3.5 \(\int x (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=116 \[ \frac{d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac{(6 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}+\frac{d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^2}+\frac{d^4 x \sqrt{d^2-e^2 x^2}}{16 e} \]

[Out]

(d^4*x*Sqrt[d^2 - e^2*x^2])/(16*e) + (d^2*x*(d^2 - e^2*x^2)^(3/2))/(24*e) - ((6*
d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*e^2) + (d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(16*e^2)

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Rubi [A]  time = 0.103243, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac{(6 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}+\frac{d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^2}+\frac{d^4 x \sqrt{d^2-e^2 x^2}}{16 e} \]

Antiderivative was successfully verified.

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

[Out]

(d^4*x*Sqrt[d^2 - e^2*x^2])/(16*e) + (d^2*x*(d^2 - e^2*x^2)^(3/2))/(24*e) - ((6*
d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*e^2) + (d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(16*e^2)

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Rubi in Sympy [A]  time = 21.708, size = 114, normalized size = 0.98 \[ \frac{d^{6} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{16 e^{2}} + \frac{d^{4} x \sqrt{d^{2} - e^{2} x^{2}}}{16 e} + \frac{d^{2} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24 e} - \frac{d \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{30 e^{2}} - \frac{\left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**6*atan(e*x/sqrt(d**2 - e**2*x**2))/(16*e**2) + d**4*x*sqrt(d**2 - e**2*x**2)/
(16*e) + d**2*x*(d**2 - e**2*x**2)**(3/2)/(24*e) - d*(d**2 - e**2*x**2)**(5/2)/(
30*e**2) - (d + e*x)*(d**2 - e**2*x**2)**(5/2)/(6*e**2)

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Mathematica [A]  time = 0.0923253, size = 103, normalized size = 0.89 \[ \frac{15 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (48 d^5+15 d^4 e x-96 d^3 e^2 x^2-70 d^2 e^3 x^3+48 d e^4 x^4+40 e^5 x^5\right )}{240 e^2} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(48*d^5 + 15*d^4*e*x - 96*d^3*e^2*x^2 - 70*d^2*e^3*x^3 +
48*d*e^4*x^4 + 40*e^5*x^5)) + 15*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^2
)

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Maple [A]  time = 0.018, size = 123, normalized size = 1.1 \[ -{\frac{d}{5\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{x}{6\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}x}{24\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{4}x}{16\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{6}}{16\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*d/e^2*(-e^2*x^2+d^2)^(5/2)-1/6*x*(-e^2*x^2+d^2)^(5/2)/e+1/24*d^2*x*(-e^2*x^
2+d^2)^(3/2)/e+1/16*d^4*x*(-e^2*x^2+d^2)^(1/2)/e+1/16*d^6/e/(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 = 0.820194, size = 155, normalized size = 1.34 \[ \frac{d^{6} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e} + \frac{\sqrt{-e^{2} x^{2} + d^{2}} d^{4} x}{16 \, e} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} x}{24 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} x}{6 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d}{5 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*d^6*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e) + 1/16*sqrt(-e^2*x^2 + d^2)*d
^4*x/e + 1/24*(-e^2*x^2 + d^2)^(3/2)*d^2*x/e - 1/6*(-e^2*x^2 + d^2)^(5/2)*x/e -
1/5*(-e^2*x^2 + d^2)^(5/2)*d/e^2

