3.2035 \(\int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=295 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d} \]

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(45045*c^5
*d^5*(d + e*x)^(7/2)) + (128*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(7/2))/(6435*c^4*d^4*(d + e*x)^(5/2)) + (32*(c*d^2 - a*e^2)^2*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(715*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d^2 -
a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(195*c^2*d^2*Sqrt[d + e*x
]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d)

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Rubi [A]  time = 0.704236, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(45045*c^5
*d^5*(d + e*x)^(7/2)) + (128*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(7/2))/(6435*c^4*d^4*(d + e*x)^(5/2)) + (32*(c*d^2 - a*e^2)^2*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(715*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d^2 -
a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(195*c^2*d^2*Sqrt[d + e*x
]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d)

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Rubi in Sympy [A]  time = 105.791, size = 279, normalized size = 0.95 \[ \frac{2 \sqrt{d + e x} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{15 c d} - \frac{16 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{195 c^{2} d^{2} \sqrt{d + e x}} + \frac{32 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{715 c^{3} d^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{128 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{6435 c^{4} d^{4} \left (d + e x\right )^{\frac{5}{2}}} + \frac{256 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{45045 c^{5} d^{5} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

2*sqrt(d + e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(15*c*d) - 16*
(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(195*c**2*d*
*2*sqrt(d + e*x)) + 32*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))**(7/2)/(715*c**3*d**3*(d + e*x)**(3/2)) - 128*(a*e**2 - c*d**2)**3*(a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(6435*c**4*d**4*(d + e*x)**(5/2)) +
256*(a*e**2 - c*d**2)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(4504
5*c**5*d**5*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.262348, size = 197, normalized size = 0.67 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (128 a^4 e^8-64 a^3 c d e^6 (15 d+7 e x)+48 a^2 c^2 d^2 e^4 \left (65 d^2+70 d e x+21 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (715 d^3+1365 d^2 e x+945 d e^2 x^2+231 e^3 x^3\right )+c^4 d^4 \left (6435 d^4+20020 d^3 e x+24570 d^2 e^2 x^2+13860 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*(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)]*(128*a^4*e^8 - 64*a^3*c*d*e^6*(
15*d + 7*e*x) + 48*a^2*c^2*d^2*e^4*(65*d^2 + 70*d*e*x + 21*e^2*x^2) - 8*a*c^3*d^
3*e^2*(715*d^3 + 1365*d^2*e*x + 945*d*e^2*x^2 + 231*e^3*x^3) + c^4*d^4*(6435*d^4
+ 20020*d^3*e*x + 24570*d^2*e^2*x^2 + 13860*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*
c^5*d^5*Sqrt[d + e*x])

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Maple [A]  time = 0.01, size = 243, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3003\,{x}^{4}{c}^{4}{d}^{4}{e}^{4}-1848\,{x}^{3}a{c}^{3}{d}^{3}{e}^{5}+13860\,{x}^{3}{c}^{4}{d}^{5}{e}^{3}+1008\,{x}^{2}{a}^{2}{c}^{2}{d}^{2}{e}^{6}-7560\,{x}^{2}a{c}^{3}{d}^{4}{e}^{4}+24570\,{x}^{2}{c}^{4}{d}^{6}{e}^{2}-448\,x{a}^{3}cd{e}^{7}+3360\,x{a}^{2}{c}^{2}{d}^{3}{e}^{5}-10920\,xa{c}^{3}{d}^{5}{e}^{3}+20020\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-960\,{a}^{3}c{d}^{2}{e}^{6}+3120\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-5720\,a{c}^{3}{d}^{6}{e}^{2}+6435\,{c}^{4}{d}^{8} \right ) }{45045\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

2/45045*(c*d*x+a*e)*(3003*c^4*d^4*e^4*x^4-1848*a*c^3*d^3*e^5*x^3+13860*c^4*d^5*e
^3*x^3+1008*a^2*c^2*d^2*e^6*x^2-7560*a*c^3*d^4*e^4*x^2+24570*c^4*d^6*e^2*x^2-448
*a^3*c*d*e^7*x+3360*a^2*c^2*d^3*e^5*x-10920*a*c^3*d^5*e^3*x+20020*c^4*d^7*e*x+12
8*a^4*e^8-960*a^3*c*d^2*e^6+3120*a^2*c^2*d^4*e^4-5720*a*c^3*d^6*e^2+6435*c^4*d^8
)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/c^5/d^5/(e*x+d)^(5/2)

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Maxima [A]  time = 0.802363, size = 605, normalized size = 2.05 \[ \frac{2 \,{\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \,{\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \,{\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \,{\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \,{\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \,{\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} +{\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{45045 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

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)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^5 + 31
20*a^5*c^2*d^4*e^7 - 960*a^6*c*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7*d^8*e^3 + 31
*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^4 + 71*a^2*c^5*d^5*e
^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636*a^2*c^5*d^6*e^5 + a^3*c^
4*d^4*e^7)*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^9*e^2 + 8814*a^2*c^5*d^7*e^4 +
60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*
c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 120*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d^2*e^9)*x^
2 + (19305*a^2*c^5*d^9*e^2 + 2860*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a
^5*c^2*d^3*e^8 - 64*a^6*c*d*e^10)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x +
c^5*d^6)

