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

Optimal. Leaf size=159 \[ -\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3}+\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2} \]

[Out]

(d^5*x*Sqrt[d^2 - e^2*x^2])/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) -
(d^2*(d^2 - e^2*x^2)^(5/2))/(5*e^3) - (d*x*(d^2 - e^2*x^2)^(5/2))/(6*e^2) + (d^2
- e^2*x^2)^(7/2)/(7*e^3) + (d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^3)

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Rubi [A]  time = 0.241565, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3}+\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^5*x*Sqrt[d^2 - e^2*x^2])/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) -
(d^2*(d^2 - e^2*x^2)^(5/2))/(5*e^3) - (d*x*(d^2 - e^2*x^2)^(5/2))/(6*e^2) + (d^2
- e^2*x^2)^(7/2)/(7*e^3) + (d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^3)

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Rubi in Sympy [A]  time = 36.98, size = 136, normalized size = 0.86 \[ \frac{d^{7} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{16 e^{3}} + \frac{d^{5} x \sqrt{d^{2} - e^{2} x^{2}}}{16 e^{2}} + \frac{d^{3} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24 e^{2}} - \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{3}} - \frac{d x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6 e^{2}} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{7 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**7*atan(e*x/sqrt(d**2 - e**2*x**2))/(16*e**3) + d**5*x*sqrt(d**2 - e**2*x**2)/
(16*e**2) + d**3*x*(d**2 - e**2*x**2)**(3/2)/(24*e**2) - d**2*(d**2 - e**2*x**2)
**(5/2)/(5*e**3) - d*x*(d**2 - e**2*x**2)**(5/2)/(6*e**2) + (d**2 - e**2*x**2)**
(7/2)/(7*e**3)

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Mathematica [A]  time = 0.102159, size = 114, normalized size = 0.72 \[ \frac{105 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (96 d^6+105 d^5 e x+48 d^4 e^2 x^2-490 d^3 e^3 x^3-384 d^2 e^4 x^4+280 d e^5 x^5+240 e^6 x^6\right )}{1680 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(96*d^6 + 105*d^5*e*x + 48*d^4*e^2*x^2 - 490*d^3*e^3*x^3
- 384*d^2*e^4*x^4 + 280*d*e^5*x^5 + 240*e^6*x^6)) + 105*d^7*ArcTan[(e*x)/Sqrt[d^
2 - e^2*x^2]])/(1680*e^3)

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Maple [A]  time = 0.11, size = 148, normalized size = 0.9 \[ -{\frac{dx}{6\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}x}{24\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}x}{16\,{e}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{7}}{16\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{x}^{2}}{7\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{d}^{2}}{35\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*d*x*(-e^2*x^2+d^2)^(5/2)/e^2+1/24*d^3*x*(-e^2*x^2+d^2)^(3/2)/e^2+1/16*d^5*x
*(-e^2*x^2+d^2)^(1/2)/e^2+1/16*d^7/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^
2+d^2)^(1/2))-1/7*x^2*(-e^2*x^2+d^2)^(5/2)/e-2/35*d^2*(-e^2*x^2+d^2)^(5/2)/e^3

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Maxima [A]  time = 0.809449, size = 189, normalized size = 1.19 \[ \frac{d^{7} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{2}} + \frac{\sqrt{-e^{2} x^{2} + d^{2}} d^{5} x}{16 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3} x}{24 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} x^{2}}{7 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x}{6 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{2}}{35 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*d^7*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) + 1/16*sqrt(-e^2*x^2 + d^2)
*d^5*x/e^2 + 1/24*(-e^2*x^2 + d^2)^(3/2)*d^3*x/e^2 - 1/7*(-e^2*x^2 + d^2)^(5/2)*
x^2/e - 1/6*(-e^2*x^2 + d^2)^(5/2)*d*x/e^2 - 2/35*(-e^2*x^2 + d^2)^(5/2)*d^2/e^3