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Fricas [A]  time = 0.282978, size = 598, normalized size = 5.16 \[ \frac{240 \, d e^{11} x^{11} + 288 \, d^{2} e^{10} x^{10} - 1940 \, d^{3} e^{9} x^{9} - 2400 \, d^{4} e^{8} x^{8} + 5310 \, d^{5} e^{7} x^{7} + 6960 \, d^{6} e^{6} x^{6} - 6330 \, d^{7} e^{5} x^{5} - 8640 \, d^{8} e^{4} x^{4} + 3200 \, d^{9} e^{3} x^{3} + 3840 \, d^{10} e^{2} x^{2} - 480 \, d^{11} e x - 30 \,{\left (d^{6} e^{6} x^{6} - 18 \, d^{8} e^{4} x^{4} + 48 \, d^{10} e^{2} x^{2} - 32 \, d^{12} + 2 \,{\left (3 \, d^{7} e^{4} x^{4} - 16 \, d^{9} e^{2} x^{2} + 16 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (40 \, e^{11} x^{11} + 48 \, d e^{10} x^{10} - 790 \, d^{2} e^{9} x^{9} - 960 \, d^{3} e^{8} x^{8} + 3195 \, d^{4} e^{7} x^{7} + 4080 \, d^{5} e^{6} x^{6} - 4910 \, d^{6} e^{5} x^{5} - 6720 \, d^{7} e^{4} x^{4} + 2960 \, d^{8} e^{3} x^{3} + 3840 \, d^{9} e^{2} x^{2} - 480 \, d^{10} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \,{\left (e^{8} x^{6} - 18 \, d^{2} e^{6} x^{4} + 48 \, d^{4} e^{4} x^{2} - 32 \, d^{6} e^{2} + 2 \,{\left (3 \, d e^{6} x^{4} - 16 \, d^{3} e^{4} x^{2} + 16 \, d^{5} e^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(240*d*e^11*x^11 + 288*d^2*e^10*x^10 - 1940*d^3*e^9*x^9 - 2400*d^4*e^8*x^8
+ 5310*d^5*e^7*x^7 + 6960*d^6*e^6*x^6 - 6330*d^7*e^5*x^5 - 8640*d^8*e^4*x^4 + 3
200*d^9*e^3*x^3 + 3840*d^10*e^2*x^2 - 480*d^11*e*x - 30*(d^6*e^6*x^6 - 18*d^8*e^
4*x^4 + 48*d^10*e^2*x^2 - 32*d^12 + 2*(3*d^7*e^4*x^4 - 16*d^9*e^2*x^2 + 16*d^11)
*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (40*e^11*x^11
+ 48*d*e^10*x^10 - 790*d^2*e^9*x^9 - 960*d^3*e^8*x^8 + 3195*d^4*e^7*x^7 + 4080*
d^5*e^6*x^6 - 4910*d^6*e^5*x^5 - 6720*d^7*e^4*x^4 + 2960*d^8*e^3*x^3 + 3840*d^9*
e^2*x^2 - 480*d^10*e*x)*sqrt(-e^2*x^2 + d^2))/(e^8*x^6 - 18*d^2*e^6*x^4 + 48*d^4
*e^4*x^2 - 32*d^6*e^2 + 2*(3*d*e^6*x^4 - 16*d^3*e^4*x^2 + 16*d^5*e^2)*sqrt(-e^2*
x^2 + d^2))

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Sympy [A]  time = 32.3319, size = 580, normalized size = 5. \[ d^{3} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + d^{2} e \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\\frac{x^{4} \sqrt{d^{2}}}{4} & \text{otherwise} \end{cases}\right ) - e^{3} \left (\begin{cases} - \frac{i d^{6} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{16 e^{5}} + \frac{i d^{5} x}{16 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d^{3} x^{3}}{48 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{5 i d x^{5}}{24 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{7}}{6 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{6} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{16 e^{5}} - \frac{d^{5} x}{16 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d^{3} x^{3}}{48 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{5 d x^{5}}{24 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{7}}{6 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*
e**2), True)) + d**2*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)) - d*e**2*Piec
ewise((-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, Tru
e)) - e**3*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1
+ e**2*x**2/d**2)) - I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x*
*5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**2*x**2/d**2)),
Abs(e**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1
- e**2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24
*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2)), True))

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GIAC/XCAS [A]  time = 0.320762, size = 113, normalized size = 0.97 \[ \frac{1}{16} \, d^{6} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-2\right )}{\rm sign}\left (d\right ) - \frac{1}{240} \,{\left (48 \, d^{5} e^{\left (-2\right )} +{\left (15 \, d^{4} e^{\left (-1\right )} - 2 \,{\left (48 \, d^{3} +{\left (35 \, d^{2} e - 4 \,{\left (5 \, x e^{3} + 6 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*d^6*arcsin(x*e/d)*e^(-2)*sign(d) - 1/240*(48*d^5*e^(-2) + (15*d^4*e^(-1) -
2*(48*d^3 + (35*d^2*e - 4*(5*x*e^3 + 6*d*e^2)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)