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Fricas [A]  time = 0.218499, size = 876, normalized size = 2.97 \[ \frac{2 \,{\left (3003 \, c^{8} d^{8} e^{5} x^{9} + 6435 \, a^{4} c^{4} d^{9} e^{4} - 5720 \, a^{5} c^{3} d^{7} e^{6} + 3120 \, a^{6} c^{2} d^{5} e^{8} - 960 \, a^{7} c d^{3} e^{10} + 128 \, a^{8} d e^{12} + 231 \,{\left (73 \, c^{8} d^{9} e^{4} + 44 \, a c^{7} d^{7} e^{6}\right )} x^{8} + 42 \,{\left (915 \, c^{8} d^{10} e^{3} + 1382 \, a c^{7} d^{8} e^{5} + 277 \, a^{2} c^{6} d^{6} e^{7}\right )} x^{7} + 98 \,{\left (455 \, c^{8} d^{11} e^{2} + 1380 \, a c^{7} d^{9} e^{4} + 693 \, a^{2} c^{6} d^{7} e^{6} + 46 \, a^{3} c^{5} d^{5} e^{8}\right )} x^{6} +{\left (26455 \, c^{8} d^{12} e + 161720 \, a c^{7} d^{10} e^{3} + 163140 \, a^{2} c^{6} d^{8} e^{5} + 27068 \, a^{3} c^{5} d^{6} e^{7} - 5 \, a^{4} c^{4} d^{4} e^{9}\right )} x^{5} +{\left (6435 \, c^{8} d^{13} + 100100 \, a c^{7} d^{11} e^{2} + 204100 \, a^{2} c^{6} d^{9} e^{4} + 67800 \, a^{3} c^{5} d^{7} e^{6} - 65 \, a^{4} c^{4} d^{5} e^{8} + 8 \, a^{5} c^{3} d^{3} e^{10}\right )} x^{4} + 2 \,{\left (12870 \, a c^{7} d^{12} e + 67925 \, a^{2} c^{6} d^{10} e^{3} + 45500 \, a^{3} c^{5} d^{8} e^{5} - 225 \, a^{4} c^{4} d^{6} e^{7} + 64 \, a^{5} c^{3} d^{4} e^{9} - 8 \, a^{6} c^{2} d^{2} e^{11}\right )} x^{3} + 2 \,{\left (19305 \, a^{2} c^{6} d^{11} e^{2} + 35750 \, a^{3} c^{5} d^{9} e^{4} - 1625 \, a^{4} c^{4} d^{7} e^{6} + 840 \, a^{5} c^{3} d^{5} e^{8} - 248 \, a^{6} c^{2} d^{3} e^{10} + 32 \, a^{7} c d e^{12}\right )} x^{2} +{\left (25740 \, a^{3} c^{5} d^{10} e^{3} + 3575 \, a^{4} c^{4} d^{8} e^{5} - 4160 \, a^{5} c^{3} d^{6} e^{7} + 2640 \, a^{6} c^{2} d^{4} e^{9} - 896 \, a^{7} c d^{2} e^{11} + 128 \, a^{8} e^{13}\right )} x\right )}}{45045 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{5} d^{5}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^8*d^8*e^5*x^9 + 6435*a^4*c^4*d^9*e^4 - 5720*a^5*c^3*d^7*e^6 + 31
20*a^6*c^2*d^5*e^8 - 960*a^7*c*d^3*e^10 + 128*a^8*d*e^12 + 231*(73*c^8*d^9*e^4 +
44*a*c^7*d^7*e^6)*x^8 + 42*(915*c^8*d^10*e^3 + 1382*a*c^7*d^8*e^5 + 277*a^2*c^6
*d^6*e^7)*x^7 + 98*(455*c^8*d^11*e^2 + 1380*a*c^7*d^9*e^4 + 693*a^2*c^6*d^7*e^6
+ 46*a^3*c^5*d^5*e^8)*x^6 + (26455*c^8*d^12*e + 161720*a*c^7*d^10*e^3 + 163140*a
^2*c^6*d^8*e^5 + 27068*a^3*c^5*d^6*e^7 - 5*a^4*c^4*d^4*e^9)*x^5 + (6435*c^8*d^13
+ 100100*a*c^7*d^11*e^2 + 204100*a^2*c^6*d^9*e^4 + 67800*a^3*c^5*d^7*e^6 - 65*a
^4*c^4*d^5*e^8 + 8*a^5*c^3*d^3*e^10)*x^4 + 2*(12870*a*c^7*d^12*e + 67925*a^2*c^6
*d^10*e^3 + 45500*a^3*c^5*d^8*e^5 - 225*a^4*c^4*d^6*e^7 + 64*a^5*c^3*d^4*e^9 - 8
*a^6*c^2*d^2*e^11)*x^3 + 2*(19305*a^2*c^6*d^11*e^2 + 35750*a^3*c^5*d^9*e^4 - 162
5*a^4*c^4*d^7*e^6 + 840*a^5*c^3*d^5*e^8 - 248*a^6*c^2*d^3*e^10 + 32*a^7*c*d*e^12
)*x^2 + (25740*a^3*c^5*d^10*e^3 + 3575*a^4*c^4*d^8*e^5 - 4160*a^5*c^3*d^6*e^7 +
2640*a^6*c^2*d^4*e^9 - 896*a^7*c*d^2*e^11 + 128*a^8*e^13)*x)/(sqrt(c*d*e*x^2 + a
*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^5*d^5)

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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)**(3/2)*(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)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Timed out