3.1934 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac{12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + e*x)^
10) + (12*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d^2 - a*e^2
)^2*(d + e*x)^9) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(4
29*(c*d^2 - a*e^2)^3*(d + e*x)^8) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(3003*(c*d^2 - a*e^2)^4*(d + e*x)^7)

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Rubi [A]  time = 0.405157, 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 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac{12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + e*x)^
10) + (12*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d^2 - a*e^2
)^2*(d + e*x)^9) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(4
29*(c*d^2 - a*e^2)^3*(d + e*x)^8) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(3003*(c*d^2 - a*e^2)^4*(d + e*x)^7)

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Rubi in Sympy [A]  time = 80.0609, size = 218, normalized size = 0.94 \[ \frac{32 c^{3} d^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{3003 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{429 \left (d + e x\right )^{8} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{12 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{143 \left (d + e x\right )^{9} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{13 \left (d + e x\right )^{10} \left (a e^{2} - c d^{2}\right )} \]

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)**(5/2)/(e*x+d)**10,x)

[Out]

32*c**3*d**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(3003*(d + e*x)**
7*(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)
)**(7/2)/(429*(d + e*x)**8*(a*e**2 - c*d**2)**3) + 12*c*d*(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))**(7/2)/(143*(d + e*x)**9*(a*e**2 - c*d**2)**2) - 2*(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(13*(d + e*x)**10*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.250451, size = 148, normalized size = 0.64 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (-231 a^3 e^6+63 a^2 c d e^4 (13 d+2 e x)-7 a c^2 d^2 e^2 \left (143 d^2+52 d e x+8 e^2 x^2\right )+c^3 d^3 \left (429 d^3+286 d^2 e x+104 d e^2 x^2+16 e^3 x^3\right )\right )}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^6 + 63*a^2*c*d*e^4*
(13*d + 2*e*x) - 7*a*c^2*d^2*e^2*(143*d^2 + 52*d*e*x + 8*e^2*x^2) + c^3*d^3*(429
*d^3 + 286*d^2*e*x + 104*d*e^2*x^2 + 16*e^3*x^3)))/(3003*(c*d^2 - a*e^2)^4*(d +
e*x)^7)

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Maple [A]  time = 0.016, size = 217, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+56\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-104\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-126\,x{a}^{2}cd{e}^{5}+364\,xa{c}^{2}{d}^{3}{e}^{3}-286\,{c}^{3}{d}^{5}ex+231\,{a}^{3}{e}^{6}-819\,{a}^{2}c{d}^{2}{e}^{4}+1001\,{c}^{2}{d}^{4}a{e}^{2}-429\,{c}^{3}{d}^{6} \right ) }{3003\, \left ( ex+d \right ) ^{9} \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2)/(e*x+d)^10,x)

[Out]

-2/3003*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+56*a*c^2*d^2*e^4*x^2-104*c^3*d^4*e^2*x^
2-126*a^2*c*d*e^5*x+364*a*c^2*d^3*e^3*x-286*c^3*d^5*e*x+231*a^3*e^6-819*a^2*c*d^
2*e^4+1001*a*c^2*d^4*e^2-429*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e
*x+d)^9/(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)

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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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 34.0528, size = 1111, normalized size = 4.81 \[ \text{result too large to display} \]

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)^10,x, algorithm="fricas")

[Out]

2/3003*(16*c^6*d^6*e^3*x^6 + 429*a^3*c^3*d^6*e^3 - 1001*a^4*c^2*d^4*e^5 + 819*a^
5*c*d^2*e^7 - 231*a^6*e^9 + 8*(13*c^6*d^7*e^2 - a*c^5*d^5*e^4)*x^5 + 2*(143*c^6*
d^8*e - 26*a*c^5*d^6*e^3 + 3*a^2*c^4*d^4*e^5)*x^4 + (429*c^6*d^9 - 143*a*c^5*d^7
*e^2 + 39*a^2*c^4*d^5*e^4 - 5*a^3*c^3*d^3*e^6)*x^3 + (1287*a*c^5*d^8*e - 2145*a^
2*c^4*d^6*e^3 + 1469*a^3*c^3*d^4*e^5 - 371*a^4*c^2*d^2*e^7)*x^2 + (1287*a^2*c^4*
d^7*e^2 - 2717*a^3*c^3*d^5*e^4 + 2093*a^4*c^2*d^3*e^6 - 567*a^5*c*d*e^8)*x)*sqrt
(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^4*d^15 - 4*a*c^3*d^13*e^2 + 6*a^2*c^2
*d^11*e^4 - 4*a^3*c*d^9*e^6 + a^4*d^7*e^8 + (c^4*d^8*e^7 - 4*a*c^3*d^6*e^9 + 6*a
^2*c^2*d^4*e^11 - 4*a^3*c*d^2*e^13 + a^4*e^15)*x^7 + 7*(c^4*d^9*e^6 - 4*a*c^3*d^
7*e^8 + 6*a^2*c^2*d^5*e^10 - 4*a^3*c*d^3*e^12 + a^4*d*e^14)*x^6 + 21*(c^4*d^10*e
^5 - 4*a*c^3*d^8*e^7 + 6*a^2*c^2*d^6*e^9 - 4*a^3*c*d^4*e^11 + a^4*d^2*e^13)*x^5
+ 35*(c^4*d^11*e^4 - 4*a*c^3*d^9*e^6 + 6*a^2*c^2*d^7*e^8 - 4*a^3*c*d^5*e^10 + a^
4*d^3*e^12)*x^4 + 35*(c^4*d^12*e^3 - 4*a*c^3*d^10*e^5 + 6*a^2*c^2*d^8*e^7 - 4*a^
3*c*d^6*e^9 + a^4*d^4*e^11)*x^3 + 21*(c^4*d^13*e^2 - 4*a*c^3*d^11*e^4 + 6*a^2*c^
2*d^9*e^6 - 4*a^3*c*d^7*e^8 + a^4*d^5*e^10)*x^2 + 7*(c^4*d^14*e - 4*a*c^3*d^12*e
^3 + 6*a^2*c^2*d^10*e^5 - 4*a^3*c*d^8*e^7 + a^4*d^6*e^9)*x)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**10,x)

[Out]

Timed out

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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)^(5/2)/(e*x + d)^10,x, algorithm="giac")

[Out]

Timed out