3.1896 \(\int (d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=388 \[ -\frac{21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac{21 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac{7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac{21 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac{3 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d} \]

[Out]

(21*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(512*c^5*d^5*e) + (7*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2))/(64*c^4*d^4) + (21*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*c^3*d^3) + (3*(c*d^2 - a*e^2)*(d + e*x)^
2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*c^2*d^2) + ((d + e*x)^3*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*c*d) - (21*(c*d^2 - a*e^2)^6*ArcT
anh[(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])])/(1024*c^(11/2)*d^(11/2)*e^(3/2))

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Rubi [A]  time = 1.08024, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac{21 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac{7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac{21 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac{3 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d} \]

Antiderivative was successfully verified.

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

[Out]

(21*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(512*c^5*d^5*e) + (7*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2))/(64*c^4*d^4) + (21*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*c^3*d^3) + (3*(c*d^2 - a*e^2)*(d + e*x)^
2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*c^2*d^2) + ((d + e*x)^3*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*c*d) - (21*(c*d^2 - a*e^2)^6*ArcT
anh[(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])])/(1024*c^(11/2)*d^(11/2)*e^(3/2))

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Rubi in Sympy [A]  time = 124.803, size = 371, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{6 c d} - \frac{3 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{20 c^{2} d^{2}} + \frac{21 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{160 c^{3} d^{3}} - \frac{7 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{64 c^{4} d^{4}} + \frac{21 \left (a e^{2} - c d^{2}\right )^{4} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{512 c^{5} d^{5} e} - \frac{21 \left (a e^{2} - c d^{2}\right )^{6} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{1024 c^{\frac{11}{2}} d^{\frac{11}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

(d + e*x)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(6*c*d) - 3*(d +
e*x)**2*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(20*
c**2*d**2) + 21*(d + e*x)*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))**(3/2)/(160*c**3*d**3) - 7*(a*e**2 - c*d**2)**3*(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))**(3/2)/(64*c**4*d**4) + 21*(a*e**2 - c*d**2)**4*(a*e**2 + c
*d**2 + 2*c*d*e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(512*c**5*d**5
*e) - 21*(a*e**2 - c*d**2)**6*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqr
t(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(1024*c**(11/2)*d*
*(11/2)*e**(3/2))

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Mathematica [A]  time = 0.768745, size = 393, normalized size = 1.01 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{630 a^5 e^9}{c^5 d^5}-\frac{3570 a^4 e^7}{c^4 d^3}+\frac{8316 a^3 e^5}{c^3 d}+\frac{32 e^2 x^3 \left (-9 a^2 e^4+50 a c d^2 e^2+759 c^2 d^4\right )}{c^2 d^2}-\frac{10116 a^2 d e^3}{c^2}+\frac{16 e x^2 \left (21 a^3 e^6-117 a^2 c d^2 e^4+267 a c^2 d^4 e^2+1429 c^3 d^6\right )}{c^3 d^3}+\frac{4 x \left (-105 a^4 e^8+588 a^3 c d^2 e^6-1350 a^2 c^2 d^4 e^4+1612 a c^3 d^6 e^2+2455 c^4 d^8\right )}{c^4 d^4}-\frac{315 \left (c d^2-a e^2\right )^6 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{11/2} d^{11/2} e^{3/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{6670 a d^3 e}{c}+\frac{256 e^3 x^4 \left (a e^2+49 c d^2\right )}{c d}+\frac{630 d^5}{e}+2560 e^4 x^5\right )}{15360} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((630*d^5)/e + (6670*a*d^3*e)/c - (10116*a^2*d*e^
3)/c^2 + (8316*a^3*e^5)/(c^3*d) - (3570*a^4*e^7)/(c^4*d^3) + (630*a^5*e^9)/(c^5*
d^5) + (4*(2455*c^4*d^8 + 1612*a*c^3*d^6*e^2 - 1350*a^2*c^2*d^4*e^4 + 588*a^3*c*
d^2*e^6 - 105*a^4*e^8)*x)/(c^4*d^4) + (16*e*(1429*c^3*d^6 + 267*a*c^2*d^4*e^2 -
117*a^2*c*d^2*e^4 + 21*a^3*e^6)*x^2)/(c^3*d^3) + (32*e^2*(759*c^2*d^4 + 50*a*c*d
^2*e^2 - 9*a^2*e^4)*x^3)/(c^2*d^2) + (256*e^3*(49*c*d^2 + a*e^2)*x^4)/(c*d) + 25
60*e^4*x^5 - (315*(c*d^2 - a*e^2)^6*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^(11/2)*d^(11/2)*e^(3/2)*Sqrt[a*
e + c*d*x]*Sqrt[d + e*x])))/15360

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Maple [B]  time = 0.035, size = 1327, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

