3.1943 \(\int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac{35 \sqrt{e} \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{9/2} d^{9/2}}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^4 d^4}+\frac{35 e (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2}-\frac{2 (d+e x)^4}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*(d + e*x)^4)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e*(c*d^
2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*c^4*d^4) + (35*e*(c
*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*c^3*d^3
) + (7*e*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2) +
(35*Sqrt[e]*(c*d^2 - a*e^2)^3*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqr
t[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*c^(9/2)*d^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.688485, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{35 \sqrt{e} \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{9/2} d^{9/2}}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^4 d^4}+\frac{35 e (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2}-\frac{2 (d+e x)^4}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

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)^(3/2),x]

[Out]

(-2*(d + e*x)^4)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e*(c*d^
2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*c^4*d^4) + (35*e*(c
*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*c^3*d^3
) + (7*e*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2) +
(35*Sqrt[e]*(c*d^2 - a*e^2)^3*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqr
t[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*c^(9/2)*d^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 102.656, size = 292, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )^{4}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{7 e \left (d + e x\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 c^{2} d^{2}} - \frac{35 e \left (d + e x\right ) \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{12 c^{3} d^{3}} + \frac{35 e \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 c^{4} d^{4}} - \frac{35 \sqrt{e} \left (a e^{2} - c d^{2}\right )^{3} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{16 c^{\frac{9}{2}} d^{\frac{9}{2}}} \]

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)**(3/2),x)

[Out]

-2*(d + e*x)**4/(c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 7*e*(d +
e*x)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3*c**2*d**2) - 35*e*(d +
e*x)*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(12*c**3*
d**3) + 35*e*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
/(8*c**4*d**4) - 35*sqrt(e)*(a*e**2 - c*d**2)**3*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))))
/(16*c**(9/2)*d**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.689195, size = 249, normalized size = 0.82 \[ \frac{\frac{2 (d+e x)^2 (a e+c d x) \left (105 a^3 e^6-35 a^2 c d e^4 (8 d-e x)+7 a c^2 d^2 e^2 \left (33 d^2-14 d e x-2 e^2 x^2\right )+c^3 d^3 \left (-48 d^3+87 d^2 e x+38 d e^2 x^2+8 e^3 x^3\right )\right )}{3 c^4 d^4}+\frac{35 \sqrt{e} (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 (a e+c d x)^{3/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{9/2} d^{9/2}}}{16 ((d+e x) (a e+c d x))^{3/2}} \]

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)^(3/2),x]

[Out]

((2*(a*e + c*d*x)*(d + e*x)^2*(105*a^3*e^6 - 35*a^2*c*d*e^4*(8*d - e*x) + 7*a*c^
2*d^2*e^2*(33*d^2 - 14*d*e*x - 2*e^2*x^2) + c^3*d^3*(-48*d^3 + 87*d^2*e*x + 38*d
*e^2*x^2 + 8*e^3*x^3)))/(3*c^4*d^4) + (35*Sqrt[e]*(c*d^2 - a*e^2)^3*(a*e + c*d*x
)^(3/2)*(d + e*x)^(3/2)*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^(9/2)*d^(9/2)))/(16*((a*e + c*d*x)*(d + e*x
))^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 1850, normalized size = 6.1 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

-125/32*d^3/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-203/24*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)^(1/2)*x*a^3+415/48*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)^(1/2)*x*a-
35/16*e^7/d^4/c^4*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^3+105/16*e^5/d^2/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^2+35/16*e^7/d^4/c^4*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-105/16*
e^5/d^2/c^3*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-14/3*e^4/d/c^2*x^2/(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-7/32*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)^(1/2)*a^4+103/32*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)^(1/2)*a^2-125/16*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)^(1/2)*x-35/32
*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)
^(1/2)*a^6+35/12*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)^(1/2)*a^5-7/12*e^5/d^2/c^2*x^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)*a+35/24*e^6/d^3/c^3*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-16/3
*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)*a^3+395/48*e^2*d/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+125/24*e^2*d/
c*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-125/32*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)^(1/2)+23/12*e^3/c*x^3/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)+2*d^5*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+
c*d^2)^2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/3*e^4*x^4/d/c/(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)-105/16*e^3/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)*a+105/16*e^3/c^2*
x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+5/12*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)^(1/2)*a-35/16*e*d^2/c*x/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)-35/32*e^8/d^5/c^5/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*a^4+245/48*e^6/d^3/c^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+35/16*e
*d^2/c*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)-28/3*e^4/d/c^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*a^2-53/24*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)^(1/2)*x*a^2-35/16*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)^(1/2)*x*a^5+385/48*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)^(1/2)*x*a^4

