3.464 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx\)

Optimal. Leaf size=371 \[ -\frac{\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 d x}-\frac{\left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{a} d^{3/2} \sqrt{e}}+\frac{1}{2} c^{3/2} d^{3/2} \sqrt{e} \left (5 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3} \]

[Out]

-((5*c^2*d^4 + 12*a*c*d^2*e^2 - a^2*e^4 - 2*c*d*e*(7*c*d^2 + a*e^2)*x)*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*d*x) - ((4*a*d*e + 3*(3*c*d^2 + a*e^2)*x)
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*d*x^3) + (c^(3/2)*d^(3/2)*Sq
rt[e]*(3*c*d^2 + 5*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 - ((5*c^3*d^6 + 45*a*c
^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 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])])/(16*Sqr
t[a]*d^(3/2)*Sqrt[e])

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Rubi [A]  time = 1.29497, antiderivative size = 371, 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 \[ -\frac{\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 d x}-\frac{\left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{a} d^{3/2} \sqrt{e}}+\frac{1}{2} c^{3/2} d^{3/2} \sqrt{e} \left (5 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)),x]

[Out]

-((5*c^2*d^4 + 12*a*c*d^2*e^2 - a^2*e^4 - 2*c*d*e*(7*c*d^2 + a*e^2)*x)*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*d*x) - ((4*a*d*e + 3*(3*c*d^2 + a*e^2)*x)
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*d*x^3) + (c^(3/2)*d^(3/2)*Sq
rt[e]*(3*c*d^2 + 5*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 - ((5*c^3*d^6 + 45*a*c
^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 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])])/(16*Sqr
t[a]*d^(3/2)*Sqrt[e])

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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)**(5/2)/x**4/(e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 1.11588, size = 399, normalized size = 1.08 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^2 e^2 \left (8 d^2+14 d e x+3 e^2 x^2\right )+2 a c d^2 e x (13 d+34 e x)+3 c^2 d^3 x^2 (11 d-8 e x)\right )+3 x^3 \log (x) \left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right )-3 x^3 \left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )+24 \sqrt{a} c^{3/2} d^3 e x^3 \left (5 a e^2+3 c d^2\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )\right )}{48 \sqrt{a} d^{3/2} \sqrt{e} x^3 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.034, size = 3144, normalized size = 8.5 \[ \text{output too large to display} \]

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^4/(e*x+d),x)

[Out]

-15/16*d*a^2*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)*c-45/16*d^3*a*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
^2+3/64/d^5*e^8*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/256/d^6*e^11*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)+15/256/d^4*e^9/c*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^4-3/64/d^5
*e^8*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-9/64/d*e^4*a*c*(c*d*e
*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/256/d^6*e^11*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)+15/256*d^2*e^3*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
/d^4*e^9*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)+625/256*d^2*a*e^3*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)-5/6/d^2/a^2/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+5/24*d^2/
a^2/e*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/12/d/a^2/e^2/x^2*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+1/64/d*a*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x*c+3/8/d^3/a*e^2*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*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)^(5/2)*x-1/16/d*e^2*c*(c*d*e*(x+d/e)^2+(a
*e^2-c*d^2)*(x+d/e))^(3/2)+3/128*d^2*e*c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e
))^(1/2)-3/64*e^3*a*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-5/16*d^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)*c^3-1/24/d^3*a*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/
8/d^2*a^2*e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+37/48/d*e^2*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)*c+5/12/d^3/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/
2)+35/24*d/a*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+493/128*d^2*e*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+25/24/a^2/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(5/2)*c^2+107/64*a*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+7/40/d^4*e^3*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/5/d^4*e^3*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2
)*(x+d/e))^(5/2)-1/8/d^4*e^5*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+93/64*d
*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^2+5/6/d/a^2*c^2*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(5/2)*x-1/3/d^2/a/e/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/
2)+19/24/d^2/a*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+5/8*d^4/a/e*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3+5/24*d^3/a^2/e^2*c^3*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)-3/8/d^4/a*e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1/8*d^2/a^
3/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+1/16/d*a^3*e^6/(a*d*e)^(1/2)*l
n((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)+15/128*a^2*e^5*c*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)+3/128/d^6*e^9*a^4/c^2*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)-9/64/d^3*e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*x*a^2-3/64/d^4*e^7/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-15/128/d^2*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^3-1/8/d^2*e^3*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))
^(3/2)*x-3/128/d^6*e^9*a^4/c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+15/
128/d^2*e^7*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*e^2*c^2*(c*d*e*(x+d/e)^
2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/256*d^4*e*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)+1/16/d^5*e^6*a^2/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+9/64/d^3*e^6
*a^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-15/128*e^5*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)+1/8/d^4*e^5*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3
/2)*x+3/64/d^4*e^7*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+5/6/a*e*c
^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/8/a^3/e^3/x*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(7/2)*c^2+1/12/d^2*e^3*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+
5/8*d^3/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3+387/256*d^4*e*c^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)-1/16/d^5*e^6*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 9.08583, 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^4),x, algorithm="fricas")

[Out]

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

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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**4/(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^4),x, algorithm="giac")

[Out]

Exception raised: TypeError