3.107 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{4} d x \left (d^2-e^2 x^2\right )^{3/2}+\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 e}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}+\frac{3}{8} d^3 x \sqrt{d^2-e^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0742063, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{4} d x \left (d^2-e^2 x^2\right )^{3/2}+\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 e}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}+\frac{3}{8} d^3 x \sqrt{d^2-e^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(d + e*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.7284, size = 83, normalized size = 0.83 \[ \frac{3 d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e} + \frac{3 d^{3} x \sqrt{d^{2} - e^{2} x^{2}}}{8} + \frac{d x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{4} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/(e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.0915401, size = 91, normalized size = 0.91 \[ \frac{15 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (8 d^4+25 d^3 e x-16 d^2 e^2 x^2-10 d e^3 x^3+8 e^4 x^4\right )}{40 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(d + e*x),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(8*d^4 + 25*d^3*e*x - 16*d^2*e^2*x^2 - 10*d*e^3*x^3 + 8*e^4
*x^4) + 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(40*e)

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Maple [A]  time = 0.009, size = 147, normalized size = 1.5 \[{\frac{1}{5\,e} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{dx}{4} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{3}x}{8}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{3\,{d}^{5}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.789761, size = 147, normalized size = 1.47 \[ -\frac{3 i \, d^{5} \arcsin \left (\frac{e x}{d} + 2\right )}{8 \, e} + \frac{3}{8} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x + \frac{3 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{4 \, e} + \frac{1}{4} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d x + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.290756, size = 504, normalized size = 5.04 \[ \frac{8 \, e^{10} x^{10} - 10 \, d e^{9} x^{9} - 120 \, d^{2} e^{8} x^{8} + 155 \, d^{3} e^{7} x^{7} + 440 \, d^{4} e^{6} x^{6} - 605 \, d^{5} e^{5} x^{5} - 640 \, d^{6} e^{4} x^{4} + 860 \, d^{7} e^{3} x^{3} + 320 \, d^{8} e^{2} x^{2} - 400 \, d^{9} e x - 30 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (8 \, d e^{8} x^{8} - 10 \, d^{2} e^{7} x^{7} - 48 \, d^{3} e^{6} x^{6} + 65 \, d^{4} e^{5} x^{5} + 96 \, d^{5} e^{4} x^{4} - 132 \, d^{6} e^{3} x^{3} - 64 \, d^{7} e^{2} x^{2} + 80 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40 \,{\left (5 \, d e^{5} x^{4} - 20 \, d^{3} e^{3} x^{2} + 16 \, d^{5} e -{\left (e^{5} x^{4} - 12 \, d^{2} e^{3} x^{2} + 16 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d),x, algorithm="fricas")

[Out]

1/40*(8*e^10*x^10 - 10*d*e^9*x^9 - 120*d^2*e^8*x^8 + 155*d^3*e^7*x^7 + 440*d^4*e
^6*x^6 - 605*d^5*e^5*x^5 - 640*d^6*e^4*x^4 + 860*d^7*e^3*x^3 + 320*d^8*e^2*x^2 -
400*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*(8*d*e^8*x^8 - 10*d^2*e^7*x^7 - 48*d^3*e^6*x^6 + 65*d^4*e^5*x^5 + 96*d
^5*e^4*x^4 - 132*d^6*e^3*x^3 - 64*d^7*e^2*x^2 + 80*d^8*e*x)*sqrt(-e^2*x^2 + d^2)
)/(5*d*e^5*x^4 - 20*d^3*e^3*x^2 + 16*d^5*e - (e^5*x^4 - 12*d^2*e^3*x^2 + 16*d^4*
e)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 21.9166, size = 435, normalized size = 4.35 \[ d^{3} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\\frac{x^{4} \sqrt{d^{2}}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(5/2)/(e*x+d),x)

[Out]

d**3*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2))
+ I*e**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**2*
asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) - d**2*e*Piecewise((x
**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - d
*e**2*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**3*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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError