3.188 \(\int \frac{x^5 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=204 \[ \frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{18 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

[Out]

(d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^
2 - e^2*x^2)^(3/2)) + (10*d^2*(d - e*x)^2)/(e^6*Sqrt[d^2 - e^2*x^2]) + (59*d^2*S
qrt[d^2 - e^2*x^2])/(3*e^6) - (2*d*x*Sqrt[d^2 - e^2*x^2])/e^5 + (x^2*Sqrt[d^2 -
e^2*x^2])/(3*e^4) + (18*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

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Rubi [A]  time = 0.930435, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{18 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

(d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^
2 - e^2*x^2)^(3/2)) + (10*d^2*(d - e*x)^2)/(e^6*Sqrt[d^2 - e^2*x^2]) + (59*d^2*S
qrt[d^2 - e^2*x^2])/(3*e^6) - (2*d*x*Sqrt[d^2 - e^2*x^2])/e^5 + (x^2*Sqrt[d^2 -
e^2*x^2])/(3*e^4) + (18*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

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Rubi in Sympy [A]  time = 55.5822, size = 175, normalized size = 0.86 \[ \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e^{6} \left (d + e x\right )^{4}} + \frac{18 d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{6}} + \frac{20 d^{3} \sqrt{d^{2} - e^{2} x^{2}}}{e^{6} \left (d + e x\right )} - \frac{8 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e^{6} \left (d + e x\right )^{3}} + \frac{10 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{e^{6}} - \frac{2 d x \sqrt{d^{2} - e^{2} x^{2}}}{e^{5}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

d**4*(d**2 - e**2*x**2)**(3/2)/(5*e**6*(d + e*x)**4) + 18*d**3*atan(e*x/sqrt(d**
2 - e**2*x**2))/e**6 + 20*d**3*sqrt(d**2 - e**2*x**2)/(e**6*(d + e*x)) - 8*d**3*
(d**2 - e**2*x**2)**(3/2)/(5*e**6*(d + e*x)**3) + 10*d**2*sqrt(d**2 - e**2*x**2)
/e**6 - 2*d*x*sqrt(d**2 - e**2*x**2)/e**5 - (d**2 - e**2*x**2)**(3/2)/(3*e**6)

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Mathematica [A]  time = 0.114647, size = 109, normalized size = 0.53 \[ \frac{270 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right )}{(d+e x)^3}}{15 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(424*d^5 + 1002*d^4*e*x + 674*d^3*e^2*x^2 + 70*d^2*e^3*x^3
- 15*d*e^4*x^4 + 5*e^5*x^5))/(d + e*x)^3 + 270*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]])/(15*e^6)

