3.1907 \(\int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=461 \[ \frac{99 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{13/2} d^{13/2} e^{5/2}}-\frac{99 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^6 d^6 e^2}+\frac{33 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^5 d^5 e}+\frac{33 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{640 c^4 d^4}+\frac{33 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac{11 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 c d} \]

[Out]

(-99*(c*d^2 - a*e^2)^6*(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])/(16384*c^6*d^6*e^2) + (33*(c*d^2 - a*e^2)^4*(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))/(2048*c^5*d^5*e) + (33*(c
*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(640*c^4*d^4) + (
33*(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)^(5/2))/(4
48*c^3*d^3) + (11*(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)^(5/2))/(112*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(5/2))/(8*c*d) + (99*(c*d^2 - a*e^2)^8*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])])/(32768*c^
(13/2)*d^(13/2)*e^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.29265, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{99 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{13/2} d^{13/2} e^{5/2}}-\frac{99 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^6 d^6 e^2}+\frac{33 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^5 d^5 e}+\frac{33 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{640 c^4 d^4}+\frac{33 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac{11 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-99*(c*d^2 - a*e^2)^6*(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])/(16384*c^6*d^6*e^2) + (33*(c*d^2 - a*e^2)^4*(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))/(2048*c^5*d^5*e) + (33*(c
*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(640*c^4*d^4) + (
33*(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)^(5/2))/(4
48*c^3*d^3) + (11*(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)^(5/2))/(112*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(5/2))/(8*c*d) + (99*(c*d^2 - a*e^2)^8*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])])/(32768*c^
(13/2)*d^(13/2)*e^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 160.472, size = 444, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{8 c d} - \frac{11 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{112 c^{2} d^{2}} + \frac{33 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{448 c^{3} d^{3}} - \frac{33 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{640 c^{4} d^{4}} + \frac{33 \left (a e^{2} - c d^{2}\right )^{4} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{2048 c^{5} d^{5} e} - \frac{99 \left (a e^{2} - c d^{2}\right )^{6} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{16384 c^{6} d^{6} e^{2}} + \frac{99 \left (a e^{2} - c d^{2}\right )^{8} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{32768 c^{\frac{13}{2}} d^{\frac{13}{2}} e^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

(d + e*x)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(8*c*d) - 11*(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))**(5/2)/(11
2*c**2*d**2) + 33*(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))**(5/2)/(448*c**3*d**3) - 33*(a*e**2 - c*d**2)**3*(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2))**(5/2)/(640*c**4*d**4) + 33*(a*e**2 - c*d**2)**4*(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)/(2048*c
**5*d**5*e) - 99*(a*e**2 - c*d**2)**6*(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))/(16384*c**6*d**6*e**2) + 99*(a*e**2 - c*d**2)
**8*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))))/(32768*c**(13/2)*d**(13/2)*e**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 1.58528, size = 538, normalized size = 1.17 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 \left (-3465 a^7 e^{14}+1155 a^6 c d e^{12} (23 d+2 e x)-231 a^5 c^2 d^2 e^{10} \left (383 d^2+76 d e x+8 e^2 x^2\right )+33 a^4 c^3 d^3 e^8 \left (5053 d^3+1754 d^2 e x+424 d e^2 x^2+48 e^3 x^3\right )-11 a^3 c^4 d^4 e^6 \left (17609 d^4+9800 d^3 e x+4176 d^2 e^2 x^2+1088 d e^3 x^3+128 e^4 x^4\right )+a^2 c^5 d^5 e^4 \left (140903 d^5+123418 d^4 e x+85136 d^3 e^2 x^2+39200 d^2 e^3 x^3+10624 d e^4 x^4+1280 e^5 x^5\right )+a c^6 d^6 e^2 \left (26565 d^6+441196 d^5 e x+1226408 d^4 e^2 x^2+1691968 d^3 e^3 x^3+1288576 d^2 e^4 x^4+519680 d e^5 x^5+87040 e^6 x^6\right )+c^7 d^7 \left (-3465 d^7+2310 d^6 e x+227528 d^5 e^2 x^2+788016 d^4 e^3 x^3+1211008 d^3 e^4 x^4+984320 d^2 e^5 x^5+414720 d e^6 x^6+71680 e^7 x^7\right )\right )}{35 c^6 d^6 e^2 (d+e x) (a e+c d x)}+\frac{99 \left (c d^2-a e^2\right )^8 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{13/2} d^{13/2} e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right )}{32768} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(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*(-3465*a^7*e^14 + 1155*a^6*c*d*e^12*(23*d +
2*e*x) - 231*a^5*c^2*d^2*e^10*(383*d^2 + 76*d*e*x + 8*e^2*x^2) + 33*a^4*c^3*d^3
*e^8*(5053*d^3 + 1754*d^2*e*x + 424*d*e^2*x^2 + 48*e^3*x^3) - 11*a^3*c^4*d^4*e^6
*(17609*d^4 + 9800*d^3*e*x + 4176*d^2*e^2*x^2 + 1088*d*e^3*x^3 + 128*e^4*x^4) +
a^2*c^5*d^5*e^4*(140903*d^5 + 123418*d^4*e*x + 85136*d^3*e^2*x^2 + 39200*d^2*e^3
*x^3 + 10624*d*e^4*x^4 + 1280*e^5*x^5) + a*c^6*d^6*e^2*(26565*d^6 + 441196*d^5*e
*x + 1226408*d^4*e^2*x^2 + 1691968*d^3*e^3*x^3 + 1288576*d^2*e^4*x^4 + 519680*d*
e^5*x^5 + 87040*e^6*x^6) + c^7*d^7*(-3465*d^7 + 2310*d^6*e*x + 227528*d^5*e^2*x^
2 + 788016*d^4*e^3*x^3 + 1211008*d^3*e^4*x^4 + 984320*d^2*e^5*x^5 + 414720*d*e^6
*x^6 + 71680*e^7*x^7)))/(35*c^6*d^6*e^2*(a*e + c*d*x)*(d + e*x)) + (99*(c*d^2 -
a*e^2)^8*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^(13/2)*d^(13/2)*e^(5/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2
))))/32768

