3.1954 \(\int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=231 \[ \frac{5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}+\frac{5 e^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^4}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^4)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (10*e*(
d + e*x)^2)/(3*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*e^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^3*d^3) + (5*e^(3/2)*(c*d^2 - a*e^2)
*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])])/(2*c^(7/2)*d^(7/2))

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Rubi [A]  time = 0.429562, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}+\frac{5 e^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^4}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^4)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (10*e*(
d + e*x)^2)/(3*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*e^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^3*d^3) + (5*e^(3/2)*(c*d^2 - a*e^2)
*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])])/(2*c^(7/2)*d^(7/2))

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Rubi in Sympy [A]  time = 68.9748, size = 224, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )^{4}}{3 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{10 e \left (d + e x\right )^{2}}{3 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{5 e^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c^{3} d^{3}} - \frac{5 e^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 c^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(d + e*x)**4/(3*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) - 10*e
*(d + e*x)**2/(3*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 5*e
**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(c**3*d**3) - 5*e**(3/2)*(a*e
**2 - c*d**2)*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqr
t(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(2*c**(7/2)*d**(7/2))

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Mathematica [A]  time = 0.697271, size = 204, normalized size = 0.88 \[ \frac{1}{2} ((d+e x) (a e+c d x))^{5/2} \left (\frac{30 a^2 e^4-20 a c d e^2 (d-2 e x)-2 c^2 d^2 \left (2 d^2+14 d e x-3 e^2 x^2\right )}{3 c^3 d^3 (d+e x)^2 (a e+c d x)^4}+\frac{5 e^{3/2} \left (c d^2-a e^2\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((30*a^2*e^4 - 20*a*c*d*e^2*(d - 2*e*x) - 2*c^2
*d^2*(2*d^2 + 14*d*e*x - 3*e^2*x^2))/(3*c^3*d^3*(a*e + c*d*x)^4*(d + e*x)^2) + (
5*e^(3/2)*(c*d^2 - a*e^2)*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x
]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(7/2)*d^(7/2)*(a*e + c*d*x)^(5/2)*(d + e*
x)^(5/2))))/2

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Maple [B]  time = 0.023, size = 3215, normalized size = 13.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19/12*e*d^8*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)-43/48*e^6/d^3/c^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+5/96*e^
8/d^5/c^5/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+73/12*e^9/c^2/(-a^2*e^4+2*
a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+1/12*e^2*d^5/
(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-97/12
*e^5*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*a^2-19/16*e^3/c^2*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+5/2*e^2*d^3/(-
a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+e^4*x^4
/d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/4*e^5/d^4/c^4/(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*a^2+5/2*e^2/d/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^
(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-35/4*e^2*d/c*x^2/(a
*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+11/4*e^4/d/c^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*a^2-5/2*e^2/d/c^2*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-71/16*e
^2*d/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-115/16*e*d^2/c*x/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)-61/6*e^10/d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-5/2*e^8/d^3/c^3/(-a^2*e^4+2*a*c*d^2
*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-5/2*e^6/d/c^2/(-a^2*
e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+5/6*e^1
2/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*x*a^5+5/48*e^11/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*x*a^5-61/48*e^9/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4+67/3*e^8*d/c/(-a^2*e^4+2*a*c*d^2*e^
2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-15/32*d^3/c/(a*e*d+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+5/4*e/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
+5/4*e*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)-5/6*e^3/c*x^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+19/96*d^7*c/(-a^2*e^4+
2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-11/48*e^3*d^4/(-a
^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+67/24*
e^7/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*x*a^3-43/3*e^6*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*a^2+19/48*e*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+5/96*e^12/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)
/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^6-7/12*e^10/d^3/c^4/(-a^2*e^4+2*a*c*d
^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5+73/96*e^8/d/c^3/(-a^
2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+1/2*e^6
*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*
a^3+19/6*e^2*d^7*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*x-5/16*e^7/d^4/c^4*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3
-97/96*e^4*d^3/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*a^2+4*e^7*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*a^3+4*e^4/d/c^2*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*a+5/2*e^4/d^3/c^3*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-11/6*e^4*d^5*c/(-
a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-43/
24*e^5*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)*x*a^2+5/2*e^4*d/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*x*a+11/16*e^5/d^2/c^3*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*a^2+5/12*e^13/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*a^6-14/3*e^11/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+2/3*e^3*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2
*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-5/4*e^6/d^3/c^3*x^2/(a*e*d+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-5/4*e^9/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^
4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-5/2*e^7/d^2/c^3/(-a^2*e^4+2*a*c*d
^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+5/2*e^3*d^2/c/(-a^2*
e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+5/6*e^5/d^2
/c^2*x^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-5/2*e^4/d^3/c^3*ln((1/2*a*e^2
+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*
c)^(1/2)*a

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0689, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \,{\left (7 \, c^{2} d^{3} e - 10 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{12 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}, \frac{15 \,{\left (a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) + 2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \,{\left (7 \, c^{2} d^{3} e - 10 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{6 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(15*(a^2*c*d^2*e^3 - a^3*e^5 + (c^3*d^4*e - a*c^2*d^2*e^3)*x^2 + 2*(a*c^2*
d^3*e^2 - a^2*c*d*e^4)*x)*sqrt(e/(c*d))*log(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 + 4*(2*c^2*d^2*e*x + c^2*d^3 + a
*c*d*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) + 4*(3*c^2*
d^2*e^2*x^2 - 2*c^2*d^4 - 10*a*c*d^2*e^2 + 15*a^2*e^4 - 2*(7*c^2*d^3*e - 10*a*c*
d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x^2 + 2*a*c^4*d^
4*e*x + a^2*c^3*d^3*e^2), 1/6*(15*(a^2*c*d^2*e^3 - a^3*e^5 + (c^3*d^4*e - a*c^2*
d^2*e^3)*x^2 + 2*(a*c^2*d^3*e^2 - a^2*c*d*e^4)*x)*sqrt(-e/(c*d))*arctan(1/2*(2*c
*d*e*x + c*d^2 + a*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*sqrt(-e
/(c*d)))) + 2*(3*c^2*d^2*e^2*x^2 - 2*c^2*d^4 - 10*a*c*d^2*e^2 + 15*a^2*e^4 - 2*(
7*c^2*d^3*e - 10*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5
*d^5*x^2 + 2*a*c^4*d^4*e*x + a^2*c^3*d^3*e^2)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265605, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done