3.1922 \(\int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=414 \[ -\frac{45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac{45 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^3}-\frac{15 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d} \]

[Out]

(45*(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^5*d^5*e^3) - (15*(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^4*d^4*e^2) + (3*(c
*d^2 - a*e^2)^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))/(128*c^3*d^3*e) + (9*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(112*c^2*d^2) + ((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(7/2))/(8*c*d) - (45*(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^
(11/2)*d^(11/2)*e^(7/2))

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Rubi [A]  time = 0.863802, antiderivative size = 414, 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 \[ -\frac{45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac{45 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^3}-\frac{15 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d} \]

Antiderivative was successfully verified.

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

[Out]

(45*(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^5*d^5*e^3) - (15*(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^4*d^4*e^2) + (3*(c
*d^2 - a*e^2)^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))/(128*c^3*d^3*e) + (9*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(112*c^2*d^2) + ((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(7/2))/(8*c*d) - (45*(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^
(11/2)*d^(11/2)*e^(7/2))

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Rubi in Sympy [A]  time = 117.859, size = 401, normalized size = 0.97 \[ \frac{\left (d + e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{8 c d} - \frac{9 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{112 c^{2} d^{2}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{128 c^{3} d^{3} e} - \frac{15 \left (a e^{2} - c d^{2}\right )^{4} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{2048 c^{4} d^{4} e^{2}} + \frac{45 \left (a e^{2} - c d^{2}\right )^{6} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{16384 c^{5} d^{5} e^{3}} - \frac{45 \left (a e^{2} - c d^{2}\right )^{8} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{32768 c^{\frac{11}{2}} d^{\frac{11}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(8*c*d) - 9*(a*e**2
- c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(112*c**2*d**2) + 3*
(a*e**2 - c*d**2)**2*(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)/(128*c**3*d**3*e) - 15*(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**4*d**4
*e**2) + 45*(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**5*d**5*e**3) - 45*(a*e**2 - c*d**2)**8*a
tanh((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**(11/2)*d**(11/2)*e**(7/2))

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Mathematica [A]  time = 2.02257, size = 539, normalized size = 1.3 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{2 \left (315 a^7 e^{14}-105 a^6 c d e^{12} (23 d+2 e x)+21 a^5 c^2 d^2 e^{10} \left (383 d^2+76 d e x+8 e^2 x^2\right )-3 a^4 c^3 d^3 e^8 \left (5053 d^3+1754 d^2 e x+424 d e^2 x^2+48 e^3 x^3\right )+a^3 c^4 d^4 e^6 \left (17609 d^4+9800 d^3 e x+4176 d^2 e^2 x^2+1088 d e^3 x^3+128 e^4 x^4\right )+3 a^2 c^5 d^5 e^4 \left (2681 d^5+31014 d^4 e x+66928 d^3 e^2 x^2+68320 d^2 e^3 x^3+34432 d e^4 x^4+6912 e^5 x^5\right )+3 a c^6 d^6 e^2 \left (-805 d^6+532 d^5 e x+32344 d^4 e^2 x^2+87744 d^3 e^3 x^3+99968 d^2 e^4 x^4+53760 d e^5 x^5+11264 e^6 x^6\right )+c^7 d^7 \left (315 d^7-210 d^6 e x+168 d^5 e^2 x^2+32624 d^4 e^3 x^3+98432 d^3 e^4 x^4+119040 d^2 e^5 x^5+66560 d e^6 x^6+14336 e^7 x^7\right )\right )}{7 c^5 d^5 e^3 (d+e x)^2 (a e+c d x)^2}-\frac{45 \left (c d^2-a e^2\right )^8 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{11/2} d^{11/2} e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right )}{32768} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2*(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*(315*a^7*e^14 - 105*a^6*c*d*e^12*(23*d + 2*
e*x) + 21*a^5*c^2*d^2*e^10*(383*d^2 + 76*d*e*x + 8*e^2*x^2) - 3*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) + 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) + 3*a^2*c^
5*d^5*e^4*(2681*d^5 + 31014*d^4*e*x + 66928*d^3*e^2*x^2 + 68320*d^2*e^3*x^3 + 34
432*d*e^4*x^4 + 6912*e^5*x^5) + 3*a*c^6*d^6*e^2*(-805*d^6 + 532*d^5*e*x + 32344*
d^4*e^2*x^2 + 87744*d^3*e^3*x^3 + 99968*d^2*e^4*x^4 + 53760*d*e^5*x^5 + 11264*e^
6*x^6) + c^7*d^7*(315*d^7 - 210*d^6*e*x + 168*d^5*e^2*x^2 + 32624*d^4*e^3*x^3 +
98432*d^3*e^4*x^4 + 119040*d^2*e^5*x^5 + 66560*d*e^6*x^6 + 14336*e^7*x^7)))/(7*c
^5*d^5*e^3*(a*e + c*d*x)^2*(d + e*x)^2) - (45*(c*d^2 - a*e^2)^8*Log[a*e^2 + 2*Sq
rt[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^(7/2)*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/32768