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Fricas [A]  time = 0.27809, size = 657, normalized size = 4.13 \[ -\frac{240 \, e^{14} x^{14} + 280 \, d e^{13} x^{13} - 6384 \, d^{2} e^{12} x^{12} - 7490 \, d^{3} e^{11} x^{11} + 34608 \, d^{4} e^{10} x^{10} + 41475 \, d^{5} e^{9} x^{9} - 75600 \, d^{6} e^{8} x^{8} - 93905 \, d^{7} e^{7} x^{7} + 73920 \, d^{8} e^{6} x^{6} + 99400 \, d^{9} e^{5} x^{5} - 26880 \, d^{10} e^{4} x^{4} - 46480 \, d^{11} e^{3} x^{3} + 6720 \, d^{13} e x + 210 \,{\left (7 \, d^{8} e^{6} x^{6} - 56 \, d^{10} e^{4} x^{4} + 112 \, d^{12} e^{2} x^{2} - 64 \, d^{14} -{\left (d^{7} e^{6} x^{6} - 24 \, d^{9} e^{4} x^{4} + 80 \, d^{11} e^{2} x^{2} - 64 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 7 \,{\left (240 \, d e^{12} x^{12} + 280 \, d^{2} e^{11} x^{11} - 2304 \, d^{3} e^{10} x^{10} - 2730 \, d^{4} e^{9} x^{9} + 6960 \, d^{5} e^{8} x^{8} + 8505 \, d^{6} e^{7} x^{7} - 8640 \, d^{7} e^{6} x^{6} - 11240 \, d^{8} e^{5} x^{5} + 3840 \, d^{9} e^{4} x^{4} + 6160 \, d^{10} e^{3} x^{3} - 960 \, d^{12} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \,{\left (7 \, d e^{9} x^{6} - 56 \, d^{3} e^{7} x^{4} + 112 \, d^{5} e^{5} x^{2} - 64 \, d^{7} e^{3} -{\left (e^{9} x^{6} - 24 \, d^{2} e^{7} x^{4} + 80 \, d^{4} e^{5} x^{2} - 64 \, d^{6} e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1680*(240*e^14*x^14 + 280*d*e^13*x^13 - 6384*d^2*e^12*x^12 - 7490*d^3*e^11*x^
11 + 34608*d^4*e^10*x^10 + 41475*d^5*e^9*x^9 - 75600*d^6*e^8*x^8 - 93905*d^7*e^7
*x^7 + 73920*d^8*e^6*x^6 + 99400*d^9*e^5*x^5 - 26880*d^10*e^4*x^4 - 46480*d^11*e
^3*x^3 + 6720*d^13*e*x + 210*(7*d^8*e^6*x^6 - 56*d^10*e^4*x^4 + 112*d^12*e^2*x^2
- 64*d^14 - (d^7*e^6*x^6 - 24*d^9*e^4*x^4 + 80*d^11*e^2*x^2 - 64*d^13)*sqrt(-e^
2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 7*(240*d*e^12*x^12 + 2
80*d^2*e^11*x^11 - 2304*d^3*e^10*x^10 - 2730*d^4*e^9*x^9 + 6960*d^5*e^8*x^8 + 85
05*d^6*e^7*x^7 - 8640*d^7*e^6*x^6 - 11240*d^8*e^5*x^5 + 3840*d^9*e^4*x^4 + 6160*
d^10*e^3*x^3 - 960*d^12*e*x)*sqrt(-e^2*x^2 + d^2))/(7*d*e^9*x^6 - 56*d^3*e^7*x^4
+ 112*d^5*e^5*x^2 - 64*d^7*e^3 - (e^9*x^6 - 24*d^2*e^7*x^4 + 80*d^4*e^5*x^2 - 6
4*d^6*e^3)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 34.3137, size = 653, normalized size = 4.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*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**2*e*Piecewise((-2*d**4*sqr
t(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)) - d*e**2*Piece
wise((-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)) - e**3*Piecewise((
-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**2 - e**2*x**2)/(
105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x
**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True))

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GIAC/XCAS [A]  time = 0.315728, size = 130, normalized size = 0.82 \[ \frac{1}{16} \, d^{7} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )}{\rm sign}\left (d\right ) - \frac{1}{1680} \,{\left (96 \, d^{6} e^{\left (-3\right )} +{\left (105 \, d^{5} e^{\left (-2\right )} + 2 \,{\left (24 \, d^{4} e^{\left (-1\right )} -{\left (245 \, d^{3} + 4 \,{\left (48 \, d^{2} e - 5 \,{\left (6 \, x e^{3} + 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*d^7*arcsin(x*e/d)*e^(-3)*sign(d) - 1/1680*(96*d^6*e^(-3) + (105*d^5*e^(-2)
+ 2*(24*d^4*e^(-1) - (245*d^3 + 4*(48*d^2*e - 5*(6*x*e^3 + 7*d*e^2)*x)*x)*x)*x)*
x)*sqrt(-x^2*e^2 + d^2)