3.445 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=389 \[ \frac{\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*d*x^4) - ((c/(a*e) - (7*e)/d^2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*x^3) + ((5*c^2*d^4 + 6*a*c*d^2*
e^2 - 35*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(96*a^2*d^3*e^2*x
^2) - ((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*Sqrt[a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*a^3*d^4*e^3*x) + ((c*d^2 - a*e^2)*(5*c
^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 35*a^3*e^6)*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])])/(128*a^(7/2)*d^(9/2)*e^(7/2))

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Rubi [A]  time = 1.64302, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^5*(d + e*x)),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*d*x^4) - ((c/(a*e) - (7*e)/d^2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*x^3) + ((5*c^2*d^4 + 6*a*c*d^2*
e^2 - 35*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(96*a^2*d^3*e^2*x
^2) - ((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*Sqrt[a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*a^3*d^4*e^3*x) + ((c*d^2 - a*e^2)*(5*c
^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 35*a^3*e^6)*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])])/(128*a^(7/2)*d^(9/2)*e^(7/2))

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

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**5/(e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 0.826194, size = 378, normalized size = 0.97 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^3 e^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+a^2 c d^2 e^2 x \left (-8 d^2+12 d e x-25 e^2 x^2\right )+a c^2 d^4 e x^2 (10 d-17 e x)-15 c^3 d^6 x^3\right )-3 x^4 \log (x) \left (-35 a^4 e^8+20 a^3 c d^2 e^6+6 a^2 c^2 d^4 e^4+4 a c^3 d^6 e^2+5 c^4 d^8\right )+3 x^4 \left (-35 a^4 e^8+20 a^3 c d^2 e^6+6 a^2 c^2 d^4 e^4+4 a c^3 d^6 e^2+5 c^4 d^8\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )\right )}{384 a^{7/2} d^{9/2} e^{7/2} x^4 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^5*(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]*(-15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(10*d - 17*e*x) + a^2*c*d^2*e^2*x
*(-8*d^2 + 12*d*e*x - 25*e^2*x^2) + a^3*e^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2
+ 105*e^3*x^3)) - 3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c
*d^2*e^6 - 35*a^4*e^8)*x^4*Log[x] + 3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d
^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*x^4*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)]))/(384*a^(7/2)*d^(9/2)*e^
(7/2)*x^4*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.033, size = 1494, normalized size = 3.8 \[ \text{result too large to display} \]

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^5/(e*x+d),x)

[Out]

13/24/d^3/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-17/32/d/a^2*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+1/32*d^2*c^3/a^2/e/(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)+5/64*d/
a^4/e^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3+5/128*d^4/a^3/e^3/(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^4-93/64/d^4/a*e^3*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-43
/64/d^2/a^2*e*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-5/64*d^2/a^4/e^3*c^4
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-7/16/d^2/a^2/e/x^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*c+5/24/d/a^2/e^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*c+19/64/d/a^3/e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2-7/32*d/a^3/e^
2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3-5/64*d^3/a^4/e^4*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*c^4-5/32/a^3/e^3/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)*c^2-1/2/d^5*e^6*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^3*e^4*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-35/128/d^4*a*e^5/(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)-1/4/d^2/a/e/x^4*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/64/a*e/(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^2-19/64/
a^3/e*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/32/d^2*c*e^3/(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)-39/32/d^3/a*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+43/64/d^3/a^2
/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-29/32/d^4/a*e/x^2*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)+93/64/d^5/a*e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
-1/2/d^3*e^4*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^5*e^6*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/d^5*e
^4*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-29/64/d^5*e^4*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{5}}\,{d x} \]

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^5),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^5), x)

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Fricas [A]  time = 1.99548, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} -{\left (8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{a d e}}{x^{2}}\right ) + 4 \,{\left (48 \, a^{3} d^{3} e^{3} +{\left (15 \, c^{3} d^{6} + 17 \, a c^{2} d^{4} e^{2} + 25 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6}\right )} x^{3} - 2 \,{\left (5 \, a c^{2} d^{5} e + 6 \, a^{2} c d^{3} e^{3} - 35 \, a^{3} d e^{5}\right )} x^{2} + 8 \,{\left (a^{2} c d^{4} e^{2} - 7 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{a d e}}{768 \, \sqrt{a d e} a^{3} d^{4} e^{3} x^{4}}, \frac{3 \,{\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} a d e}\right ) - 2 \,{\left (48 \, a^{3} d^{3} e^{3} +{\left (15 \, c^{3} d^{6} + 17 \, a c^{2} d^{4} e^{2} + 25 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6}\right )} x^{3} - 2 \,{\left (5 \, a c^{2} d^{5} e + 6 \, a^{2} c d^{3} e^{3} - 35 \, a^{3} d e^{5}\right )} x^{2} + 8 \,{\left (a^{2} c d^{4} e^{2} - 7 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-a d e}}{384 \, \sqrt{-a d e} a^{3} d^{4} e^{3} x^{4}}\right ] \]

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^5),x, algorithm="fricas")

[Out]

[-1/768*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 -
35*a^4*e^8)*x^4*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*(48*a^3*d^3*e^3
+ (15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*x^3 - 2*(5*a
*c^2*d^5*e + 6*a^2*c*d^3*e^3 - 35*a^3*d*e^5)*x^2 + 8*(a^2*c*d^4*e^2 - 7*a^3*d^2*
e^4)*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^
3*d^4*e^3*x^4), 1/384*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a
^3*c*d^2*e^6 - 35*a^4*e^8)*x^4*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*(48*a^3*d^3*e^3 +
(15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*x^3 - 2*(5*a*c^
2*d^5*e + 6*a^2*c*d^3*e^3 - 35*a^3*d*e^5)*x^2 + 8*(a^2*c*d^4*e^2 - 7*a^3*d^2*e^4
)*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^3
*d^4*e^3*x^4)]

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**5/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 3.42025, 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^5),x, algorithm="giac")

[Out]

Done