3.2026 \(\int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=233 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d} \]

[Out]

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

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Rubi [A]  time = 0.502107, 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 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d} \]

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

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Rubi in Sympy [A]  time = 79.3828, size = 219, normalized size = 0.94 \[ \frac{2 \sqrt{d + e x} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{11 c d} - \frac{4 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{33 c^{2} d^{2} \sqrt{d + e x}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{231 c^{3} d^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{32 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{1155 c^{4} d^{4} \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.217892, size = 132, normalized size = 0.57 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (-16 a^3 e^6+8 a^2 c d e^4 (11 d+5 e x)-2 a c^2 d^2 e^2 \left (99 d^2+110 d e x+35 e^2 x^2\right )+c^3 d^3 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \]

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)^(3/2),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(11*d + 5*e*x) -
2*a*c^2*d^2*e^2*(99*d^2 + 110*d*e*x + 35*e^2*x^2) + c^3*d^3*(231*d^3 + 495*d^2*
e*x + 385*d*e^2*x^2 + 105*e^3*x^3)))/(1155*c^4*d^4*(d + e*x)^(5/2))

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Maple [A]  time = 0.009, size = 168, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -105\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+70\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-385\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-40\,x{a}^{2}cd{e}^{5}+220\,xa{c}^{2}{d}^{3}{e}^{3}-495\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-88\,{a}^{2}c{d}^{2}{e}^{4}+198\,{c}^{2}{d}^{4}a{e}^{2}-231\,{c}^{3}{d}^{6} \right ) }{1155\,{c}^{4}{d}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.774043, size = 369, normalized size = 1.58 \[ \frac{2 \,{\left (105 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 198 \, a^{3} c^{2} d^{4} e^{4} + 88 \, a^{4} c d^{2} e^{6} - 16 \, a^{5} e^{8} + 35 \,{\left (11 \, c^{5} d^{6} e^{2} + 4 \, a c^{4} d^{4} e^{4}\right )} x^{4} + 5 \,{\left (99 \, c^{5} d^{7} e + 110 \, a c^{4} d^{5} e^{3} + a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (77 \, c^{5} d^{8} + 264 \, a c^{4} d^{6} e^{2} + 11 \, a^{2} c^{3} d^{4} e^{4} - 2 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} +{\left (462 \, a c^{4} d^{7} e + 99 \, a^{2} c^{3} d^{5} e^{3} - 44 \, a^{3} c^{2} d^{3} e^{5} + 8 \, a^{4} c d e^{7}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{1155 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]

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

[Out]

2/1155*(105*c^5*d^5*e^3*x^5 + 231*a^2*c^3*d^6*e^2 - 198*a^3*c^2*d^4*e^4 + 88*a^4
*c*d^2*e^6 - 16*a^5*e^8 + 35*(11*c^5*d^6*e^2 + 4*a*c^4*d^4*e^4)*x^4 + 5*(99*c^5*
d^7*e + 110*a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*x^3 + 3*(77*c^5*d^8 + 264*a*c^4*d^6
*e^2 + 11*a^2*c^3*d^4*e^4 - 2*a^3*c^2*d^2*e^6)*x^2 + (462*a*c^4*d^7*e + 99*a^2*c
^3*d^5*e^3 - 44*a^3*c^2*d^3*e^5 + 8*a^4*c*d*e^7)*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 = 0.218371, size = 581, normalized size = 2.49 \[ \frac{2 \,{\left (105 \, c^{6} d^{6} e^{4} x^{7} + 231 \, a^{3} c^{3} d^{7} e^{3} - 198 \, a^{4} c^{2} d^{5} e^{5} + 88 \, a^{5} c d^{3} e^{7} - 16 \, a^{6} d e^{9} + 245 \,{\left (2 \, c^{6} d^{7} e^{3} + a c^{5} d^{5} e^{5}\right )} x^{6} + 5 \,{\left (176 \, c^{6} d^{8} e^{2} + 236 \, a c^{5} d^{6} e^{4} + 29 \, a^{2} c^{4} d^{4} e^{6}\right )} x^{5} +{\left (726 \, c^{6} d^{9} e + 2222 \, a c^{5} d^{7} e^{3} + 728 \, a^{2} c^{4} d^{5} e^{5} - a^{3} c^{3} d^{3} e^{7}\right )} x^{4} +{\left (231 \, c^{6} d^{10} + 1980 \, a c^{5} d^{8} e^{2} + 1474 \, a^{2} c^{4} d^{6} e^{4} - 12 \, a^{3} c^{3} d^{4} e^{6} + 2 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{3} +{\left (693 \, a c^{5} d^{9} e + 1584 \, a^{2} c^{4} d^{7} e^{3} - 110 \, a^{3} c^{3} d^{5} e^{5} + 46 \, a^{4} c^{2} d^{3} e^{7} - 8 \, a^{5} c d e^{9}\right )} x^{2} +{\left (693 \, a^{2} c^{4} d^{8} e^{2} + 132 \, a^{3} c^{3} d^{6} e^{4} - 154 \, a^{4} c^{2} d^{4} e^{6} + 80 \, a^{5} c d^{2} e^{8} - 16 \, a^{6} e^{10}\right )} x\right )}}{1155 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

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

[Out]

2/1155*(105*c^6*d^6*e^4*x^7 + 231*a^3*c^3*d^7*e^3 - 198*a^4*c^2*d^5*e^5 + 88*a^5
*c*d^3*e^7 - 16*a^6*d*e^9 + 245*(2*c^6*d^7*e^3 + a*c^5*d^5*e^5)*x^6 + 5*(176*c^6
*d^8*e^2 + 236*a*c^5*d^6*e^4 + 29*a^2*c^4*d^4*e^6)*x^5 + (726*c^6*d^9*e + 2222*a
*c^5*d^7*e^3 + 728*a^2*c^4*d^5*e^5 - a^3*c^3*d^3*e^7)*x^4 + (231*c^6*d^10 + 1980
*a*c^5*d^8*e^2 + 1474*a^2*c^4*d^6*e^4 - 12*a^3*c^3*d^4*e^6 + 2*a^4*c^2*d^2*e^8)*
x^3 + (693*a*c^5*d^9*e + 1584*a^2*c^4*d^7*e^3 - 110*a^3*c^3*d^5*e^5 + 46*a^4*c^2
*d^3*e^7 - 8*a^5*c*d*e^9)*x^2 + (693*a^2*c^4*d^8*e^2 + 132*a^3*c^3*d^6*e^4 - 154
*a^4*c^2*d^4*e^6 + 80*a^5*c*d^2*e^8 - 16*a^6*e^10)*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)**(3/2)*(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)^(3/2),x, algorithm="giac")

[Out]

Timed out