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Maple [A]  time = 0.032, size = 297, normalized size = 1.5 \[ -{\frac{1}{3\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-2\,{\frac{dx\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{{e}^{5}}}-2\,{\frac{{d}^{3}}{{e}^{5}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }+20\,{\frac{{d}^{2}}{{e}^{6}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+20\,{\frac{{d}^{3}}{{e}^{5}\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ) }+10\,{\frac{{d}^{2}}{{e}^{8}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}-{\frac{8\,{d}^{3}}{5\,{e}^{9}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}+{\frac{{d}^{4}}{5\,{e}^{10}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/3/e^6*(-e^2*x^2+d^2)^(3/2)-2*d*x*(-e^2*x^2+d^2)^(1/2)/e^5-2/e^5*d^3/(e^2)^(1/
2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+20/e^6*d^2*(-(x+d/e)^2*e^2+2*d*e*(
x+d/e))^(1/2)+20/e^5*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(1/2))+10/e^8*d^2/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-8/5/e^
9*d^3/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+1/5*d^4/e^10/(x+d/e)^4*(-(x
+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*x^5/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.3122, size = 845, normalized size = 4.14 \[ -\frac{5 \, e^{11} x^{11} + 10 \, d e^{10} x^{10} - 95 \, d^{2} e^{9} x^{9} + 770 \, d^{3} e^{8} x^{8} + 4924 \, d^{4} e^{7} x^{7} + 2710 \, d^{5} e^{6} x^{6} - 17532 \, d^{6} e^{5} x^{5} - 23400 \, d^{7} e^{4} x^{4} + 5760 \, d^{8} e^{3} x^{3} + 21600 \, d^{9} e^{2} x^{2} + 8640 \, d^{10} e x + 540 \,{\left (d^{3} e^{8} x^{8} - 3 \, d^{4} e^{7} x^{7} - 27 \, d^{5} e^{6} x^{6} - 21 \, d^{6} e^{5} x^{5} + 70 \, d^{7} e^{4} x^{4} + 100 \, d^{8} e^{3} x^{3} - 16 \, d^{9} e^{2} x^{2} - 80 \, d^{10} e x - 32 \, d^{11} +{\left (d^{3} e^{7} x^{7} + 8 \, d^{4} e^{6} x^{6} + d^{5} e^{5} x^{5} - 50 \, d^{6} e^{4} x^{4} - 60 \, d^{7} e^{3} x^{3} + 32 \, d^{8} e^{2} x^{2} + 80 \, d^{9} e x + 32 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (5 \, e^{10} x^{10} - 45 \, d e^{9} x^{9} + 100 \, d^{2} e^{8} x^{8} + 1018 \, d^{3} e^{7} x^{7} - 890 \, d^{4} e^{6} x^{6} - 11412 \, d^{5} e^{5} x^{5} - 12600 \, d^{6} e^{4} x^{4} + 10080 \, d^{7} e^{3} x^{3} + 21600 \, d^{8} e^{2} x^{2} + 8640 \, d^{9} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{14} x^{8} - 3 \, d e^{13} x^{7} - 27 \, d^{2} e^{12} x^{6} - 21 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 100 \, d^{5} e^{9} x^{3} - 16 \, d^{6} e^{8} x^{2} - 80 \, d^{7} e^{7} x - 32 \, d^{8} e^{6} +{\left (e^{13} x^{7} + 8 \, d e^{12} x^{6} + d^{2} e^{11} x^{5} - 50 \, d^{3} e^{10} x^{4} - 60 \, d^{4} e^{9} x^{3} + 32 \, d^{5} e^{8} x^{2} + 80 \, d^{6} e^{7} x + 32 \, d^{7} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*x^5/(e*x + d)^4,x, algorithm="fricas")

[Out]

-1/15*(5*e^11*x^11 + 10*d*e^10*x^10 - 95*d^2*e^9*x^9 + 770*d^3*e^8*x^8 + 4924*d^
4*e^7*x^7 + 2710*d^5*e^6*x^6 - 17532*d^6*e^5*x^5 - 23400*d^7*e^4*x^4 + 5760*d^8*
e^3*x^3 + 21600*d^9*e^2*x^2 + 8640*d^10*e*x + 540*(d^3*e^8*x^8 - 3*d^4*e^7*x^7 -
27*d^5*e^6*x^6 - 21*d^6*e^5*x^5 + 70*d^7*e^4*x^4 + 100*d^8*e^3*x^3 - 16*d^9*e^2
*x^2 - 80*d^10*e*x - 32*d^11 + (d^3*e^7*x^7 + 8*d^4*e^6*x^6 + d^5*e^5*x^5 - 50*d
^6*e^4*x^4 - 60*d^7*e^3*x^3 + 32*d^8*e^2*x^2 + 80*d^9*e*x + 32*d^10)*sqrt(-e^2*x
^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (5*e^10*x^10 - 45*d*e^9*x
^9 + 100*d^2*e^8*x^8 + 1018*d^3*e^7*x^7 - 890*d^4*e^6*x^6 - 11412*d^5*e^5*x^5 -
12600*d^6*e^4*x^4 + 10080*d^7*e^3*x^3 + 21600*d^8*e^2*x^2 + 8640*d^9*e*x)*sqrt(-
e^2*x^2 + d^2))/(e^14*x^8 - 3*d*e^13*x^7 - 27*d^2*e^12*x^6 - 21*d^3*e^11*x^5 + 7
0*d^4*e^10*x^4 + 100*d^5*e^9*x^3 - 16*d^6*e^8*x^2 - 80*d^7*e^7*x - 32*d^8*e^6 +
(e^13*x^7 + 8*d*e^12*x^6 + d^2*e^11*x^5 - 50*d^3*e^10*x^4 - 60*d^4*e^9*x^3 + 32*
d^5*e^8*x^2 + 80*d^6*e^7*x + 32*d^7*e^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(x**5*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.324126, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*x^5/(e*x + d)^4,x, algorithm="giac")

[Out]

Done