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

Optimal. Leaf size=136 \[ -\frac{d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac{\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac{d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^2}-\frac{d^3 x \sqrt{d^2-e^2 x^2}}{4 e} \]

[Out]

-(d^3*x*Sqrt[d^2 - e^2*x^2])/(4*e) - (d*x*(d^2 - e^2*x^2)^(3/2))/(6*e) - (2*(d^2
- e^2*x^2)^(5/2))/(15*e^2) - (d^2 - e^2*x^2)^(7/2)/(3*e^2*(d + e*x)^2) - (d^5*A
rcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^2)

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Rubi [A]  time = 0.150591, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac{\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac{d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^2}-\frac{d^3 x \sqrt{d^2-e^2 x^2}}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

-(d^3*x*Sqrt[d^2 - e^2*x^2])/(4*e) - (d*x*(d^2 - e^2*x^2)^(3/2))/(6*e) - (2*(d^2
- e^2*x^2)^(5/2))/(15*e^2) - (d^2 - e^2*x^2)^(7/2)/(3*e^2*(d + e*x)^2) - (d^5*A
rcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^2)

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Rubi in Sympy [A]  time = 22.1974, size = 116, normalized size = 0.85 \[ - \frac{d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{4 e^{2}} - \frac{d^{3} x \sqrt{d^{2} - e^{2} x^{2}}}{4 e} - \frac{d x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{6 e} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{15 e^{2}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{3 e^{2} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**5*atan(e*x/sqrt(d**2 - e**2*x**2))/(4*e**2) - d**3*x*sqrt(d**2 - e**2*x**2)/
(4*e) - d*x*(d**2 - e**2*x**2)**(3/2)/(6*e) - 2*(d**2 - e**2*x**2)**(5/2)/(15*e*
*2) - (d**2 - e**2*x**2)**(7/2)/(3*e**2*(d + e*x)**2)

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Mathematica [A]  time = 0.0859724, size = 91, normalized size = 0.67 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-28 d^4+15 d^3 e x+16 d^2 e^2 x^2-30 d e^3 x^3+12 e^4 x^4\right )-15 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{60 e^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-28*d^4 + 15*d^3*e*x + 16*d^2*e^2*x^2 - 30*d*e^3*x^3 + 12*
e^4*x^4) - 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(60*e^2)

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Maple [A]  time = 0.015, size = 198, normalized size = 1.5 \[ -{\frac{2}{15\,{e}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{dx}{6\,e} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{3}x}{4\,e}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{{d}^{5}}{4\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{3\,{e}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-1/6/e*d*(-(x+d/e)^2*e^2+2*d*e*(x+
d/e))^(3/2)*x-1/4/e*d^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-1/4/e*d^5/(e^2)^(
1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-1/3/e^4/(x+d/e)^
2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)

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Maxima [A]  time = 0.792095, size = 225, normalized size = 1.65 \[ \frac{i \, d^{5} \arcsin \left (\frac{e x}{d} + 2\right )}{4 \, e^{2}} - \frac{\sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{4 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d}{4 \,{\left (e^{3} x + d e^{2}\right )}} - \frac{\sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{2 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d x}{4 \, e} - \frac{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2}}{12 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{5 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*I*d^5*arcsin(e*x/d + 2)/e^2 - 1/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^3*x/e -
1/4*(-e^2*x^2 + d^2)^(5/2)*d/(e^3*x + d*e^2) - 1/2*sqrt(e^2*x^2 + 4*d*e*x + 3*d^
2)*d^4/e^2 + 1/4*(-e^2*x^2 + d^2)^(3/2)*d*x/e - 5/12*(-e^2*x^2 + d^2)^(3/2)*d^2/
e^2 + 1/5*(-e^2*x^2 + d^2)^(5/2)/e^2

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Fricas [A]  time = 0.295152, size = 509, normalized size = 3.74 \[ \frac{12 \, e^{10} x^{10} - 30 \, d e^{9} x^{9} - 140 \, d^{2} e^{8} x^{8} + 405 \, d^{3} e^{7} x^{7} + 100 \, d^{4} e^{6} x^{6} - 1035 \, d^{5} e^{5} x^{5} + 480 \, d^{6} e^{4} x^{4} + 900 \, d^{7} e^{3} x^{3} - 480 \, d^{8} e^{2} x^{2} - 240 \, d^{9} e x + 30 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (12 \, d e^{8} x^{8} - 30 \, d^{2} e^{7} x^{7} - 32 \, d^{3} e^{6} x^{6} + 135 \, d^{4} e^{5} x^{5} - 48 \, d^{5} e^{4} x^{4} - 156 \, d^{6} e^{3} x^{3} + 96 \, d^{7} e^{2} x^{2} + 48 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{60 \,{\left (5 \, d e^{6} x^{4} - 20 \, d^{3} e^{4} x^{2} + 16 \, d^{5} e^{2} -{\left (e^{6} x^{4} - 12 \, d^{2} e^{4} x^{2} + 16 \, d^{4} e^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(12*e^10*x^10 - 30*d*e^9*x^9 - 140*d^2*e^8*x^8 + 405*d^3*e^7*x^7 + 100*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 - 480*d^8*e^2*x^2
- 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 - 1
2*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*(12*d*e^8*x^8 - 30*d^2*e^7*x^7 - 32*d^3*e^6*x^6 + 135*d^4*e^5*x^5 -
48*d^5*e^4*x^4 - 156*d^6*e^3*x^3 + 96*d^7*e^2*x^2 + 48*d^8*e*x)*sqrt(-e^2*x^2 +
d^2))/(5*d*e^6*x^4 - 20*d^3*e^4*x^2 + 16*d^5*e^2 - (e^6*x^4 - 12*d^2*e^4*x^2 + 1
6*d^4*e^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 21.0022, size = 321, normalized size = 2.36 \[ d^{2} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\\frac{x^{4} \sqrt{d^{2}}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*
e**2), True)) - 2*d*e*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**2*Piecewi
se((-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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out