3.2025 \(\int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=295 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d} \]

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(15015*c^5
*d^5*(d + e*x)^(5/2)) + (128*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(5/2))/(3003*c^4*d^4*(d + e*x)^(3/2)) + (32*(c*d^2 - a*e^2)^2*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(429*c^3*d^3*Sqrt[d + e*x]) + (16*(c*d^2 - a
*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(143*c^2*d^2)
+ (2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d)

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Rubi [A]  time = 0.706701, 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 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),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)^(5/2))/(15015*c^5
*d^5*(d + e*x)^(5/2)) + (128*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(5/2))/(3003*c^4*d^4*(d + e*x)^(3/2)) + (32*(c*d^2 - a*e^2)^2*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(429*c^3*d^3*Sqrt[d + e*x]) + (16*(c*d^2 - a
*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(143*c^2*d^2)
+ (2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d)

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Rubi in Sympy [A]  time = 105.818, size = 279, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{13 c d} - \frac{16 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{143 c^{2} d^{2}} + \frac{32 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{429 c^{3} d^{3} \sqrt{d + e x}} - \frac{128 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{3003 c^{4} d^{4} \left (d + e x\right )^{\frac{3}{2}}} + \frac{256 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{15015 c^{5} d^{5} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**(3/2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(13*c*d) -
16*sqrt(d + e*x)*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(
5/2)/(143*c**2*d**2) + 32*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))**(5/2)/(429*c**3*d**3*sqrt(d + e*x)) - 128*(a*e**2 - c*d**2)**3*(a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(3003*c**4*d**4*(d + e*x)**(3/2)) +
256*(a*e**2 - c*d**2)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(1501
5*c**5*d**5*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.289004, size = 187, normalized size = 0.63 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (128 a^4 e^8-64 a^3 c d e^6 (13 d+5 e x)+16 a^2 c^2 d^2 e^4 \left (143 d^2+130 d e x+35 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (429 d^3+715 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right )+c^4 d^4 \left (3003 d^4+8580 d^3 e x+10010 d^2 e^2 x^2+5460 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(13*d + 5*e*x)
+ 16*a^2*c^2*d^2*e^4*(143*d^2 + 130*d*e*x + 35*e^2*x^2) - 8*a*c^3*d^3*e^2*(429*d
^3 + 715*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3) + c^4*d^4*(3003*d^4 + 8580*d^3*e
*x + 10010*d^2*e^2*x^2 + 5460*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*c^5*d^5*(d + e*
x)^(5/2))

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Maple [A]  time = 0.01, size = 243, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1155\,{x}^{4}{c}^{4}{d}^{4}{e}^{4}-840\,{x}^{3}a{c}^{3}{d}^{3}{e}^{5}+5460\,{x}^{3}{c}^{4}{d}^{5}{e}^{3}+560\,{x}^{2}{a}^{2}{c}^{2}{d}^{2}{e}^{6}-3640\,{x}^{2}a{c}^{3}{d}^{4}{e}^{4}+10010\,{x}^{2}{c}^{4}{d}^{6}{e}^{2}-320\,x{a}^{3}cd{e}^{7}+2080\,x{a}^{2}{c}^{2}{d}^{3}{e}^{5}-5720\,xa{c}^{3}{d}^{5}{e}^{3}+8580\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-832\,{a}^{3}c{d}^{2}{e}^{6}+2288\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-3432\,a{c}^{3}{d}^{6}{e}^{2}+3003\,{c}^{4}{d}^{8} \right ) }{15015\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(c*d*x+a*e)*(1155*c^4*d^4*e^4*x^4-840*a*c^3*d^3*e^5*x^3+5460*c^4*d^5*e^3
*x^3+560*a^2*c^2*d^2*e^6*x^2-3640*a*c^3*d^4*e^4*x^2+10010*c^4*d^6*e^2*x^2-320*a^
3*c*d*e^7*x+2080*a^2*c^2*d^3*e^5*x-5720*a*c^3*d^5*e^3*x+8580*c^4*d^7*e*x+128*a^4
*e^8-832*a^3*c*d^2*e^6+2288*a^2*c^2*d^4*e^4-3432*a*c^3*d^6*e^2+3003*c^4*d^8)*(c*
d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/c^5/d^5/(e*x+d)^(3/2)

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Maxima [A]  time = 0.764775, size = 504, normalized size = 1.71 \[ \frac{2 \,{\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \,{\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \,{\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \,{\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \,{\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \,{\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{15015 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

