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

Optimal. Leaf size=352 \[ -\frac{1}{2} a^{3/2} \sqrt{d} e^{3/2} \left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )+\frac{\left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e}-\frac{\left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{c} \sqrt{d} e^{3/2}}-\frac{(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x} \]

[Out]

((c^2*d^4 + 28*a*c*d^2*e^2 + 19*a^2*e^4 + 2*c*d*e*(c*d^2 + 7*a*e^2)*x)*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e) - ((3*a*e - c*d*x)*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(3*x) - ((c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*
e^4 - 5*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*Sqrt[c]*Sqrt[d]*e^(3/2)) - (a
^(3/2)*Sqrt[d]*e^(3/2)*(5*c*d^2 + 3*a*e^2)*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])])/2

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Rubi [A]  time = 1.22396, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175 \[ -\frac{1}{2} a^{3/2} \sqrt{d} e^{3/2} \left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )+\frac{\left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e}-\frac{\left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{c} \sqrt{d} e^{3/2}}-\frac{(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x} \]

Antiderivative was successfully verified.

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

[Out]

((c^2*d^4 + 28*a*c*d^2*e^2 + 19*a^2*e^4 + 2*c*d*e*(c*d^2 + 7*a*e^2)*x)*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e) - ((3*a*e - c*d*x)*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(3*x) - ((c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*
e^4 - 5*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*Sqrt[c]*Sqrt[d]*e^(3/2)) - (a
^(3/2)*Sqrt[d]*e^(3/2)*(5*c*d^2 + 3*a*e^2)*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])])/2

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Rubi in Sympy [A]  time = 144.324, size = 345, normalized size = 0.98 \[ - \frac{a^{\frac{3}{2}} \sqrt{d} e^{\frac{3}{2}} \left (3 a e^{2} + 5 c d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2} - \frac{\left (3 a e - c d x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 x} + \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (\frac{19 a^{2} e^{4}}{2} + 14 a c d^{2} e^{2} + \frac{c^{2} d^{4}}{2} + c d e x \left (7 a e^{2} + c d^{2}\right )\right )}{4 e} + \frac{\left (5 a^{3} e^{6} + 45 a^{2} c d^{2} e^{4} + 15 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{16 \sqrt{c} \sqrt{d} e^{\frac{3}{2}}} \]

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)**(5/2)/x**2/(e*x+d),x)

[Out]

-a**(3/2)*sqrt(d)*e**(3/2)*(3*a*e**2 + 5*c*d**2)*atanh((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))
))/2 - (3*a*e - c*d*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*x) +
 sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(19*a**2*e**4/2 + 14*a*c*d**2*e*
*2 + c**2*d**4/2 + c*d*e*x*(7*a*e**2 + c*d**2))/(4*e) + (5*a**3*e**6 + 45*a**2*c
*d**2*e**4 + 15*a*c**2*d**4*e**2 - c**3*d**6)*atanh((a*e**2 + c*d**2 + 2*c*d*e*x
)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(1
6*sqrt(c)*sqrt(d)*e**(3/2))

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Mathematica [A]  time = 1.17545, size = 376, normalized size = 1.07 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (24 a^{3/2} \sqrt{c} d e^3 x \log (x) \left (3 a e^2+5 c d^2\right )-24 a^{3/2} \sqrt{c} d e^3 x \left (3 a e^2+5 c d^2\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )+2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (3 a^2 e^3 (11 e x-8 d)+2 a c d e^2 x (34 d+13 e x)+c^2 d^2 x \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )-3 x \left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )\right )}{48 \sqrt{c} \sqrt{d} e^{3/2} x (d+e x)^{3/2} (a e+c d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sq
rt[d + e*x]*(3*a^2*e^3*(-8*d + 11*e*x) + 2*a*c*d*e^2*x*(34*d + 13*e*x) + c^2*d^2
*x*(3*d^2 + 14*d*e*x + 8*e^2*x^2)) + 24*a^(3/2)*Sqrt[c]*d*e^3*(5*c*d^2 + 3*a*e^2
)*x*Log[x] - 24*a^(3/2)*Sqrt[c]*d*e^3*(5*c*d^2 + 3*a*e^2)*x*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)] - 3*(c^3*
d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*x*Log[a*e^2 + 2*Sqrt[c]*S
qrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)]))/(48*Sqrt[c]*
Sqrt[d]*e^(3/2)*x*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))

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Maple [B]  time = 0.025, size = 2364, normalized size = 6.7 \[ \text{result too large to display} \]

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)^(5/2)/x^2/(e*x+d),x)

[Out]

