1/2/d^4*e^5*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*
d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)*a-1/2/d^
2*e^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e
^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)*c+1/d^4*e^3*(c*d*e*(x+d/e)^2+(a*e^2-c*d^
2)*(x+d/e))^(1/2)+3/8/d^4*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x^4),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x^4), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.811004, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{a d e}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} d^{2} e^{2} -{\left (3 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c d^{3} e - 5 \, a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{a d e}}{96 \, \sqrt{a d e} a^{2} d^{3} e^{2} x^{3}}, -\frac{3 \,{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} a d e}\right ) + 2 \,{\left (8 \, a^{2} d^{2} e^{2} -{\left (3 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c d^{3} e - 5 \, a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-a d e}}{48 \, \sqrt{-a d e} a^{2} d^{3} e^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x^4),x, algorithm="fricas")

[Out]

[-1/96*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x^3*log((4*(2*
a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2
)*x) + (8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 8*(a*c*d^3*e +
a^2*d*e^3)*x)*sqrt(a*d*e))/x^2) + 4*(8*a^2*d^2*e^2 - (3*c^2*d^4 + 4*a*c*d^2*e^2
- 15*a^2*e^4)*x^2 + 2*(a*c*d^3*e - 5*a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)*sqrt(a*d*e))/(sqrt(a*d*e)*a^2*d^3*e^2*x^3), -1/48*(3*(c^3*d^6 +
a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x^3*arctan(1/2*(2*a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(-a*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*a*d*e)) + 2*
(8*a^2*d^2*e^2 - (3*c^2*d^4 + 4*a*c*d^2*e^2 - 15*a^2*e^4)*x^2 + 2*(a*c*d^3*e - 5
*a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e))/(sqrt(-
a*d*e)*a^2*d^3*e^2*x^3)]

_______________________________________________________________________________________

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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**4/(e*x+d),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.1782, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x^4),x, algorithm="giac")

[Out]

Done