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Maple [B]  time = 0.018, size = 2044, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/128*d^3/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+45/2048*d^4*(a*e*d+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)*a+3/64*d^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+45/
4096*d^9/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-45/512*d*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*x*a^2+3/64*e^4/d^2/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a
^2+45/8192*e^10/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6+15/256*e^5
*a^3/c^2/d*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-135/4096*e^8/d^2/c^3*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5-45/32768*e^13/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^8+45/4096*e^11/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^7-15/1024*e^7/d^3/c^3*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4-225/2048*d^2*e^4/c*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*x*a^3+315/4096*d*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^5-1575/1638
4*d^3*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^4-315/8192*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)*a^2
-315/8192/d*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^6-135/4096*d^6*c*(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*x*a+15/256*d^3*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*
a+23/112/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+315/4096*d^5*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^3-225/16384*d^7/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-225/16
384*d^3*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+45/8192*d^8/e^2*c^2*(a
*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+405/16384/d*e^7/c^3*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*a^5-45/32768*d^11/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)-225/16384*d*
e^5/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-3/128*d*e/c*(a*e*d+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(5/2)*a+675/8192*d^4*e^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2)*x*a^2-3/128*e^3/d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2+45/2048*e^6
/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-9/112*e^2/d^2/c^2*(a*e*d+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a+3/128*e^5/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)*a^3-15/2048*e^8/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5+4
5/16384*e^11/d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7-225/16384*e^9/d
^3/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6+1/8*e*x*(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(7/2)/d/c-3/32*e^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a-15
/1024*d^5/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-15/1024*d^2*e^2/c*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+675/8192*e^6/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*a^4-15/2048*d^6/e^2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+4
05/16384*d^5*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-15/1024*e^4/c^2*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+45/16384*d^9/e^3*c^2*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.370829, 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)^2,x, algorithm="fricas")

[Out]