-15/128*e^3*d^2*a^2*c*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)+15/256*e*d^4*a*c^2*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)-15/256*e^7/d^2*a^4/c*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)+225/256*d^4*e*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*c^2+121/64*d*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*a*c*e^2+15/256/d^2/c*e^7*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^4+375/128*d^2*
c*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)*a^2-5/2*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-3/256*e^
9/d^4*a^5/c^2*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)+3/64*e^6/d^3*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x-3/64*e^6/d^3*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2
)*x-9/64*e^2*d*a*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/256*e^9/d^4
*a^5/c^2*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)-1/8*e*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2
)*(x+d/e))^(3/2)*x+3/128/e*d^4*c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)
+15/128*e^5*a^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)+3/64*d^3*c^2*(c*d*e*(x+d/e)^2+
(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+13/64*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*x*c^2+13/128*d^4/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+1/a/e*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+9/8*e*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*
x+25/128*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^3+1/d*a*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)+1/5*e/d^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-1/16*d*c*(c*d*e
*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+19/8*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)*a^2+4/5/d^2*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+67/48*d*c*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/d^2/a/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
7/2)-13/256*d^6*c^3/e*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)-3/64/d^2/c*e^5*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*a^3+227/64*d^2*c*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-
3/2*d*a^3*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)-1/8*e^3/d^2*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*x-1/16*e^4/d^3*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/128*e^7
/d^4*a^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/64*e^5/d^2*a^3/c*(c*d*e*(
x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/8*e^3/d^2*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^
2)*(x+d/e))^(3/2)*x+1/16*e^4/d^3*a^2/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(
3/2)+9/64*e^4/d*a^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/128*e^7/d^
4*a^4/c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-3/64*e*d^2*a*c*(c*d*e*(x
+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-3/256/e*d^6*c^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)-9/64/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2*e^4+1/d*c/a*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x

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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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 18.817, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^2),x, algorithm="fricas")

[Out]

[1/96*(24*(5*a*c*d^2*e^2 + 3*a^2*e^4)*sqrt(a*d*e)*sqrt(c*d*e)*x*log((8*a^2*d^2*e
^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^
3)*x)/x^2) - 3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*x*log
(4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a
*e^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e
 + a*c*d*e^3)*x)*sqrt(c*d*e)) + 4*(8*c^2*d^2*e^2*x^3 - 24*a^2*d*e^3 + 2*(7*c^2*d
^3*e + 13*a*c*d*e^3)*x^2 + (3*c^2*d^4 + 68*a*c*d^2*e^2 + 33*a^2*e^4)*x)*sqrt(c*d
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e))/(sqrt(c*d*e)*e*x), 1/48*(12*(5*
a*c*d^2*e^2 + 3*a^2*e^4)*sqrt(a*d*e)*sqrt(-c*d*e)*x*log((8*a^2*d^2*e^2 + (c^2*d^
4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) -
 3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*x*arctan(1/2*(2*c
*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*c*d*e)) + 2*(8*c^2*d^2*e^2*x^3 - 24*a^2*d*e^3 + 2*(7*c^2*d^3*e + 13*a*c*d*e^3)
*x^2 + (3*c^2*d^4 + 68*a*c*d^2*e^2 + 33*a^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)*sqrt(-c*d*e))/(sqrt(-c*d*e)*e*x), -1/96*(48*(5*a*c*d^2*e^2 + 3*a
^2*e^4)*sqrt(-a*d*e)*sqrt(c*d*e)*x*arctan(1/2*(2*a*d*e + (c*d^2 + a*e^2)*x)/(sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e))) + 3*(c^3*d^6 - 15*a*c^2*
d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*x*log(4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a
*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2
*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d*e)) - 4*(
8*c^2*d^2*e^2*x^3 - 24*a^2*d*e^3 + 2*(7*c^2*d^3*e + 13*a*c*d*e^3)*x^2 + (3*c^2*d
^4 + 68*a*c*d^2*e^2 + 33*a^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*sqrt(c*d*e))/(sqrt(c*d*e)*e*x), -1/48*(24*(5*a*c*d^2*e^2 + 3*a^2*e^4)*sqrt(-a*d
*e)*sqrt(-c*d*e)*x*arctan(1/2*(2*a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e))) + 3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2
*c*d^2*e^4 - 5*a^3*e^6)*x*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)) - 2*(8*c^2*d^2*e^2*x^3 - 24*a
^2*d*e^3 + 2*(7*c^2*d^3*e + 13*a*c*d*e^3)*x^2 + (3*c^2*d^4 + 68*a*c*d^2*e^2 + 33
*a^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e))/(sqrt(-c*
d*e)*e*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**2/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError

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