3.1923 \(\int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=356 \[ -\frac{5 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^3}-\frac{5 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d} \]

[Out]

(5*(c*d^2 - a*e^2)^5*(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])/(1024*c^4*d^4*e^3) - (5*(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(384*c^3*d^3*e^2) + ((c*d^2 -
a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2
))/(24*c^2*d^2*e) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(7*c*d) - (5*(
c*d^2 - a*e^2)^7*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])])/(2048*c^(9/2)*d^(9/2)*e^(7/2))

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Rubi [A]  time = 0.575601, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ -\frac{5 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^3}-\frac{5 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d} \]

Antiderivative was successfully verified.

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

[Out]

(5*(c*d^2 - a*e^2)^5*(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])/(1024*c^4*d^4*e^3) - (5*(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(384*c^3*d^3*e^2) + ((c*d^2 -
a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2
))/(24*c^2*d^2*e) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(7*c*d) - (5*(
c*d^2 - a*e^2)^7*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])])/(2048*c^(9/2)*d^(9/2)*e^(7/2))

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Rubi in Sympy [A]  time = 81.8169, size = 342, normalized size = 0.96 \[ \left (- \frac{a e}{24 c^{2} d^{2}} + \frac{1}{24 c e}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} + \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{7 c d} + \frac{5 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{384 c^{3} d^{3} e^{2}} - \frac{5 \left (a e^{2} - c d^{2}\right )^{5} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{1024 c^{4} d^{4} e^{3}} + \frac{5 \left (a e^{2} - c d^{2}\right )^{7} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2048 c^{\frac{9}{2}} d^{\frac{9}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a*e/(24*c**2*d**2) + 1/(24*c*e))*(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c*d*e*
x**2 + x*(a*e**2 + c*d**2))**(5/2) + (a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*
*(7/2)/(7*c*d) + 5*(a*e**2 - c*d**2)**3*(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c
*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(384*c**3*d**3*e**2) - 5*(a*e**2 - c*d**
2)**5*(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
))/(1024*c**4*d**4*e**3) + 5*(a*e**2 - c*d**2)**7*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)))
)/(2048*c**(9/2)*d**(9/2)*e**(7/2))

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Mathematica [A]  time = 1.34459, size = 450, normalized size = 1.26 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{2 \left (-105 a^6 e^{12}+70 a^5 c d e^{10} (10 d+e x)-7 a^4 c^2 d^2 e^8 \left (283 d^2+66 d e x+8 e^2 x^2\right )+4 a^3 c^3 d^3 e^6 \left (768 d^3+323 d^2 e x+92 d e^2 x^2+12 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (1981 d^4+17140 d^3 e x+27648 d^2 e^2 x^2+18800 d e^3 x^3+4736 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (-350 d^5+231 d^4 e x+9032 d^3 e^2 x^2+18248 d^2 e^3 x^3+13824 d e^4 x^4+3712 e^5 x^5\right )+c^6 d^6 \left (105 d^6-70 d^5 e x+56 d^4 e^2 x^2+6096 d^3 e^3 x^3+13696 d^2 e^4 x^4+11008 d e^5 x^5+3072 e^6 x^6\right )\right )}{21 c^4 d^4 e^3 (d+e x)^2 (a e+c d x)^2}-\frac{5 \left (c d^2-a e^2\right )^7 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{9/2} d^{9/2} e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right )}{2048} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(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)*((2*(-105*a^6*e^12 + 70*a^5*c*d*e^10*(10*d + e*
x) - 7*a^4*c^2*d^2*e^8*(283*d^2 + 66*d*e*x + 8*e^2*x^2) + 4*a^3*c^3*d^3*e^6*(768
*d^3 + 323*d^2*e*x + 92*d*e^2*x^2 + 12*e^3*x^3) + a^2*c^4*d^4*e^4*(1981*d^4 + 17
140*d^3*e*x + 27648*d^2*e^2*x^2 + 18800*d*e^3*x^3 + 4736*e^4*x^4) + 2*a*c^5*d^5*
e^2*(-350*d^5 + 231*d^4*e*x + 9032*d^3*e^2*x^2 + 18248*d^2*e^3*x^3 + 13824*d*e^4
*x^4 + 3712*e^5*x^5) + c^6*d^6*(105*d^6 - 70*d^5*e*x + 56*d^4*e^2*x^2 + 6096*d^3
*e^3*x^3 + 13696*d^2*e^4*x^4 + 11008*d*e^5*x^5 + 3072*e^6*x^6)))/(21*c^4*d^4*e^3
*(a*e + c*d*x)^2*(d + e*x)^2) - (5*(c*d^2 - a*e^2)^7*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)*
e^(7/2)*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/2048