_______________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.983172, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} +{\left (c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \,{\left (8 \, c^{3} d^{3} e^{3} x^{3} - 48 \, c^{3} d^{6} + 231 \, a c^{2} d^{4} e^{2} - 280 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (19 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (87 \, c^{3} d^{5} e - 98 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{96 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, \frac{105 \,{\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} +{\left (c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) + 2 \,{\left (8 \, c^{3} d^{3} e^{3} x^{3} - 48 \, c^{3} d^{6} + 231 \, a c^{2} d^{4} e^{2} - 280 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (19 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (87 \, c^{3} d^{5} e - 98 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{48 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(105*(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7 + (c^4*d
^7 - 3*a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - a^3*c*d*e^6)*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*(8*c^3*d^3*e^3*x^3 - 48*c^3*d^6 + 231*a*c^2*d^4*e^2 -
280*a^2*c*d^2*e^4 + 105*a^3*e^6 + 2*(19*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (87
*c^3*d^5*e - 98*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d
^2 + a*e^2)*x))/(c^5*d^5*x + a*c^4*d^4*e), 1/48*(105*(a*c^3*d^6*e - 3*a^2*c^2*d^
4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7 + (c^4*d^7 - 3*a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e
^4 - a^3*c*d*e^6)*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*(8*c^3*d^3*e^3*
x^3 - 48*c^3*d^6 + 231*a*c^2*d^4*e^2 - 280*a^2*c*d^2*e^4 + 105*a^3*e^6 + 2*(19*c
^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (87*c^3*d^5*e - 98*a*c^2*d^3*e^3 + 35*a^2*c*
d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x + a*c^4*d^4*e)
]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((d + e*x)**5/((d + e*x)*(a*e + c*d*x))**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274169, size = 840, normalized size = 2.78 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (c^{5} d^{7} e^{7} - 2 \, a c^{4} d^{5} e^{9} + a^{2} c^{3} d^{3} e^{11}\right )} x}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}} + \frac{23 \, c^{5} d^{8} e^{6} - 53 \, a c^{4} d^{6} e^{8} + 37 \, a^{2} c^{3} d^{4} e^{10} - 7 \, a^{3} c^{2} d^{2} e^{12}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}\right )} x + \frac{125 \, c^{5} d^{9} e^{5} - 362 \, a c^{4} d^{7} e^{7} + 384 \, a^{2} c^{3} d^{5} e^{9} - 182 \, a^{3} c^{2} d^{3} e^{11} + 35 \, a^{4} c d e^{13}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}\right )} x + \frac{39 \, c^{5} d^{10} e^{4} + 55 \, a c^{4} d^{8} e^{6} - 472 \, a^{2} c^{3} d^{6} e^{8} + 728 \, a^{3} c^{2} d^{4} e^{10} - 455 \, a^{4} c d^{2} e^{12} + 105 \, a^{5} e^{14}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}\right )} x - \frac{48 \, c^{5} d^{11} e^{3} - 327 \, a c^{4} d^{9} e^{5} + 790 \, a^{2} c^{3} d^{7} e^{7} - 896 \, a^{3} c^{2} d^{5} e^{9} + 490 \, a^{4} c d^{3} e^{11} - 105 \, a^{5} d e^{13}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}}{24 \, \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{35 \,{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{16 \, c^{5} d^{5}} \]

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)^(3/2),x, algorithm="giac")

[Out]

1/24*(((2*(4*(c^5*d^7*e^7 - 2*a*c^4*d^5*e^9 + a^2*c^3*d^3*e^11)*x/(c^6*d^8*e^3 -
2*a*c^5*d^6*e^5 + a^2*c^4*d^4*e^7) + (23*c^5*d^8*e^6 - 53*a*c^4*d^6*e^8 + 37*a^
2*c^3*d^4*e^10 - 7*a^3*c^2*d^2*e^12)/(c^6*d^8*e^3 - 2*a*c^5*d^6*e^5 + a^2*c^4*d^
4*e^7))*x + (125*c^5*d^9*e^5 - 362*a*c^4*d^7*e^7 + 384*a^2*c^3*d^5*e^9 - 182*a^3
*c^2*d^3*e^11 + 35*a^4*c*d*e^13)/(c^6*d^8*e^3 - 2*a*c^5*d^6*e^5 + a^2*c^4*d^4*e^
7))*x + (39*c^5*d^10*e^4 + 55*a*c^4*d^8*e^6 - 472*a^2*c^3*d^6*e^8 + 728*a^3*c^2*
d^4*e^10 - 455*a^4*c*d^2*e^12 + 105*a^5*e^14)/(c^6*d^8*e^3 - 2*a*c^5*d^6*e^5 + a
^2*c^4*d^4*e^7))*x - (48*c^5*d^11*e^3 - 327*a*c^4*d^9*e^5 + 790*a^2*c^3*d^7*e^7
- 896*a^3*c^2*d^5*e^9 + 490*a^4*c*d^3*e^11 - 105*a^5*d*e^13)/(c^6*d^8*e^3 - 2*a*
c^5*d^6*e^5 + a^2*c^4*d^4*e^7))/sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x) - 35
/16*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*sqrt(c*d)*e^(-1/2)
*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + a*d
*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5)