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)^(3/2)*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^4 + 22
88*a^4*c^2*d^4*e^6 - 832*a^5*c*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6*d^7*e^3 + 7*
a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4 + a^2*c^4*d^4*e^6)*
x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4*d^5*e^5 - 2*a^3*c^3*d^3
*e^7)*x^3 + 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^
3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2 + 2*(3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*
e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^2*d^3*e^7 - 32*a^5*c*d*e^9)*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 = 0.223881, size = 753, normalized size = 2.55 \[ \frac{2 \,{\left (1155 \, c^{7} d^{7} e^{5} x^{8} + 3003 \, a^{3} c^{4} d^{9} e^{3} - 3432 \, a^{4} c^{3} d^{7} e^{5} + 2288 \, a^{5} c^{2} d^{5} e^{7} - 832 \, a^{6} c d^{3} e^{9} + 128 \, a^{7} d e^{11} + 105 \,{\left (63 \, c^{7} d^{8} e^{4} + 25 \, a c^{6} d^{6} e^{6}\right )} x^{7} + 35 \,{\left (442 \, c^{7} d^{9} e^{3} + 439 \, a c^{6} d^{7} e^{5} + 43 \, a^{2} c^{5} d^{5} e^{7}\right )} x^{6} + 5 \,{\left (3718 \, c^{7} d^{10} e^{2} + 7410 \, a c^{6} d^{8} e^{4} + 1809 \, a^{2} c^{5} d^{6} e^{6} - a^{3} c^{4} d^{4} e^{8}\right )} x^{5} +{\left (11583 \, c^{7} d^{11} e + 46618 \, a c^{6} d^{9} e^{3} + 22698 \, a^{2} c^{5} d^{7} e^{5} - 57 \, a^{3} c^{4} d^{5} e^{7} + 8 \, a^{4} c^{3} d^{3} e^{9}\right )} x^{4} +{\left (3003 \, c^{7} d^{12} + 31317 \, a c^{6} d^{10} e^{2} + 30602 \, a^{2} c^{5} d^{8} e^{4} - 338 \, a^{3} c^{4} d^{6} e^{6} + 112 \, a^{4} c^{3} d^{4} e^{8} - 16 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} +{\left (9009 \, a c^{6} d^{11} e + 24453 \, a^{2} c^{5} d^{9} e^{3} - 2002 \, a^{3} c^{4} d^{7} e^{5} + 1248 \, a^{4} c^{3} d^{5} e^{7} - 432 \, a^{5} c^{2} d^{3} e^{9} + 64 \, a^{6} c d e^{11}\right )} x^{2} +{\left (9009 \, a^{2} c^{5} d^{10} e^{2} + 1287 \, a^{3} c^{4} d^{8} e^{4} - 2288 \, a^{4} c^{3} d^{6} e^{6} + 1872 \, a^{5} c^{2} d^{4} e^{8} - 768 \, a^{6} c d^{2} e^{10} + 128 \, a^{7} e^{12}\right )} x\right )}}{15015 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{5} d^{5}} \]

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)^(3/2)*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/15015*(1155*c^7*d^7*e^5*x^8 + 3003*a^3*c^4*d^9*e^3 - 3432*a^4*c^3*d^7*e^5 + 22
88*a^5*c^2*d^5*e^7 - 832*a^6*c*d^3*e^9 + 128*a^7*d*e^11 + 105*(63*c^7*d^8*e^4 +
25*a*c^6*d^6*e^6)*x^7 + 35*(442*c^7*d^9*e^3 + 439*a*c^6*d^7*e^5 + 43*a^2*c^5*d^5
*e^7)*x^6 + 5*(3718*c^7*d^10*e^2 + 7410*a*c^6*d^8*e^4 + 1809*a^2*c^5*d^6*e^6 - a
^3*c^4*d^4*e^8)*x^5 + (11583*c^7*d^11*e + 46618*a*c^6*d^9*e^3 + 22698*a^2*c^5*d^
7*e^5 - 57*a^3*c^4*d^5*e^7 + 8*a^4*c^3*d^3*e^9)*x^4 + (3003*c^7*d^12 + 31317*a*c
^6*d^10*e^2 + 30602*a^2*c^5*d^8*e^4 - 338*a^3*c^4*d^6*e^6 + 112*a^4*c^3*d^4*e^8
- 16*a^5*c^2*d^2*e^10)*x^3 + (9009*a*c^6*d^11*e + 24453*a^2*c^5*d^9*e^3 - 2002*a
^3*c^4*d^7*e^5 + 1248*a^4*c^3*d^5*e^7 - 432*a^5*c^2*d^3*e^9 + 64*a^6*c*d*e^11)*x
^2 + (9009*a^2*c^5*d^10*e^2 + 1287*a^3*c^4*d^8*e^4 - 2288*a^4*c^3*d^6*e^6 + 1872
*a^5*c^2*d^4*e^8 - 768*a^6*c*d^2*e^10 + 128*a^7*e^12)*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)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),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)^(3/2)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Timed out