3.1973 \(\int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx\)

Optimal. Leaf size=83 \[ -\frac{4 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )}{9 e^3}+\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^3}+\frac{2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \]

[Out]

(2*(c*d^2 - a*e^2)^2*(d + e*x)^(7/2))/(7*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)
^(9/2))/(9*e^3) + (2*c^2*d^2*(d + e*x)^(11/2))/(11*e^3)

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Rubi [A]  time = 0.161669, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ -\frac{4 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )}{9 e^3}+\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^3}+\frac{2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \]

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)^2,x]

[Out]

(2*(c*d^2 - a*e^2)^2*(d + e*x)^(7/2))/(7*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)
^(9/2))/(9*e^3) + (2*c^2*d^2*(d + e*x)^(11/2))/(11*e^3)

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Rubi in Sympy [A]  time = 35.6086, size = 76, normalized size = 0.92 \[ \frac{2 c^{2} d^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{3}} + \frac{4 c d \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} - c d^{2}\right )}{9 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{7 e^{3}} \]

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)**2*(e*x+d)**(1/2),x)

[Out]

2*c**2*d**2*(d + e*x)**(11/2)/(11*e**3) + 4*c*d*(d + e*x)**(9/2)*(a*e**2 - c*d**
2)/(9*e**3) + 2*(d + e*x)**(7/2)*(a*e**2 - c*d**2)**2/(7*e**3)

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Mathematica [A]  time = 0.114689, size = 67, normalized size = 0.81 \[ \frac{2 (d+e x)^{7/2} \left (99 a^2 e^4-22 a c d e^2 (2 d-7 e x)+c^2 d^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

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)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(99*a^2*e^4 - 22*a*c*d*e^2*(2*d - 7*e*x) + c^2*d^2*(8*d^2 - 2
8*d*e*x + 63*e^2*x^2)))/(693*e^3)

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Maple [A]  time = 0.011, size = 73, normalized size = 0.9 \[{\frac{126\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+308\,xacd{e}^{3}-56\,x{c}^{2}{d}^{3}e+198\,{a}^{2}{e}^{4}-88\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

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)^2*(e*x+d)^(1/2),x)

[Out]

2/693*(e*x+d)^(7/2)*(63*c^2*d^2*e^2*x^2+154*a*c*d*e^3*x-28*c^2*d^3*e*x+99*a^2*e^
4-44*a*c*d^2*e^2+8*c^2*d^4)/e^3

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Maxima [A]  time = 0.767021, size = 108, normalized size = 1.3 \[ \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} d^{2} - 154 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \]

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

[Out]

2/693*(63*(e*x + d)^(11/2)*c^2*d^2 - 154*(c^2*d^3 - a*c*d*e^2)*(e*x + d)^(9/2) +
99*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(7/2))/e^3

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Fricas [A]  time = 0.224179, size = 248, normalized size = 2.99 \[ \frac{2 \,{\left (63 \, c^{2} d^{2} e^{5} x^{5} + 8 \, c^{2} d^{7} - 44 \, a c d^{5} e^{2} + 99 \, a^{2} d^{3} e^{4} + 7 \,{\left (23 \, c^{2} d^{3} e^{4} + 22 \, a c d e^{6}\right )} x^{4} +{\left (113 \, c^{2} d^{4} e^{3} + 418 \, a c d^{2} e^{5} + 99 \, a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} + 110 \, a c d^{3} e^{4} + 99 \, a^{2} d e^{6}\right )} x^{2} -{\left (4 \, c^{2} d^{6} e - 22 \, a c d^{4} e^{3} - 297 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \]

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

[Out]

2/693*(63*c^2*d^2*e^5*x^5 + 8*c^2*d^7 - 44*a*c*d^5*e^2 + 99*a^2*d^3*e^4 + 7*(23*
c^2*d^3*e^4 + 22*a*c*d*e^6)*x^4 + (113*c^2*d^4*e^3 + 418*a*c*d^2*e^5 + 99*a^2*e^
7)*x^3 + 3*(c^2*d^5*e^2 + 110*a*c*d^3*e^4 + 99*a^2*d*e^6)*x^2 - (4*c^2*d^6*e - 2
2*a*c*d^4*e^3 - 297*a^2*d^2*e^5)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 3.80064, size = 97, normalized size = 1.17 \[ \frac{2 \left (\frac{c^{2} d^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{2}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (2 a c d e^{2} - 2 c^{2} d^{3}\right )}{9 e^{2}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{7 e^{2}}\right )}{e} \]

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)**2*(e*x+d)**(1/2),x)

[Out]

2*(c**2*d**2*(d + e*x)**(11/2)/(11*e**2) + (d + e*x)**(9/2)*(2*a*c*d*e**2 - 2*c*
*2*d**3)/(9*e**2) + (d + e*x)**(7/2)*(a**2*e**4 - 2*a*c*d**2*e**2 + c**2*d**4)/(
7*e**2))/e

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GIAC/XCAS [A]  time = 0.211221, size = 594, normalized size = 7.16 \[ \frac{2}{3465} \,{\left (33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c^{2} d^{4} e^{\left (-14\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} c^{2} d^{3} e^{\left (-26\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d^{2} e^{2} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a c d^{3} + 132 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a c d^{2} e^{\left (-12\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} c^{2} d^{2} e^{\left (-42\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} d e^{2} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} a c d e^{\left (-24\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a^{2} e^{\left (-10\right )}\right )} e^{\left (-1\right )} \]

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

[Out]

2/3465*(33*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(
3/2)*d^2*e^12)*c^2*d^4*e^(-14) + 22*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/
2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*c^2*d^3
*e^(-26) + 1155*(x*e + d)^(3/2)*a^2*d^2*e^2 + 462*(3*(x*e + d)^(5/2) - 5*(x*e +
d)^(3/2)*d)*a*c*d^3 + 132*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 +
35*(x*e + d)^(3/2)*d^2*e^12)*a*c*d^2*e^(-12) + (315*(x*e + d)^(11/2)*e^40 - 154
0*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*
d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*c^2*d^2*e^(-42) + 462*(3*(x*e + d)^(5/
2) - 5*(x*e + d)^(3/2)*d)*a^2*d*e^2 + 22*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d
)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*a*
c*d*e^(-24) + 33*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e
+ d)^(3/2)*d^2*e^12)*a^2*e^(-10))*e^(-1)