3.2036 \(\int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=233 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}} \]

[Out]

(32*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*c^4*d
^4*(d + e*x)^(7/2)) + (16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(7/2))/(429*c^3*d^3*(d + e*x)^(5/2)) + (12*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(143*c^2*d^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(7/2))/(13*c*d*Sqrt[d + e*x])

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Rubi [A]  time = 0.501685, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(32*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*c^4*d
^4*(d + e*x)^(7/2)) + (16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(7/2))/(429*c^3*d^3*(d + e*x)^(5/2)) + (12*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(143*c^2*d^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(7/2))/(13*c*d*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 81.3588, size = 219, normalized size = 0.94 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{13 c d \sqrt{d + e x}} - \frac{12 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{143 c^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{429 c^{3} d^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{32 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{3003 c^{4} d^{4} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(13*c*d*sqrt(d + e*x)) - 12*
(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(143*c**2*d*
*2*(d + e*x)**(3/2)) + 16*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))**(7/2)/(429*c**3*d**3*(d + e*x)**(5/2)) - 32*(a*e**2 - c*d**2)**3*(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(3003*c**4*d**4*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.217789, size = 142, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (143 d^2+182 d e x+63 e^2 x^2\right )+c^3 d^3 \left (429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(1
3*d + 7*e*x) - 2*a*c^2*d^2*e^2*(143*d^2 + 182*d*e*x + 63*e^2*x^2) + c^3*d^3*(429
*d^3 + 1001*d^2*e*x + 819*d*e^2*x^2 + 231*e^3*x^3)))/(3003*c^4*d^4*Sqrt[d + e*x]
)

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Maple [A]  time = 0.012, size = 168, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -231\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+126\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-819\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-56\,x{a}^{2}cd{e}^{5}+364\,xa{c}^{2}{d}^{3}{e}^{3}-1001\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-104\,{a}^{2}c{d}^{2}{e}^{4}+286\,{c}^{2}{d}^{4}a{e}^{2}-429\,{c}^{3}{d}^{6} \right ) }{3003\,{c}^{4}{d}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.799875, size = 452, normalized size = 1.94 \[ \frac{2 \,{\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \,{\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \,{\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} +{\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \,{\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{3003 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]

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)^(5/2)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/3003*(231*c^6*d^6*e^3*x^6 + 429*a^3*c^3*d^6*e^3 - 286*a^4*c^2*d^4*e^5 + 104*a^
5*c*d^2*e^7 - 16*a^6*e^9 + 63*(13*c^6*d^7*e^2 + 9*a*c^5*d^5*e^4)*x^5 + 7*(143*c^
6*d^8*e + 299*a*c^5*d^6*e^3 + 53*a^2*c^4*d^4*e^5)*x^4 + (429*c^6*d^9 + 2717*a*c^
5*d^7*e^2 + 1469*a^2*c^4*d^5*e^4 + 5*a^3*c^3*d^3*e^6)*x^3 + 3*(429*a*c^5*d^8*e +
715*a^2*c^4*d^6*e^3 + 13*a^3*c^3*d^4*e^5 - 2*a^4*c^2*d^2*e^7)*x^2 + (1287*a^2*c
^4*d^7*e^2 + 143*a^3*c^3*d^5*e^4 - 52*a^4*c^2*d^3*e^6 + 8*a^5*c*d*e^8)*x)*sqrt(c
*d*x + a*e)*(e*x + d)/(c^4*d^4*e*x + c^4*d^5)

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Fricas [A]  time = 0.214258, size = 687, normalized size = 2.95 \[ \frac{2 \,{\left (231 \, c^{7} d^{7} e^{4} x^{8} + 429 \, a^{4} c^{3} d^{7} e^{4} - 286 \, a^{5} c^{2} d^{5} e^{6} + 104 \, a^{6} c d^{3} e^{8} - 16 \, a^{7} d e^{10} + 42 \,{\left (25 \, c^{7} d^{8} e^{3} + 19 \, a c^{6} d^{6} e^{5}\right )} x^{7} + 14 \,{\left (130 \, c^{7} d^{9} e^{2} + 265 \, a c^{6} d^{7} e^{4} + 67 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{6} + 2 \,{\left (715 \, c^{7} d^{10} e + 3315 \, a c^{6} d^{8} e^{3} + 2250 \, a^{2} c^{5} d^{6} e^{5} + 188 \, a^{3} c^{4} d^{4} e^{7}\right )} x^{5} +{\left (429 \, c^{7} d^{11} + 5434 \, a c^{6} d^{9} e^{2} + 8424 \, a^{2} c^{5} d^{7} e^{4} + 1884 \, a^{3} c^{4} d^{5} e^{6} - a^{4} c^{3} d^{3} e^{8}\right )} x^{4} + 2 \,{\left (858 \, a c^{6} d^{10} e + 3718 \, a^{2} c^{5} d^{8} e^{3} + 1898 \, a^{3} c^{4} d^{6} e^{5} - 7 \, a^{4} c^{3} d^{4} e^{7} + a^{5} c^{2} d^{2} e^{9}\right )} x^{3} + 2 \,{\left (1287 \, a^{2} c^{5} d^{9} e^{2} + 2002 \, a^{3} c^{4} d^{7} e^{4} - 78 \, a^{4} c^{3} d^{5} e^{6} + 27 \, a^{5} c^{2} d^{3} e^{8} - 4 \, a^{6} c d e^{10}\right )} x^{2} + 2 \,{\left (858 \, a^{3} c^{4} d^{8} e^{3} + 143 \, a^{4} c^{3} d^{6} e^{5} - 117 \, a^{5} c^{2} d^{4} e^{7} + 48 \, a^{6} c d^{2} e^{9} - 8 \, a^{7} e^{11}\right )} x\right )}}{3003 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

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)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/3003*(231*c^7*d^7*e^4*x^8 + 429*a^4*c^3*d^7*e^4 - 286*a^5*c^2*d^5*e^6 + 104*a^
6*c*d^3*e^8 - 16*a^7*d*e^10 + 42*(25*c^7*d^8*e^3 + 19*a*c^6*d^6*e^5)*x^7 + 14*(1
30*c^7*d^9*e^2 + 265*a*c^6*d^7*e^4 + 67*a^2*c^5*d^5*e^6)*x^6 + 2*(715*c^7*d^10*e
+ 3315*a*c^6*d^8*e^3 + 2250*a^2*c^5*d^6*e^5 + 188*a^3*c^4*d^4*e^7)*x^5 + (429*c
^7*d^11 + 5434*a*c^6*d^9*e^2 + 8424*a^2*c^5*d^7*e^4 + 1884*a^3*c^4*d^5*e^6 - a^4
*c^3*d^3*e^8)*x^4 + 2*(858*a*c^6*d^10*e + 3718*a^2*c^5*d^8*e^3 + 1898*a^3*c^4*d^
6*e^5 - 7*a^4*c^3*d^4*e^7 + a^5*c^2*d^2*e^9)*x^3 + 2*(1287*a^2*c^5*d^9*e^2 + 200
2*a^3*c^4*d^7*e^4 - 78*a^4*c^3*d^5*e^6 + 27*a^5*c^2*d^3*e^8 - 4*a^6*c*d*e^10)*x^
2 + 2*(858*a^3*c^4*d^8*e^3 + 143*a^4*c^3*d^6*e^5 - 117*a^5*c^2*d^4*e^7 + 48*a^6*
c*d^2*e^9 - 8*a^7*e^11)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x
+ d)*c^4*d^4)

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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)**(1/2)*(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 [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)*sqrt(e*x + d),x, algorithm="giac")

[Out]

Timed out