3.1908 \(\int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=401 \[ \frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}-\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d} \]

[Out]

(-9*(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^5*d^5*e^2) + (3*(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))/(128*c^4*d^4*e) + (3*(c*d^2
- a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(40*c^3*d^3) + (3*(c*d
^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(28*c^2*d^2
) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d) + (9*(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]*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2048*c^(11/2)*d^(11/2)*e^(5/2))

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Rubi [A]  time = 0.912346, antiderivative size = 401, 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 \[ \frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}-\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-9*(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^5*d^5*e^2) + (3*(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))/(128*c^4*d^4*e) + (3*(c*d^2
- a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(40*c^3*d^3) + (3*(c*d
^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(28*c^2*d^2
) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d) + (9*(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]*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2048*c^(11/2)*d^(11/2)*e^(5/2))

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Rubi in Sympy [A]  time = 120.151, size = 386, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 c d} - \frac{3 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{28 c^{2} d^{2}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{40 c^{3} d^{3}} - \frac{3 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{128 c^{4} d^{4} e} + \frac{9 \left (a e^{2} - c d^{2}\right )^{5} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{1024 c^{5} d^{5} e^{2}} - \frac{9 \left (a e^{2} - c d^{2}\right )^{7} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2048 c^{\frac{11}{2}} d^{\frac{11}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

(d + e*x)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(7*c*d) - 3*(d +
e*x)*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(28*c**
2*d**2) + 3*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/
2)/(40*c**3*d**3) - 3*(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)/(128*c**4*d**4*e) + 9*(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**5*d**5*e**2) - 9*(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**(11/2)*d**(11/2)*e**(5/2))

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Mathematica [A]  time = 1.38356, size = 450, normalized size = 1.12 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 \left (315 a^6 e^{12}-210 a^5 c d e^{10} (10 d+e x)+21 a^4 c^2 d^2 e^8 \left (283 d^2+66 d e x+8 e^2 x^2\right )-12 a^3 c^3 d^3 e^6 \left (768 d^3+323 d^2 e x+92 d e^2 x^2+12 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (8393 d^4+5924 d^3 e x+3072 d^2 e^2 x^2+944 d e^3 x^3+128 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (1050 d^5+13643 d^4 e x+30248 d^3 e^2 x^2+31272 d^2 e^3 x^3+15872 d e^4 x^4+3200 e^5 x^5\right )+c^6 d^6 \left (-315 d^6+210 d^5 e x+14168 d^4 e^2 x^2+39056 d^3 e^3 x^3+44928 d^2 e^4 x^4+24320 d e^5 x^5+5120 e^6 x^6\right )\right )}{35 c^5 d^5 e^2 (d+e x) (a e+c d x)}+\frac{9 \left (c d^2-a e^2\right )^7 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{11/2} d^{11/2} e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right )}{2048} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((2*(315*a^6*e^12 - 210*a^5*c*d*e^10*(10*d + e*
x) + 21*a^4*c^2*d^2*e^8*(283*d^2 + 66*d*e*x + 8*e^2*x^2) - 12*a^3*c^3*d^3*e^6*(7
68*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*(8393*d^4 +
5924*d^3*e*x + 3072*d^2*e^2*x^2 + 944*d*e^3*x^3 + 128*e^4*x^4) + 2*a*c^5*d^5*e^2
*(1050*d^5 + 13643*d^4*e*x + 30248*d^3*e^2*x^2 + 31272*d^2*e^3*x^3 + 15872*d*e^4
*x^4 + 3200*e^5*x^5) + c^6*d^6*(-315*d^6 + 210*d^5*e*x + 14168*d^4*e^2*x^2 + 390
56*d^3*e^3*x^3 + 44928*d^2*e^4*x^4 + 24320*d*e^5*x^5 + 5120*e^6*x^6)))/(35*c^5*d
^5*e^2*(a*e + c*d*x)*(d + e*x)) + (9*(c*d^2 - a*e^2)^7*Log[a*e^2 + 2*Sqrt[c]*Sqr
t[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^(5/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/2048

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Maple [B]  time = 0.018, size = 1586, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

3/64*d^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+3/128*d^4/e*(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)+13/40*d/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+9/256*d
^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+3/40*e^4/d^3/c^3*(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(5/2)*a^2-9/256*e^8/d^3/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*a^5+45/1024*e^6/d/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-9/35*e^2/d
/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+45/256*e^5/c^2*a^3*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/128*e^7/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)*a^4+3/64*e^5/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+1/7*e^2*x
^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/c-45/1024*d^3*e^2/c*(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2)*a^2+189/2048*d^5*e^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^2-3/64*d^2
*e/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+45/512*d^4*e*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*x*a-9/512*d^6/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-
63/2048*d^7*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+9/2048*d^9/e^2*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)+9
/1024*e^10/d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6+11/28*e/c*x*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-9/1024*d^7/e^2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)-189/2048*e^8/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^5-9/2048*e^12/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^7-3/28*e^3/d^2/c^2*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)*a-9/64*e^2*d/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-45/512*e^7/d^2/c^3*
a^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+63/2048*e^10/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^6+9/512*e^9/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5-3/6
4*e^6/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^3+9/64*e^4/d/c^2*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-45/256*d^2*e^3/c*(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x*a^2+315/2048*d*e^6/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^4-315/2048*d^
3*e^4/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^3

