3.34 \(\int \frac{x^3 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^4}-\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4} \]

[Out]

(-3*d^2*x^2*Sqrt[d^2 - e^2*x^2])/(5*e^2) - (d*x^3*Sqrt[d^2 - e^2*x^2])/(2*e) - (
x^4*Sqrt[d^2 - e^2*x^2])/5 - (3*d^3*(8*d + 5*e*x)*Sqrt[d^2 - e^2*x^2])/(20*e^4)
+ (3*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^4)

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Rubi [A]  time = 0.446325, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^4}-\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]

[Out]

(-3*d^2*x^2*Sqrt[d^2 - e^2*x^2])/(5*e^2) - (d*x^3*Sqrt[d^2 - e^2*x^2])/(2*e) - (
x^4*Sqrt[d^2 - e^2*x^2])/5 - (3*d^3*(8*d + 5*e*x)*Sqrt[d^2 - e^2*x^2])/(20*e^4)
+ (3*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^4)

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Rubi in Sympy [A]  time = 55.6679, size = 136, normalized size = 0.94 \[ \frac{3 d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{4 e^{4}} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{e^{4}} - \frac{3 d^{3} x \sqrt{d^{2} - e^{2} x^{2}}}{4 e^{3}} + \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{e^{4}} - \frac{d x^{3} \sqrt{d^{2} - e^{2} x^{2}}}{2 e} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x+d)**2/(-e**2*x**2+d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0984249, size = 92, normalized size = 0.64 \[ \frac{15 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (24 d^4+15 d^3 e x+12 d^2 e^2 x^2+10 d e^3 x^3+4 e^4 x^4\right )}{20 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(24*d^4 + 15*d^3*e*x + 12*d^2*e^2*x^2 + 10*d*e^3*x^3 + 4*
e^4*x^4)) + 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(20*e^4)

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Maple [A]  time = 0.014, size = 149, normalized size = 1. \[ -{\frac{3\,{d}^{2}{x}^{2}}{5\,{e}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{6\,{d}^{4}}{5\,{e}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{x}^{4}}{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{d{x}^{3}}{2\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{d}^{3}x}{4\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{3\,{d}^{5}}{4\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x+d)^2/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-3/5*d^2*x^2*(-e^2*x^2+d^2)^(1/2)/e^2-6/5*d^4*(-e^2*x^2+d^2)^(1/2)/e^4-1/5*x^4*(
-e^2*x^2+d^2)^(1/2)-1/2*d*x^3*(-e^2*x^2+d^2)^(1/2)/e-3/4*d^3*x*(-e^2*x^2+d^2)^(1
/2)/e^3+3/4*d^5/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

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Maxima [A]  time = 0.791558, size = 190, normalized size = 1.32 \[ -\frac{1}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} x^{4} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} d x^{3}}{2 \, e} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x^{2}}{5 \, e^{2}} + \frac{3 \, d^{5} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{4 \, \sqrt{e^{2}} e^{3}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x}{4 \, e^{3}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^3/sqrt(-e^2*x^2 + d^2),x, algorithm="maxima")

[Out]

-1/5*sqrt(-e^2*x^2 + d^2)*x^4 - 1/2*sqrt(-e^2*x^2 + d^2)*d*x^3/e - 3/5*sqrt(-e^2
*x^2 + d^2)*d^2*x^2/e^2 + 3/4*d^5*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^3) -
3/4*sqrt(-e^2*x^2 + d^2)*d^3*x/e^3 - 6/5*sqrt(-e^2*x^2 + d^2)*d^4/e^4