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Maple [B]  time = 0.017, size = 1533, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/1024*d^4*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-25/1024*e^5/c^2*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+5/1024*d^8/e^3*c^2*(a*e*d+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)-5/384*d^5/e^2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/7*(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d+1/24*d^2/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(5/2)+1/12*d*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+5/192*d^3*(a*e*d+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+5/2048/d^4*e^11/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^7-1
/12/d*e^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a-5/512/d^3*e^8/c^3*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5+5/192/d^2*e^5/c^2*(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(3/2)*x*a^3+25/512/d*e^6/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*x*a^4-25/256*d*e^4/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-35/2048/d^2*
e^9/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^6-175/2048*d^2*e^5/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^4-1
05/2048*d^6*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)*a^2+35/2048*d^8/e*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+5/512*d^7/e^2*c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-5/64*e^3/c*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2+5/64*d^2*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)*x*a-5/1024/d^4*e^9/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6+5
/384/d^3*e^6/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-1/24/d^2*e^3/c^2*(a
*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2+5/256/d^2*e^7/c^3*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*a^5-5/2048*d^10/e^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)-5/192/d*e^4/c^
2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+25/256*d^3*e^2*(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*x*a^2-25/512*d^5*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*
x*a+175/2048*d^4*e^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^3-5/192*d^4/e*c*(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)*x-5/256*d^6/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+1
05/2048*e^7/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^5

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.35345, 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, algorithm="fricas")

[Out]

