3.1942 \(\int \frac{1}{(d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*(c*d^2 - a*e^2)*(d + e*x)^4)
+ (12*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d^2 - a*e^2)^2*(d
+ e*x)^3) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d^2
- a*e^2)^3*(d + e*x)^2) + (32*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])/(35*(c*d^2 - a*e^2)^4*(d + e*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.402697, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*(c*d^2 - a*e^2)*(d + e*x)^4)
+ (12*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d^2 - a*e^2)^2*(d
+ e*x)^3) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d^2
- a*e^2)^3*(d + e*x)^2) + (32*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])/(35*(c*d^2 - a*e^2)^4*(d + e*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 80.4371, size = 216, normalized size = 0.94 \[ \frac{32 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{4}} - \frac{16 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{12 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{7 \left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32*c**3*d**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(35*(d + e*x)*(a*e**
2 - c*d**2)**4) - 16*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3
5*(d + e*x)**2*(a*e**2 - c*d**2)**3) + 12*c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**
2 + c*d**2))/(35*(d + e*x)**3*(a*e**2 - c*d**2)**2) - 2*sqrt(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2))/(7*(d + e*x)**4*(a*e**2 - c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.199917, size = 138, normalized size = 0.6 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-5 a^3 e^6+3 a^2 c d e^4 (7 d+2 e x)-a c^2 d^2 e^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )+c^3 d^3 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )\right )}{35 (d+e x)^4 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-5*a^3*e^6 + 3*a^2*c*d*e^4*(7*d + 2*e*x) - a*c
^2*d^2*e^2*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + c^3*d^3*(35*d^3 + 70*d^2*e*x + 56*d
*e^2*x^2 + 16*e^3*x^3)))/(35*(c*d^2 - a*e^2)^4*(d + e*x)^4)

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 217, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+8\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-56\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-6\,x{a}^{2}cd{e}^{5}+28\,xa{c}^{2}{d}^{3}{e}^{3}-70\,{c}^{3}{d}^{5}ex+5\,{a}^{3}{e}^{6}-21\,{a}^{2}c{d}^{2}{e}^{4}+35\,{c}^{2}{d}^{4}a{e}^{2}-35\,{c}^{3}{d}^{6} \right ) }{35\, \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \right ) \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+8*a*c^2*d^2*e^4*x^2-56*c^3*d^4*e^2*x^2-6*
a^2*c*d*e^5*x+28*a*c^2*d^3*e^3*x-70*c^3*d^5*e*x+5*a^3*e^6-21*a^2*c*d^2*e^4+35*a*
c^2*d^4*e^2-35*c^3*d^6)/(e*x+d)^3/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a
*c^3*d^6*e^2+c^4*d^8)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 1.25634, size = 610, normalized size = 2.64 \[ \frac{2 \,{\left (16 \, c^{3} d^{3} e^{3} x^{3} + 35 \, c^{3} d^{6} - 35 \, a c^{2} d^{4} e^{2} + 21 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} + 8 \,{\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (35 \, c^{3} d^{5} e - 14 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{35 \,{\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} +{\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \,{\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \,{\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \,{\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4),x, algorithm="fricas")

[Out]

2/35*(16*c^3*d^3*e^3*x^3 + 35*c^3*d^6 - 35*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 - 5*
a^3*e^6 + 8*(7*c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 2*(35*c^3*d^5*e - 14*a*c^2*d^3
*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^4*d^12 -
 4*a*c^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 - 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8
*e^4 - 4*a*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 - 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 +
4*(c^4*d^9*e^3 - 4*a*c^3*d^7*e^5 + 6*a^2*c^2*d^5*e^7 - 4*a^3*c*d^3*e^9 + a^4*d*e
^11)*x^3 + 6*(c^4*d^10*e^2 - 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 - 4*a^3*c*d^4*e
^8 + a^4*d^2*e^10)*x^2 + 4*(c^4*d^11*e - 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 - 4
*a^3*c*d^5*e^7 + a^4*d^3*e^9)*x)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.316328, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4),x, algorithm="giac")

[Out]

Done