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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)^(3/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.316669, 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)^(3/2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/143360*(4*(5120*c^6*d^6*e^6*x^6 - 315*c^6*d^12 + 2100*a*c^5*d^10*e^2 + 8393*a
^2*c^4*d^8*e^4 - 9216*a^3*c^3*d^6*e^6 + 5943*a^4*c^2*d^4*e^8 - 2100*a^5*c*d^2*e^
10 + 315*a^6*e^12 + 1280*(19*c^6*d^7*e^5 + 5*a*c^5*d^5*e^7)*x^5 + 128*(351*c^6*d
^8*e^4 + 248*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 16*(2441*c^6*d^9*e^3 + 3909*
a*c^5*d^7*e^5 + 59*a^2*c^4*d^5*e^7 - 9*a^3*c^3*d^3*e^9)*x^3 + 8*(1771*c^6*d^10*e
^2 + 7562*a*c^5*d^8*e^4 + 384*a^2*c^4*d^6*e^6 - 138*a^3*c^3*d^4*e^8 + 21*a^4*c^2
*d^2*e^10)*x^2 + 2*(105*c^6*d^11*e + 13643*a*c^5*d^9*e^3 + 2962*a^2*c^4*d^7*e^5
- 1938*a^3*c^3*d^5*e^7 + 693*a^4*c^2*d^3*e^9 - 105*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) + 315*(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)*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^2), 1/71680*(2*(5120*c^6*d^6*e^6*x^6 - 315*c^6*d^12 + 2100*a*c^5*d^
10*e^2 + 8393*a^2*c^4*d^8*e^4 - 9216*a^3*c^3*d^6*e^6 + 5943*a^4*c^2*d^4*e^8 - 21
00*a^5*c*d^2*e^10 + 315*a^6*e^12 + 1280*(19*c^6*d^7*e^5 + 5*a*c^5*d^5*e^7)*x^5 +
128*(351*c^6*d^8*e^4 + 248*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 16*(2441*c^6*
d^9*e^3 + 3909*a*c^5*d^7*e^5 + 59*a^2*c^4*d^5*e^7 - 9*a^3*c^3*d^3*e^9)*x^3 + 8*(
1771*c^6*d^10*e^2 + 7562*a*c^5*d^8*e^4 + 384*a^2*c^4*d^6*e^6 - 138*a^3*c^3*d^4*e
^8 + 21*a^4*c^2*d^2*e^10)*x^2 + 2*(105*c^6*d^11*e + 13643*a*c^5*d^9*e^3 + 2962*a
^2*c^4*d^7*e^5 - 1938*a^3*c^3*d^5*e^7 + 693*a^4*c^2*d^3*e^9 - 105*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) + 315*(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)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))/(s
qrt(-c*d*e)*c^5*d^5*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)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259345, size = 824, normalized size = 2.05 \[ \frac{1}{35840} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c d x e^{4} + \frac{{\left (19 \, c^{7} d^{8} e^{9} + 5 \, a c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (351 \, c^{7} d^{9} e^{8} + 248 \, a c^{6} d^{7} e^{10} + a^{2} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (2441 \, c^{7} d^{10} e^{7} + 3909 \, a c^{6} d^{8} e^{9} + 59 \, a^{2} c^{5} d^{6} e^{11} - 9 \, a^{3} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (1771 \, c^{7} d^{11} e^{6} + 7562 \, a c^{6} d^{9} e^{8} + 384 \, a^{2} c^{5} d^{7} e^{10} - 138 \, a^{3} c^{4} d^{5} e^{12} + 21 \, a^{4} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (105 \, c^{7} d^{12} e^{5} + 13643 \, a c^{6} d^{10} e^{7} + 2962 \, a^{2} c^{5} d^{8} e^{9} - 1938 \, a^{3} c^{4} d^{6} e^{11} + 693 \, a^{4} c^{3} d^{4} e^{13} - 105 \, a^{5} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac{{\left (315 \, c^{7} d^{13} e^{4} - 2100 \, a c^{6} d^{11} e^{6} - 8393 \, a^{2} c^{5} d^{9} e^{8} + 9216 \, a^{3} c^{4} d^{7} e^{10} - 5943 \, a^{4} c^{3} d^{5} e^{12} + 2100 \, a^{5} c^{2} d^{3} e^{14} - 315 \, a^{6} c d e^{16}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} - \frac{9 \,{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2048 \, c^{6} d^{6}} \]

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)^(3/2)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/35840*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*(4*c*d*x*e^4
+ (19*c^7*d^8*e^9 + 5*a*c^6*d^6*e^11)*e^(-6)/(c^6*d^6))*x + (351*c^7*d^9*e^8 +
248*a*c^6*d^7*e^10 + a^2*c^5*d^5*e^12)*e^(-6)/(c^6*d^6))*x + (2441*c^7*d^10*e^7
+ 3909*a*c^6*d^8*e^9 + 59*a^2*c^5*d^6*e^11 - 9*a^3*c^4*d^4*e^13)*e^(-6)/(c^6*d^6
))*x + (1771*c^7*d^11*e^6 + 7562*a*c^6*d^9*e^8 + 384*a^2*c^5*d^7*e^10 - 138*a^3*
c^4*d^5*e^12 + 21*a^4*c^3*d^3*e^14)*e^(-6)/(c^6*d^6))*x + (105*c^7*d^12*e^5 + 13
643*a*c^6*d^10*e^7 + 2962*a^2*c^5*d^8*e^9 - 1938*a^3*c^4*d^6*e^11 + 693*a^4*c^3*
d^4*e^13 - 105*a^5*c^2*d^2*e^15)*e^(-6)/(c^6*d^6))*x - (315*c^7*d^13*e^4 - 2100*
a*c^6*d^11*e^6 - 8393*a^2*c^5*d^9*e^8 + 9216*a^3*c^4*d^7*e^10 - 5943*a^4*c^3*d^5
*e^12 + 2100*a^5*c^2*d^3*e^14 - 315*a^6*c*d*e^16)*e^(-6)/(c^6*d^6)) - 9/2048*(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^(-5/2
)*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^6*d^6)