3.1 \(\int x^2 (d+e x) \sqrt{d^2-e^2 x^2} \, dx\)

Optimal. Leaf size=132 \[ -\frac{d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac{d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}+\frac{d^3 x \sqrt{d^2-e^2 x^2}}{8 e^2} \]

[Out]

(d^3*x*Sqrt[d^2 - e^2*x^2])/(8*e^2) - (d^2*(d^2 - e^2*x^2)^(3/2))/(3*e^3) - (d*x
*(d^2 - e^2*x^2)^(3/2))/(4*e^2) + (d^2 - e^2*x^2)^(5/2)/(5*e^3) + (d^5*ArcTan[(e
*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)

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Rubi [A]  time = 0.20001, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac{d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}+\frac{d^3 x \sqrt{d^2-e^2 x^2}}{8 e^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(d + e*x)*Sqrt[d^2 - e^2*x^2],x]

[Out]

(d^3*x*Sqrt[d^2 - e^2*x^2])/(8*e^2) - (d^2*(d^2 - e^2*x^2)^(3/2))/(3*e^3) - (d*x
*(d^2 - e^2*x^2)^(3/2))/(4*e^2) + (d^2 - e^2*x^2)^(5/2)/(5*e^3) + (d^5*ArcTan[(e
*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)

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Rubi in Sympy [A]  time = 31.8788, size = 112, normalized size = 0.85 \[ \frac{d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e^{3}} + \frac{d^{3} x \sqrt{d^{2} - e^{2} x^{2}}}{8 e^{2}} - \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{3}} - \frac{d x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{4 e^{2}} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(e*x+d)*(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**5*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e**3) + d**3*x*sqrt(d**2 - e**2*x**2)/(
8*e**2) - d**2*(d**2 - e**2*x**2)**(3/2)/(3*e**3) - d*x*(d**2 - e**2*x**2)**(3/2
)/(4*e**2) + (d**2 - e**2*x**2)**(5/2)/(5*e**3)

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Mathematica [A]  time = 0.0862242, size = 91, normalized size = 0.69 \[ \frac{15 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-16 d^4-15 d^3 e x-8 d^2 e^2 x^2+30 d e^3 x^3+24 e^4 x^4\right )}{120 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(d + e*x)*Sqrt[d^2 - e^2*x^2],x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-16*d^4 - 15*d^3*e*x - 8*d^2*e^2*x^2 + 30*d*e^3*x^3 + 24*e
^4*x^4) + 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(120*e^3)

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Maple [A]  time = 0.057, size = 125, normalized size = 1. \[ -{\frac{dx}{4\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{3}x}{8\,{e}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{5}}{8\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{x}^{2}}{5\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{d}^{2}}{15\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x+d)*(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/4*d*x*(-e^2*x^2+d^2)^(3/2)/e^2+1/8*d^3*x*(-e^2*x^2+d^2)^(1/2)/e^2+1/8*d^5/e^2
/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/5*x^2*(-e^2*x^2+d^2)^(
3/2)/e-2/15*d^2*(-e^2*x^2+d^2)^(3/2)/e^3

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Maxima [A]  time = 0.81739, size = 158, normalized size = 1.2 \[ \frac{d^{5} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{2}} + \frac{\sqrt{-e^{2} x^{2} + d^{2}} d^{3} x}{8 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} x^{2}}{5 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d x}{4 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2}}{15 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^2,x, algorithm="maxima")

[Out]

1/8*d^5*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) + 1/8*sqrt(-e^2*x^2 + d^2)*d
^3*x/e^2 - 1/5*(-e^2*x^2 + d^2)^(3/2)*x^2/e - 1/4*(-e^2*x^2 + d^2)^(3/2)*d*x/e^2
- 2/15*(-e^2*x^2 + d^2)^(3/2)*d^2/e^3

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Fricas [A]  time = 0.278723, size = 479, normalized size = 3.63 \[ \frac{24 \, e^{10} x^{10} + 30 \, d e^{9} x^{9} - 320 \, d^{2} e^{8} x^{8} - 405 \, d^{3} e^{7} x^{7} + 760 \, d^{4} e^{6} x^{6} + 1035 \, d^{5} e^{5} x^{5} - 480 \, d^{6} e^{4} x^{4} - 900 \, d^{7} e^{3} x^{3} + 240 \, d^{9} e x - 30 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (24 \, d e^{8} x^{8} + 30 \, d^{2} e^{7} x^{7} - 104 \, d^{3} e^{6} x^{6} - 135 \, d^{4} e^{5} x^{5} + 96 \, d^{5} e^{4} x^{4} + 156 \, d^{6} e^{3} x^{3} - 48 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \,{\left (5 \, d e^{7} x^{4} - 20 \, d^{3} e^{5} x^{2} + 16 \, d^{5} e^{3} -{\left (e^{7} x^{4} - 12 \, d^{2} e^{5} x^{2} + 16 \, d^{4} e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^2,x, algorithm="fricas")

[Out]

1/120*(24*e^10*x^10 + 30*d*e^9*x^9 - 320*d^2*e^8*x^8 - 405*d^3*e^7*x^7 + 760*d^4
*e^6*x^6 + 1035*d^5*e^5*x^5 - 480*d^6*e^4*x^4 - 900*d^7*e^3*x^3 + 240*d^9*e*x -
30*(5*d^6*e^4*x^4 - 20*d^8*e^2*x^2 + 16*d^10 - (d^5*e^4*x^4 - 12*d^7*e^2*x^2 + 1
6*d^9)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 5*(24*d
*e^8*x^8 + 30*d^2*e^7*x^7 - 104*d^3*e^6*x^6 - 135*d^4*e^5*x^5 + 96*d^5*e^4*x^4 +
156*d^6*e^3*x^3 - 48*d^8*e*x)*sqrt(-e^2*x^2 + d^2))/(5*d*e^7*x^4 - 20*d^3*e^5*x
^2 + 16*d^5*e^3 - (e^7*x^4 - 12*d^2*e^5*x^2 + 16*d^4*e^3)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 14.6335, size = 279, normalized size = 2.11 \[ d \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\\frac{x^{4} \sqrt{d^{2}}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(e*x+d)*(-e**2*x**2+d**2)**(1/2),x)

[Out]

d*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**
2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d)/(8*e**3) - d**3*x
/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**
2*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise((-2*d**4*sqrt(d**2 -
e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d
**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True))

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GIAC/XCAS [A]  time = 0.333623, size = 100, normalized size = 0.76 \[ \frac{1}{8} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )}{\rm sign}\left (d\right ) - \frac{1}{120} \,{\left (16 \, d^{4} e^{\left (-3\right )} +{\left (15 \, d^{3} e^{\left (-2\right )} + 2 \,{\left (4 \, d^{2} e^{\left (-1\right )} - 3 \,{\left (4 \, x e + 5 \, d\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^2,x, algorithm="giac")

[Out]

1/8*d^5*arcsin(x*e/d)*e^(-3)*sign(d) - 1/120*(16*d^4*e^(-3) + (15*d^3*e^(-2) + 2
*(4*d^2*e^(-1) - 3*(4*x*e + 5*d)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)