1/6*e^3*x^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/c+63/128*e^4/c^2*(a*e*d+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-237/320*e^2/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*a+13/20*e^2/c*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+107/192*d
^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+21/256*d^4*(a*e*d+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*x+21/512*d^5/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-9/16*e^3/
d/c^2*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+21/256*e^8/d^4/c^4*(a*e*d+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-21/64*e^6/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*x*a^3-21/1024*e^11/d^5/c^5*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^6+63/512*e^9/d^3
/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^5-3/20*e^4/d^2/c^2*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)*a+21/160*e^5/d^3/c^3*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+1
05/256*e^5*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)*a^3-315/1024*e^7/d/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^4-315/1024*d^3*e^3/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)*a^2-21/64*e^2*d^2/c*(a*e*d+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+147/320*e^4/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*a^2-7/64*e^6/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+21
/256*e^5/d/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+21/256*e^3*d/c^2*(a*e
*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+21/512*e^9/d^5/c^5*(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*a^5-63/512*e^7/d^3/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)*a^4+149/160*e*d/c*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-63/512*d^3*e/c*(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+63/512*d^5*e*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-21/1
024*d^7/e*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)

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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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289886, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^4,x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*c^5*d^5*e^5*x^5 + 315*c^5*d^10 + 3335*a*c^4*d^8*e^2 - 5058*a^2
*c^3*d^6*e^4 + 4158*a^3*c^2*d^4*e^6 - 1785*a^4*c*d^2*e^8 + 315*a^5*e^10 + 128*(4
9*c^5*d^6*e^4 + a*c^4*d^4*e^6)*x^4 + 16*(759*c^5*d^7*e^3 + 50*a*c^4*d^5*e^5 - 9*
a^2*c^3*d^3*e^7)*x^3 + 8*(1429*c^5*d^8*e^2 + 267*a*c^4*d^6*e^4 - 117*a^2*c^3*d^4
*e^6 + 21*a^3*c^2*d^2*e^8)*x^2 + 2*(2455*c^5*d^9*e + 1612*a*c^4*d^7*e^3 - 1350*a
^2*c^3*d^5*e^5 + 588*a^3*c^2*d^3*e^7 - 105*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 315*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c
^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^
12)*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)))/(sqrt(c*d*e)*c^5*d^5*e), 1/15360*(2*(1280
*c^5*d^5*e^5*x^5 + 315*c^5*d^10 + 3335*a*c^4*d^8*e^2 - 5058*a^2*c^3*d^6*e^4 + 41
58*a^3*c^2*d^4*e^6 - 1785*a^4*c*d^2*e^8 + 315*a^5*e^10 + 128*(49*c^5*d^6*e^4 + a
*c^4*d^4*e^6)*x^4 + 16*(759*c^5*d^7*e^3 + 50*a*c^4*d^5*e^5 - 9*a^2*c^3*d^3*e^7)*
x^3 + 8*(1429*c^5*d^8*e^2 + 267*a*c^4*d^6*e^4 - 117*a^2*c^3*d^4*e^6 + 21*a^3*c^2
*d^2*e^8)*x^2 + 2*(2455*c^5*d^9*e + 1612*a*c^4*d^7*e^3 - 1350*a^2*c^3*d^5*e^5 +
588*a^3*c^2*d^3*e^7 - 105*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)*sqrt(-c*d*e) - 315*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*
a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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)))/(sqrt(-c*d*e)*c^5*d^5*e)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250624, size = 655, normalized size = 1.69 \[ \frac{1}{7680} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, x e^{4} + \frac{{\left (49 \, c^{5} d^{6} e^{8} + a c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (759 \, c^{5} d^{7} e^{7} + 50 \, a c^{4} d^{5} e^{9} - 9 \, a^{2} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (1429 \, c^{5} d^{8} e^{6} + 267 \, a c^{4} d^{6} e^{8} - 117 \, a^{2} c^{3} d^{4} e^{10} + 21 \, a^{3} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (2455 \, c^{5} d^{9} e^{5} + 1612 \, a c^{4} d^{7} e^{7} - 1350 \, a^{2} c^{3} d^{5} e^{9} + 588 \, a^{3} c^{2} d^{3} e^{11} - 105 \, a^{4} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (315 \, c^{5} d^{10} e^{4} + 3335 \, a c^{4} d^{8} e^{6} - 5058 \, a^{2} c^{3} d^{6} e^{8} + 4158 \, a^{3} c^{2} d^{4} e^{10} - 1785 \, a^{4} c d^{2} e^{12} + 315 \, a^{5} e^{14}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac{21 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{c d} e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{1024 \, c^{6} d^{6}} \]

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)^4,x, algorithm="giac")

[Out]

1/7680*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*x*e^4 + (49*c
^5*d^6*e^8 + a*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x + (759*c^5*d^7*e^7 + 50*a*c^4*d
^5*e^9 - 9*a^2*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (1429*c^5*d^8*e^6 + 267*a*c^4
*d^6*e^8 - 117*a^2*c^3*d^4*e^10 + 21*a^3*c^2*d^2*e^12)*e^(-5)/(c^5*d^5))*x + (24
55*c^5*d^9*e^5 + 1612*a*c^4*d^7*e^7 - 1350*a^2*c^3*d^5*e^9 + 588*a^3*c^2*d^3*e^1
1 - 105*a^4*c*d*e^13)*e^(-5)/(c^5*d^5))*x + (315*c^5*d^10*e^4 + 3335*a*c^4*d^8*e
^6 - 5058*a^2*c^3*d^6*e^8 + 4158*a^3*c^2*d^4*e^10 - 1785*a^4*c*d^2*e^12 + 315*a^
5*e^14)*e^(-5)/(c^5*d^5)) + 21/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^
8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*s
qrt(c*d)*e^(-3/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqr
t(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^6*d^6
)