[1/458752*(4*(14336*c^7*d^7*e^7*x^7 + 315*c^7*d^14 - 2415*a*c^6*d^12*e^2 + 8043*
a^2*c^5*d^10*e^4 + 17609*a^3*c^4*d^8*e^6 - 15159*a^4*c^3*d^6*e^8 + 8043*a^5*c^2*
d^4*e^10 - 2415*a^6*c*d^2*e^12 + 315*a^7*e^14 + 1024*(65*c^7*d^8*e^6 + 33*a*c^6*
d^6*e^8)*x^6 + 768*(155*c^7*d^9*e^5 + 210*a*c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)*x^
5 + 128*(769*c^7*d^10*e^4 + 2343*a*c^6*d^8*e^6 + 807*a^2*c^5*d^6*e^8 + a^3*c^4*d
^4*e^10)*x^4 + 16*(2039*c^7*d^11*e^3 + 16452*a*c^6*d^9*e^5 + 12810*a^2*c^5*d^7*e
^7 + 68*a^3*c^4*d^5*e^9 - 9*a^4*c^3*d^3*e^11)*x^3 + 24*(7*c^7*d^12*e^2 + 4043*a*
c^6*d^10*e^4 + 8366*a^2*c^5*d^8*e^6 + 174*a^3*c^4*d^6*e^8 - 53*a^4*c^3*d^4*e^10
+ 7*a^5*c^2*d^2*e^12)*x^2 - 2*(105*c^7*d^13*e - 798*a*c^6*d^11*e^3 - 46521*a^2*c
^5*d^9*e^5 - 4900*a^3*c^4*d^7*e^7 + 2631*a^4*c^3*d^5*e^9 - 798*a^5*c^2*d^3*e^11
+ 105*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) +
315*(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^1
4 + 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^5*d^5*e^3), 1/22
9376*(2*(14336*c^7*d^7*e^7*x^7 + 315*c^7*d^14 - 2415*a*c^6*d^12*e^2 + 8043*a^2*c
^5*d^10*e^4 + 17609*a^3*c^4*d^8*e^6 - 15159*a^4*c^3*d^6*e^8 + 8043*a^5*c^2*d^4*e
^10 - 2415*a^6*c*d^2*e^12 + 315*a^7*e^14 + 1024*(65*c^7*d^8*e^6 + 33*a*c^6*d^6*e
^8)*x^6 + 768*(155*c^7*d^9*e^5 + 210*a*c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)*x^5 + 1
28*(769*c^7*d^10*e^4 + 2343*a*c^6*d^8*e^6 + 807*a^2*c^5*d^6*e^8 + a^3*c^4*d^4*e^
10)*x^4 + 16*(2039*c^7*d^11*e^3 + 16452*a*c^6*d^9*e^5 + 12810*a^2*c^5*d^7*e^7 +
68*a^3*c^4*d^5*e^9 - 9*a^4*c^3*d^3*e^11)*x^3 + 24*(7*c^7*d^12*e^2 + 4043*a*c^6*d
^10*e^4 + 8366*a^2*c^5*d^8*e^6 + 174*a^3*c^4*d^6*e^8 - 53*a^4*c^3*d^4*e^10 + 7*a
^5*c^2*d^2*e^12)*x^2 - 2*(105*c^7*d^13*e - 798*a*c^6*d^11*e^3 - 46521*a^2*c^5*d^
9*e^5 - 4900*a^3*c^4*d^7*e^7 + 2631*a^4*c^3*d^5*e^9 - 798*a^5*c^2*d^3*e^11 + 105
*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) - 315
*(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)*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^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)**2*(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 = 0.257778, size = 1006, normalized size = 2.43 \[ \text{result too large to display} \]

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)^2,x, algorithm="giac")

[Out]

1/114688*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(4*(14*c^2*d
^2*x*e^4 + (65*c^9*d^10*e^10 + 33*a*c^8*d^8*e^12)*e^(-7)/(c^7*d^7))*x + 3*(155*c
^9*d^11*e^9 + 210*a*c^8*d^9*e^11 + 27*a^2*c^7*d^7*e^13)*e^(-7)/(c^7*d^7))*x + (7
69*c^9*d^12*e^8 + 2343*a*c^8*d^10*e^10 + 807*a^2*c^7*d^8*e^12 + a^3*c^6*d^6*e^14
)*e^(-7)/(c^7*d^7))*x + (2039*c^9*d^13*e^7 + 16452*a*c^8*d^11*e^9 + 12810*a^2*c^
7*d^9*e^11 + 68*a^3*c^6*d^7*e^13 - 9*a^4*c^5*d^5*e^15)*e^(-7)/(c^7*d^7))*x + 3*(
7*c^9*d^14*e^6 + 4043*a*c^8*d^12*e^8 + 8366*a^2*c^7*d^10*e^10 + 174*a^3*c^6*d^8*
e^12 - 53*a^4*c^5*d^6*e^14 + 7*a^5*c^4*d^4*e^16)*e^(-7)/(c^7*d^7))*x - (105*c^9*
d^15*e^5 - 798*a*c^8*d^13*e^7 - 46521*a^2*c^7*d^11*e^9 - 4900*a^3*c^6*d^9*e^11 +
2631*a^4*c^5*d^7*e^13 - 798*a^5*c^4*d^5*e^15 + 105*a^6*c^3*d^3*e^17)*e^(-7)/(c^
7*d^7))*x + (315*c^9*d^16*e^4 - 2415*a*c^8*d^14*e^6 + 8043*a^2*c^7*d^12*e^8 + 17
609*a^3*c^6*d^10*e^10 - 15159*a^4*c^5*d^8*e^12 + 8043*a^5*c^4*d^6*e^14 - 2415*a^
6*c^3*d^4*e^16 + 315*a^7*c^2*d^2*e^18)*e^(-7)/(c^7*d^7)) + 45/32768*(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)*sqr
t(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^6*d^6)