[1/86016*(4*(3072*c^6*d^6*e^6*x^6 + 105*c^6*d^12 - 700*a*c^5*d^10*e^2 + 1981*a^2
*c^4*d^8*e^4 + 3072*a^3*c^3*d^6*e^6 - 1981*a^4*c^2*d^4*e^8 + 700*a^5*c*d^2*e^10
- 105*a^6*e^12 + 256*(43*c^6*d^7*e^5 + 29*a*c^5*d^5*e^7)*x^5 + 128*(107*c^6*d^8*
e^4 + 216*a*c^5*d^6*e^6 + 37*a^2*c^4*d^4*e^8)*x^4 + 16*(381*c^6*d^9*e^3 + 2281*a
*c^5*d^7*e^5 + 1175*a^2*c^4*d^5*e^7 + 3*a^3*c^3*d^3*e^9)*x^3 + 8*(7*c^6*d^10*e^2
+ 2258*a*c^5*d^8*e^4 + 3456*a^2*c^4*d^6*e^6 + 46*a^3*c^3*d^4*e^8 - 7*a^4*c^2*d^
2*e^10)*x^2 - 2*(35*c^6*d^11*e - 231*a*c^5*d^9*e^3 - 8570*a^2*c^4*d^7*e^5 - 646*
a^3*c^3*d^5*e^7 + 231*a^4*c^2*d^3*e^9 - 35*a^5*c*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d
*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*
c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7
*a^6*c*d^2*e^12 - a^7*e^14)*log(-4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqr
t(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^4*
d^4*e^3), 1/43008*(2*(3072*c^6*d^6*e^6*x^6 + 105*c^6*d^12 - 700*a*c^5*d^10*e^2 +
1981*a^2*c^4*d^8*e^4 + 3072*a^3*c^3*d^6*e^6 - 1981*a^4*c^2*d^4*e^8 + 700*a^5*c*
d^2*e^10 - 105*a^6*e^12 + 256*(43*c^6*d^7*e^5 + 29*a*c^5*d^5*e^7)*x^5 + 128*(107
*c^6*d^8*e^4 + 216*a*c^5*d^6*e^6 + 37*a^2*c^4*d^4*e^8)*x^4 + 16*(381*c^6*d^9*e^3
+ 2281*a*c^5*d^7*e^5 + 1175*a^2*c^4*d^5*e^7 + 3*a^3*c^3*d^3*e^9)*x^3 + 8*(7*c^6
*d^10*e^2 + 2258*a*c^5*d^8*e^4 + 3456*a^2*c^4*d^6*e^6 + 46*a^3*c^3*d^4*e^8 - 7*a
^4*c^2*d^2*e^10)*x^2 - 2*(35*c^6*d^11*e - 231*a*c^5*d^9*e^3 - 8570*a^2*c^4*d^7*e
^5 - 646*a^3*c^3*d^5*e^7 + 231*a^4*c^2*d^3*e^9 - 35*a^5*c*d*e^11)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 105*(c^7*d^14 - 7*a*c^6*d^12*e^2
+ 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^
4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)*sqr
t(-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^
4*d^4*e^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((e*x+d)*(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 = 0.255031, size = 836, normalized size = 2.35 \[ \frac{1}{21504} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \, c^{2} d^{2} x e^{3} + \frac{{\left (43 \, c^{8} d^{9} e^{8} + 29 \, a c^{7} d^{7} e^{10}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (107 \, c^{8} d^{10} e^{7} + 216 \, a c^{7} d^{8} e^{9} + 37 \, a^{2} c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (381 \, c^{8} d^{11} e^{6} + 2281 \, a c^{7} d^{9} e^{8} + 1175 \, a^{2} c^{6} d^{7} e^{10} + 3 \, a^{3} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (7 \, c^{8} d^{12} e^{5} + 2258 \, a c^{7} d^{10} e^{7} + 3456 \, a^{2} c^{6} d^{8} e^{9} + 46 \, a^{3} c^{5} d^{6} e^{11} - 7 \, a^{4} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac{{\left (35 \, c^{8} d^{13} e^{4} - 231 \, a c^{7} d^{11} e^{6} - 8570 \, a^{2} c^{6} d^{9} e^{8} - 646 \, a^{3} c^{5} d^{7} e^{10} + 231 \, a^{4} c^{4} d^{5} e^{12} - 35 \, a^{5} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (105 \, c^{8} d^{14} e^{3} - 700 \, a c^{7} d^{12} e^{5} + 1981 \, a^{2} c^{6} d^{10} e^{7} + 3072 \, a^{3} c^{5} d^{8} e^{9} - 1981 \, a^{4} c^{4} d^{6} e^{11} + 700 \, a^{5} c^{3} d^{4} e^{13} - 105 \, a^{6} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} + \frac{5 \,{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt{c d} e^{\left (-\frac{7}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2048 \, c^{5} d^{5}} \]

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, algorithm="giac")

[Out]

1/21504*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(12*c^2*d^2*x
*e^3 + (43*c^8*d^9*e^8 + 29*a*c^7*d^7*e^10)*e^(-6)/(c^6*d^6))*x + (107*c^8*d^10*
e^7 + 216*a*c^7*d^8*e^9 + 37*a^2*c^6*d^6*e^11)*e^(-6)/(c^6*d^6))*x + (381*c^8*d^
11*e^6 + 2281*a*c^7*d^9*e^8 + 1175*a^2*c^6*d^7*e^10 + 3*a^3*c^5*d^5*e^12)*e^(-6)
/(c^6*d^6))*x + (7*c^8*d^12*e^5 + 2258*a*c^7*d^10*e^7 + 3456*a^2*c^6*d^8*e^9 + 4
6*a^3*c^5*d^6*e^11 - 7*a^4*c^4*d^4*e^13)*e^(-6)/(c^6*d^6))*x - (35*c^8*d^13*e^4
- 231*a*c^7*d^11*e^6 - 8570*a^2*c^6*d^9*e^8 - 646*a^3*c^5*d^7*e^10 + 231*a^4*c^4
*d^5*e^12 - 35*a^5*c^3*d^3*e^14)*e^(-6)/(c^6*d^6))*x + (105*c^8*d^14*e^3 - 700*a
*c^7*d^12*e^5 + 1981*a^2*c^6*d^10*e^7 + 3072*a^3*c^5*d^8*e^9 - 1981*a^4*c^4*d^6*
e^11 + 700*a^5*c^3*d^4*e^13 - 105*a^6*c^2*d^2*e^15)*e^(-6)/(c^6*d^6)) + 5/2048*(
c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*
c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*sqrt(c*d)*e^(-7
/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 +
c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5)