_______________________________________________________________________________________

Maple [B]  time = 0.035, size = 2065, normalized size = 4.5 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

53/112*e^2/c*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+495/16384*e^6/c^3*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-99/16384*d^8/e^2*c*(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)-1793/4480*e^2/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+33
/1024*d^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+33/2048*d^5/e*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)+495/16384*d^6*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*
a+223/640*d^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+297/4096*d^5*e*(a*e*d+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-99/4096*d^8*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+99/3276
8*d^10/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)+1/8*e^3*x^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(5/2)/d/c-693/4096*e^8/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^5-99/16384*e^12/d^6/c^6*(a
*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7+495/16384*e^10/d^4/c^5*(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2)*a^6-891/16384*e^8/d^2/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)*a^5+33/2048*e^9/d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5-9
9/2048*e^7/d^3/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+1023/4480*e^4/d^2
/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2+289/448*e*d/c*x*(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(5/2)-33/640*e^6/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5
/2)*a^3+33/1024*e^5/d/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+33/1024*e^
3*d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+99/512*e^4/c^2*(a*e*d+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-99/2048*d^3*e/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)*a+495/16384*d^2*e^4/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-891
/16384*d^4*e^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+693/8192*d^6*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^2-99/8192*d^7/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*
x+3465/16384*d^2*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^4-693/4096*d^4*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^3-1485/8192*d^3*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^
2+495/2048*e^5*d/c^2*a^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-99/8192*e^11/
d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6+99/32768*e^14/d^6/c^6*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^8-99/4096*e^12/d^4/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+693/8192*e^
10/d^2/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^6-33/256*e^2*d^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*x*a-11/112*e^4/d^2/c^2*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
*a-11/32*e^3/d/c^2*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+33/448*e^5/d^3/c^
3*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2+33/1024*e^8/d^4/c^4*(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4-33/256*e^6/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*x*a^3+297/4096*e^9/d^3/c^4*a^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*x-1485/8192*e^7/d/c^3*a^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x

_______________________________________________________________________________________

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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.35644, 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)^4,x, algorithm="fricas")

[Out]

