3.1919 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^
8) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d^2 - a*e^2)^2
*(d + e*x)^7) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*
(c*d^2 - a*e^2)^3*(d + e*x)^6) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2)^(5/2))/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.401916, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^
8) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d^2 - a*e^2)^2
*(d + e*x)^7) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*
(c*d^2 - a*e^2)^3*(d + e*x)^6) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2)^(5/2))/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 79.5066, size = 218, normalized size = 0.94 \[ \frac{32 c^{3} d^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{1155 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{231 \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{33 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{11 \left (d + e x\right )^{8} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**8,x)

[Out]

32*c**3*d**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(1155*(d + e*x)**
5*(a*e**2 - c*d**2)**4) - 16*c**2*d**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)
)**(5/2)/(231*(d + e*x)**6*(a*e**2 - c*d**2)**3) + 4*c*d*(a*d*e + c*d*e*x**2 + x
*(a*e**2 + c*d**2))**(5/2)/(33*(d + e*x)**7*(a*e**2 - c*d**2)**2) - 2*(a*d*e + c
*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(11*(d + e*x)**8*(a*e**2 - c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.247787, size = 138, normalized size = 0.6 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (-105 a^3 e^6+35 a^2 c d e^4 (11 d+2 e x)-5 a c^2 d^2 e^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )+c^3 d^3 \left (231 d^3+198 d^2 e x+88 d e^2 x^2+16 e^3 x^3\right )\right )}{1155 (d+e x)^8 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-105*a^3*e^6 + 35*a^2*c*d*e^4*(11*d + 2*e*x)
- 5*a*c^2*d^2*e^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2) + c^3*d^3*(231*d^3 + 198*d^2*
e*x + 88*d*e^2*x^2 + 16*e^3*x^3)))/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^8)

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 217, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+40\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-88\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-70\,x{a}^{2}cd{e}^{5}+220\,xa{c}^{2}{d}^{3}{e}^{3}-198\,{c}^{3}{d}^{5}ex+105\,{a}^{3}{e}^{6}-385\,{a}^{2}c{d}^{2}{e}^{4}+495\,{c}^{2}{d}^{4}a{e}^{2}-231\,{c}^{3}{d}^{6} \right ) }{1155\, \left ( ex+d \right ) ^{7} \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^8,x)

[Out]

-2/1155*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+40*a*c^2*d^2*e^4*x^2-88*c^3*d^4*e^2*x^2
-70*a^2*c*d*e^5*x+220*a*c^2*d^3*e^3*x-198*c^3*d^5*e*x+105*a^3*e^6-385*a^2*c*d^2*
e^4+495*a*c^2*d^4*e^2-231*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/(e*x+
d)^7/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)

_______________________________________________________________________________________

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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 13.9808, size = 944, normalized size = 4.09 \[ \frac{2 \,{\left (16 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 495 \, a^{3} c^{2} d^{4} e^{4} + 385 \, a^{4} c d^{2} e^{6} - 105 \, a^{5} e^{8} + 8 \,{\left (11 \, c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4}\right )} x^{4} + 2 \,{\left (99 \, c^{5} d^{7} e - 22 \, a c^{4} d^{5} e^{3} + 3 \, a^{2} c^{3} d^{3} e^{5}\right )} x^{3} +{\left (231 \, c^{5} d^{8} - 99 \, a c^{4} d^{6} e^{2} + 33 \, a^{2} c^{3} d^{4} e^{4} - 5 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (231 \, a c^{4} d^{7} e - 396 \, a^{2} c^{3} d^{5} e^{3} + 275 \, a^{3} c^{2} d^{3} e^{5} - 70 \, a^{4} c d e^{7}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{1155 \,{\left (c^{4} d^{14} - 4 \, a c^{3} d^{12} e^{2} + 6 \, a^{2} c^{2} d^{10} e^{4} - 4 \, a^{3} c d^{8} e^{6} + a^{4} d^{6} e^{8} +{\left (c^{4} d^{8} e^{6} - 4 \, a c^{3} d^{6} e^{8} + 6 \, a^{2} c^{2} d^{4} e^{10} - 4 \, a^{3} c d^{2} e^{12} + a^{4} e^{14}\right )} x^{6} + 6 \,{\left (c^{4} d^{9} e^{5} - 4 \, a c^{3} d^{7} e^{7} + 6 \, a^{2} c^{2} d^{5} e^{9} - 4 \, a^{3} c d^{3} e^{11} + a^{4} d e^{13}\right )} x^{5} + 15 \,{\left (c^{4} d^{10} e^{4} - 4 \, a c^{3} d^{8} e^{6} + 6 \, a^{2} c^{2} d^{6} e^{8} - 4 \, a^{3} c d^{4} e^{10} + a^{4} d^{2} e^{12}\right )} x^{4} + 20 \,{\left (c^{4} d^{11} e^{3} - 4 \, a c^{3} d^{9} e^{5} + 6 \, a^{2} c^{2} d^{7} e^{7} - 4 \, a^{3} c d^{5} e^{9} + a^{4} d^{3} e^{11}\right )} x^{3} + 15 \,{\left (c^{4} d^{12} e^{2} - 4 \, a c^{3} d^{10} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{6} - 4 \, a^{3} c d^{6} e^{8} + a^{4} d^{4} e^{10}\right )} x^{2} + 6 \,{\left (c^{4} d^{13} e - 4 \, a c^{3} d^{11} e^{3} + 6 \, a^{2} c^{2} d^{9} e^{5} - 4 \, a^{3} c d^{7} e^{7} + a^{4} d^{5} e^{9}\right )} x\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

2/1155*(16*c^5*d^5*e^3*x^5 + 231*a^2*c^3*d^6*e^2 - 495*a^3*c^2*d^4*e^4 + 385*a^4
*c*d^2*e^6 - 105*a^5*e^8 + 8*(11*c^5*d^6*e^2 - a*c^4*d^4*e^4)*x^4 + 2*(99*c^5*d^
7*e - 22*a*c^4*d^5*e^3 + 3*a^2*c^3*d^3*e^5)*x^3 + (231*c^5*d^8 - 99*a*c^4*d^6*e^
2 + 33*a^2*c^3*d^4*e^4 - 5*a^3*c^2*d^2*e^6)*x^2 + 2*(231*a*c^4*d^7*e - 396*a^2*c
^3*d^5*e^3 + 275*a^3*c^2*d^3*e^5 - 70*a^4*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (
c*d^2 + a*e^2)*x)/(c^4*d^14 - 4*a*c^3*d^12*e^2 + 6*a^2*c^2*d^10*e^4 - 4*a^3*c*d^
8*e^6 + a^4*d^6*e^8 + (c^4*d^8*e^6 - 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 - 4*a^
3*c*d^2*e^12 + a^4*e^14)*x^6 + 6*(c^4*d^9*e^5 - 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*
e^9 - 4*a^3*c*d^3*e^11 + a^4*d*e^13)*x^5 + 15*(c^4*d^10*e^4 - 4*a*c^3*d^8*e^6 +
6*a^2*c^2*d^6*e^8 - 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 - 4*
a*c^3*d^9*e^5 + 6*a^2*c^2*d^7*e^7 - 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^
4*d^12*e^2 - 4*a*c^3*d^10*e^4 + 6*a^2*c^2*d^8*e^6 - 4*a^3*c*d^6*e^8 + a^4*d^4*e^
10)*x^2 + 6*(c^4*d^13*e - 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^5 - 4*a^3*c*d^7*e^7
+ a^4*d^5*e^9)*x)

_______________________________________________________________________________________

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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^8,x, algorithm="giac")

[Out]

Timed out