3.1978 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 c^2 d^2 \sqrt{d+e x}}{e^3} \]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(3*e^3*(d + e*x)^(3/2)) + (4*c*d*(c*d^2 - a*e^2))/(e^3*Sq
rt[d + e*x]) + (2*c^2*d^2*Sqrt[d + e*x])/e^3

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Rubi [A]  time = 0.119144, antiderivative size = 79, 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 \[ \frac{4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 c^2 d^2 \sqrt{d+e x}}{e^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(9/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(3*e^3*(d + e*x)^(3/2)) + (4*c*d*(c*d^2 - a*e^2))/(e^3*Sq
rt[d + e*x]) + (2*c^2*d^2*Sqrt[d + e*x])/e^3

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Rubi in Sympy [A]  time = 35.0666, size = 73, normalized size = 0.92 \[ \frac{2 c^{2} d^{2} \sqrt{d + e x}}{e^{3}} - \frac{4 c d \left (a e^{2} - c d^{2}\right )}{e^{3} \sqrt{d + e x}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{2}}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(9/2),x)

[Out]

2*c**2*d**2*sqrt(d + e*x)/e**3 - 4*c*d*(a*e**2 - c*d**2)/(e**3*sqrt(d + e*x)) -
2*(a*e**2 - c*d**2)**2/(3*e**3*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.0820225, size = 68, normalized size = 0.86 \[ \frac{-2 a^2 e^4-4 a c d e^2 (2 d+3 e x)+2 c^2 d^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(9/2),x]

[Out]

(-2*a^2*e^4 - 4*a*c*d*e^2*(2*d + 3*e*x) + 2*c^2*d^2*(8*d^2 + 12*d*e*x + 3*e^2*x^
2))/(3*e^3*(d + e*x)^(3/2))

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Maple [A]  time = 0.01, size = 72, normalized size = 0.9 \[ -{\frac{-6\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+12\,xacd{e}^{3}-24\,x{c}^{2}{d}^{3}e+2\,{a}^{2}{e}^{4}+8\,ac{d}^{2}{e}^{2}-16\,{c}^{2}{d}^{4}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(9/2),x)

[Out]

-2/3/(e*x+d)^(3/2)*(-3*c^2*d^2*e^2*x^2+6*a*c*d*e^3*x-12*c^2*d^3*e*x+a^2*e^4+4*a*
c*d^2*e^2-8*c^2*d^4)/e^3

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Maxima [A]  time = 0.756479, size = 113, normalized size = 1.43 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} c^{2} d^{2}}{e^{2}} - \frac{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 6 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \]

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/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

2/3*(3*sqrt(e*x + d)*c^2*d^2/e^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4 - 6*(c^2*d
^3 - a*c*d*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^2))/e

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Fricas [A]  time = 0.219893, size = 113, normalized size = 1.43 \[ \frac{2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} - a^{2} e^{4} + 6 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \]

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/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

2/3*(3*c^2*d^2*e^2*x^2 + 8*c^2*d^4 - 4*a*c*d^2*e^2 - a^2*e^4 + 6*(2*c^2*d^3*e -
a*c*d*e^3)*x)/((e^4*x + d*e^3)*sqrt(e*x + d))

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Sympy [A]  time = 25.9397, size = 264, normalized size = 3.34 \[ \begin{cases} - \frac{2 a^{2} e^{4}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{8 a c d^{2} e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{12 a c d e^{3} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 c^{2} d^{4}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 c^{2} d^{3} e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 c^{2} d^{2} e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{2} x^{3}}{3 \sqrt{d}} & \text{otherwise} \end{cases} \]

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)**(9/2),x)

[Out]

Piecewise((-2*a**2*e**4/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 8*a*
c*d**2*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 12*a*c*d*e**3*x/
(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 16*c**2*d**4/(3*d*e**3*sqrt(
d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 24*c**2*d**3*e*x/(3*d*e**3*sqrt(d + e*x) +
3*e**4*x*sqrt(d + e*x)) + 6*c**2*d**2*e**2*x**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4
*x*sqrt(d + e*x)), Ne(e, 0)), (c**2*x**3/(3*sqrt(d)), True))

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GIAC/XCAS [A]  time = 0.208777, size = 150, normalized size = 1.9 \[ 2 \, \sqrt{x e + d} c^{2} d^{2} e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )}^{3} c^{2} d^{3} -{\left (x e + d\right )}^{2} c^{2} d^{4} - 6 \,{\left (x e + d\right )}^{3} a c d e^{2} + 2 \,{\left (x e + d\right )}^{2} a c d^{2} e^{2} -{\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{7}{2}}} \]

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/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*c^2*d^2*e^(-3) + 2/3*(6*(x*e + d)^3*c^2*d^3 - (x*e + d)^2*c^2*d^
4 - 6*(x*e + d)^3*a*c*d*e^2 + 2*(x*e + d)^2*a*c*d^2*e^2 - (x*e + d)^2*a^2*e^4)*e
^(-3)/(x*e + d)^(7/2)