[1/2293760*(4*(71680*c^7*d^7*e^7*x^7 - 3465*c^7*d^14 + 26565*a*c^6*d^12*e^2 + 14
0903*a^2*c^5*d^10*e^4 - 193699*a^3*c^4*d^8*e^6 + 166749*a^4*c^3*d^6*e^8 - 88473*
a^5*c^2*d^4*e^10 + 26565*a^6*c*d^2*e^12 - 3465*a^7*e^14 + 5120*(81*c^7*d^8*e^6 +
17*a*c^6*d^6*e^8)*x^6 + 1280*(769*c^7*d^9*e^5 + 406*a*c^6*d^7*e^7 + a^2*c^5*d^5
*e^9)*x^5 + 128*(9461*c^7*d^10*e^4 + 10067*a*c^6*d^8*e^6 + 83*a^2*c^5*d^6*e^8 -
11*a^3*c^4*d^4*e^10)*x^4 + 16*(49251*c^7*d^11*e^3 + 105748*a*c^6*d^9*e^5 + 2450*
a^2*c^5*d^7*e^7 - 748*a^3*c^4*d^5*e^9 + 99*a^4*c^3*d^3*e^11)*x^3 + 8*(28441*c^7*
d^12*e^2 + 153301*a*c^6*d^10*e^4 + 10642*a^2*c^5*d^8*e^6 - 5742*a^3*c^4*d^6*e^8
+ 1749*a^4*c^3*d^4*e^10 - 231*a^5*c^2*d^2*e^12)*x^2 + 2*(1155*c^7*d^13*e + 22059
8*a*c^6*d^11*e^3 + 61709*a^2*c^5*d^9*e^5 - 53900*a^3*c^4*d^7*e^7 + 28941*a^4*c^3
*d^5*e^9 - 8778*a^5*c^2*d^3*e^11 + 1155*a^6*c*d*e^13)*x)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 3465*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^
6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28
*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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^6*d^6*e^2), 1/1146880*(2*(71680*c^7*d^7*e^7*x^7 - 3465*c^7*d^1
4 + 26565*a*c^6*d^12*e^2 + 140903*a^2*c^5*d^10*e^4 - 193699*a^3*c^4*d^8*e^6 + 16
6749*a^4*c^3*d^6*e^8 - 88473*a^5*c^2*d^4*e^10 + 26565*a^6*c*d^2*e^12 - 3465*a^7*
e^14 + 5120*(81*c^7*d^8*e^6 + 17*a*c^6*d^6*e^8)*x^6 + 1280*(769*c^7*d^9*e^5 + 40
6*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^5 + 128*(9461*c^7*d^10*e^4 + 10067*a*c^6*d^
8*e^6 + 83*a^2*c^5*d^6*e^8 - 11*a^3*c^4*d^4*e^10)*x^4 + 16*(49251*c^7*d^11*e^3 +
105748*a*c^6*d^9*e^5 + 2450*a^2*c^5*d^7*e^7 - 748*a^3*c^4*d^5*e^9 + 99*a^4*c^3*
d^3*e^11)*x^3 + 8*(28441*c^7*d^12*e^2 + 153301*a*c^6*d^10*e^4 + 10642*a^2*c^5*d^
8*e^6 - 5742*a^3*c^4*d^6*e^8 + 1749*a^4*c^3*d^4*e^10 - 231*a^5*c^2*d^2*e^12)*x^2
+ 2*(1155*c^7*d^13*e + 220598*a*c^6*d^11*e^3 + 61709*a^2*c^5*d^9*e^5 - 53900*a^
3*c^4*d^7*e^7 + 28941*a^4*c^3*d^5*e^9 - 8778*a^5*c^2*d^3*e^11 + 1155*a^6*c*d*e^1
3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) + 3465*(c^8*d^16
- 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*
e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*a
rctan(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^6*d^6*e^2)]

_______________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272017, size = 992, normalized size = 2.15 \[ \text{result too large to display} \]

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)^4,x, algorithm="giac")

[Out]

1/573440*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*(4*(14*c*d*
x*e^5 + (81*c^8*d^9*e^11 + 17*a*c^7*d^7*e^13)*e^(-7)/(c^7*d^7))*x + (769*c^8*d^1
0*e^10 + 406*a*c^7*d^8*e^12 + a^2*c^6*d^6*e^14)*e^(-7)/(c^7*d^7))*x + (9461*c^8*
d^11*e^9 + 10067*a*c^7*d^9*e^11 + 83*a^2*c^6*d^7*e^13 - 11*a^3*c^5*d^5*e^15)*e^(
-7)/(c^7*d^7))*x + (49251*c^8*d^12*e^8 + 105748*a*c^7*d^10*e^10 + 2450*a^2*c^6*d
^8*e^12 - 748*a^3*c^5*d^6*e^14 + 99*a^4*c^4*d^4*e^16)*e^(-7)/(c^7*d^7))*x + (284
41*c^8*d^13*e^7 + 153301*a*c^7*d^11*e^9 + 10642*a^2*c^6*d^9*e^11 - 5742*a^3*c^5*
d^7*e^13 + 1749*a^4*c^4*d^5*e^15 - 231*a^5*c^3*d^3*e^17)*e^(-7)/(c^7*d^7))*x + (
1155*c^8*d^14*e^6 + 220598*a*c^7*d^12*e^8 + 61709*a^2*c^6*d^10*e^10 - 53900*a^3*
c^5*d^8*e^12 + 28941*a^4*c^4*d^6*e^14 - 8778*a^5*c^3*d^4*e^16 + 1155*a^6*c^2*d^2
*e^18)*e^(-7)/(c^7*d^7))*x - (3465*c^8*d^15*e^5 - 26565*a*c^7*d^13*e^7 - 140903*
a^2*c^6*d^11*e^9 + 193699*a^3*c^5*d^9*e^11 - 166749*a^4*c^4*d^7*e^13 + 88473*a^5
*c^3*d^5*e^15 - 26565*a^6*c^2*d^3*e^17 + 3465*a^7*c*d*e^19)*e^(-7)/(c^7*d^7)) -
99/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^
6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2
*e^14 + a^8*e^16)*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^7*d^7)