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Fricas [A]  time = 0.277345, size = 479, normalized size = 3.33 \[ -\frac{4 \, e^{10} x^{10} + 10 \, d e^{9} x^{9} - 40 \, d^{2} e^{8} x^{8} - 115 \, d^{3} e^{7} x^{7} - 20 \, d^{4} e^{6} x^{6} + 85 \, d^{5} e^{5} x^{5} + 80 \, d^{6} e^{4} x^{4} + 260 \, d^{7} e^{3} x^{3} - 240 \, d^{9} e x + 30 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (4 \, d e^{8} x^{8} + 10 \, d^{2} e^{7} x^{7} - 4 \, d^{3} e^{6} x^{6} - 25 \, d^{4} e^{5} x^{5} - 16 \, d^{5} e^{4} x^{4} - 28 \, d^{6} e^{3} x^{3} + 48 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{20 \,{\left (5 \, d e^{8} x^{4} - 20 \, d^{3} e^{6} x^{2} + 16 \, d^{5} e^{4} -{\left (e^{8} x^{4} - 12 \, d^{2} e^{6} x^{2} + 16 \, d^{4} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^3/sqrt(-e^2*x^2 + d^2),x, algorithm="fricas")

[Out]

-1/20*(4*e^10*x^10 + 10*d*e^9*x^9 - 40*d^2*e^8*x^8 - 115*d^3*e^7*x^7 - 20*d^4*e^
6*x^6 + 85*d^5*e^5*x^5 + 80*d^6*e^4*x^4 + 260*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 + 16*d^9)
*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 5*(4*d*e^8*x^
8 + 10*d^2*e^7*x^7 - 4*d^3*e^6*x^6 - 25*d^4*e^5*x^5 - 16*d^5*e^4*x^4 - 28*d^6*e^
3*x^3 + 48*d^8*e*x)*sqrt(-e^2*x^2 + d^2))/(5*d*e^8*x^4 - 20*d^3*e^6*x^2 + 16*d^5
*e^4 - (e^8*x^4 - 12*d^2*e^6*x^2 + 16*d^4*e^4)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 19.0817, size = 357, normalized size = 2.48 \[ d^{2} \left (\begin{cases} - \frac{2 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac{x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \sqrt{d^{2}}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{3 i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{5}} + \frac{3 i d^{3} x}{8 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d x^{3}}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{3 d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{5}} - \frac{3 d^{3} x}{8 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d x^{3}}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{8 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6}} - \frac{4 d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{6}}{6 \sqrt{d^{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x+d)**2/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**2*Piecewise((-2*d**2*sqrt(d**2 - e**2*x**2)/(3*e**4) - x**2*sqrt(d**2 - e**2*
x**2)/(3*e**2), Ne(e, 0)), (x**4/(4*sqrt(d**2)), True)) + 2*d*e*Piecewise((-3*I*
d**4*acosh(e*x/d)/(8*e**5) + 3*I*d**3*x/(8*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d
*x**3/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - I*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)
), Abs(e**2*x**2/d**2) > 1), (3*d**4*asin(e*x/d)/(8*e**5) - 3*d**3*x/(8*e**4*sqr
t(1 - e**2*x**2/d**2)) + d*x**3/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + x**5/(4*d*sq
rt(1 - e**2*x**2/d**2)), True)) + e**2*Piecewise((-8*d**4*sqrt(d**2 - e**2*x**2)
/(15*e**6) - 4*d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**4) - x**4*sqrt(d**2 - e**
2*x**2)/(5*e**2), Ne(e, 0)), (x**6/(6*sqrt(d**2)), True))

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GIAC/XCAS [A]  time = 0.294023, size = 99, normalized size = 0.69 \[ \frac{3}{4} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )}{\rm sign}\left (d\right ) - \frac{1}{20} \,{\left (24 \, d^{4} e^{\left (-4\right )} +{\left (15 \, d^{3} e^{\left (-3\right )} + 2 \,{\left (6 \, d^{2} e^{\left (-2\right )} +{\left (5 \, d e^{\left (-1\right )} + 2 \, x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^3/sqrt(-e^2*x^2 + d^2),x, algorithm="giac")

[Out]

3/4*d^5*arcsin(x*e/d)*e^(-4)*sign(d) - 1/20*(24*d^4*e^(-4) + (15*d^3*e^(-3) + 2*
(6*d^2*e^(-2) + (5*d*e^(-1) + 2*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)