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

Optimal. Leaf size=229 \[ -\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}-\frac{5 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{64 e^6}-\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3} \]

[Out]

(-5*d^7*x*Sqrt[d^2 - e^2*x^2])/(64*e^5) - (4*d^4*x^2*(d^2 - e^2*x^2)^(3/2))/(21*
e^4) + (5*d^3*x^3*(d^2 - e^2*x^2)^(3/2))/(24*e^3) - (5*d^2*x^4*(d^2 - e^2*x^2)^(
3/2))/(21*e^2) + (d*x^5*(d^2 - e^2*x^2)^(3/2))/(4*e) - (x^6*(d^2 - e^2*x^2)^(3/2
))/9 - (d^5*(256*d - 315*e*x)*(d^2 - e^2*x^2)^(3/2))/(2016*e^6) - (5*d^9*ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]])/(64*e^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.814862, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}-\frac{5 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{64 e^6}-\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d^7*x*Sqrt[d^2 - e^2*x^2])/(64*e^5) - (4*d^4*x^2*(d^2 - e^2*x^2)^(3/2))/(21*
e^4) + (5*d^3*x^3*(d^2 - e^2*x^2)^(3/2))/(24*e^3) - (5*d^2*x^4*(d^2 - e^2*x^2)^(
3/2))/(21*e^2) + (d*x^5*(d^2 - e^2*x^2)^(3/2))/(4*e) - (x^6*(d^2 - e^2*x^2)^(3/2
))/9 - (d^5*(256*d - 315*e*x)*(d^2 - e^2*x^2)^(3/2))/(2016*e^6) - (5*d^9*ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]])/(64*e^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 89.4965, size = 212, normalized size = 0.93 \[ - \frac{5 d^{9} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{64 e^{6}} + \frac{5 d^{7} x \sqrt{d^{2} - e^{2} x^{2}}}{64 e^{5}} - \frac{2 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{6}} + \frac{5 d^{5} x^{3} \sqrt{d^{2} - e^{2} x^{2}}}{96 e^{3}} + \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{e^{6}} + \frac{d^{3} x^{5} \sqrt{d^{2} - e^{2} x^{2}}}{24 e} - \frac{4 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{7 e^{6}} - \frac{d e x^{7} \sqrt{d^{2} - e^{2} x^{2}}}{4} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{9 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*d**9*atan(e*x/sqrt(d**2 - e**2*x**2))/(64*e**6) + 5*d**7*x*sqrt(d**2 - e**2*x
**2)/(64*e**5) - 2*d**6*(d**2 - e**2*x**2)**(3/2)/(3*e**6) + 5*d**5*x**3*sqrt(d*
*2 - e**2*x**2)/(96*e**3) + d**4*(d**2 - e**2*x**2)**(5/2)/e**6 + d**3*x**5*sqrt
(d**2 - e**2*x**2)/(24*e) - 4*d**2*(d**2 - e**2*x**2)**(7/2)/(7*e**6) - d*e*x**7
*sqrt(d**2 - e**2*x**2)/4 + (d**2 - e**2*x**2)**(9/2)/(9*e**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.134909, size = 135, normalized size = 0.59 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-512 d^8+315 d^7 e x-256 d^6 e^2 x^2+210 d^5 e^3 x^3-192 d^4 e^4 x^4+168 d^3 e^5 x^5+512 d^2 e^6 x^6-1008 d e^7 x^7+448 e^8 x^8\right )-315 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4032 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-512*d^8 + 315*d^7*e*x - 256*d^6*e^2*x^2 + 210*d^5*e^3*x^3
- 192*d^4*e^4*x^4 + 168*d^3*e^5*x^5 + 512*d^2*e^6*x^6 - 1008*d*e^7*x^7 + 448*e^
8*x^8) - 315*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4032*e^6)

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 375, normalized size = 1.6 \[ -{\frac{{x}^{2}}{9\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{29\,{d}^{2}}{63\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{17\,{d}^{3}x}{24\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{85\,{d}^{5}x}{96\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{85\,{d}^{7}x}{64\,{e}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{85\,{d}^{9}}{64\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{dx}{4\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{d}^{4}}{3\,{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{5}x}{6\,{e}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{7}x}{4\,{e}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{5\,{d}^{9}}{4\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{4}}{3\,{e}^{8}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9/e^4*x^2*(-e^2*x^2+d^2)^(7/2)-29/63*d^2/e^6*(-e^2*x^2+d^2)^(7/2)-17/24*d^3/e
^5*x*(-e^2*x^2+d^2)^(5/2)-85/96*d^5/e^5*x*(-e^2*x^2+d^2)^(3/2)-85/64*d^7*x*(-e^2
*x^2+d^2)^(1/2)/e^5-85/64*d^9/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2
)^(1/2))+1/4*d/e^5*x*(-e^2*x^2+d^2)^(7/2)+2/3/e^6*d^4*(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(5/2)+5/6/e^5*d^5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+5/4/e^5*d^7*(-(x+d
/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x+5/4/e^5*d^9/(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*d^4/e^8/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(7/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.294488, size = 806, normalized size = 3.52 \[ \frac{448 \, e^{18} x^{18} - 1008 \, d e^{17} x^{17} - 17856 \, d^{2} e^{16} x^{16} + 41496 \, d^{3} e^{15} x^{15} + 104256 \, d^{4} e^{14} x^{14} - 288918 \, d^{5} e^{13} x^{13} - 157248 \, d^{6} e^{12} x^{12} + 732249 \, d^{7} e^{11} x^{11} - 80640 \, d^{8} e^{10} x^{10} - 779331 \, d^{9} e^{9} x^{9} + 322560 \, d^{10} e^{8} x^{8} + 320040 \, d^{11} e^{7} x^{7} - 172032 \, d^{12} e^{6} x^{6} - 111888 \, d^{13} e^{5} x^{5} + 168000 \, d^{15} e^{3} x^{3} - 80640 \, d^{17} e x + 630 \,{\left (9 \, d^{10} e^{8} x^{8} - 120 \, d^{12} e^{6} x^{6} + 432 \, d^{14} e^{4} x^{4} - 576 \, d^{16} e^{2} x^{2} + 256 \, d^{18} -{\left (d^{9} e^{8} x^{8} - 40 \, d^{11} e^{6} x^{6} + 240 \, d^{13} e^{4} x^{4} - 448 \, d^{15} e^{2} x^{2} + 256 \, d^{17}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (1344 \, d e^{16} x^{16} - 3024 \, d^{2} e^{15} x^{15} - 16384 \, d^{3} e^{14} x^{14} + 40824 \, d^{4} e^{13} x^{13} + 43456 \, d^{5} e^{12} x^{12} - 151242 \, d^{6} e^{11} x^{11} - 5376 \, d^{7} e^{10} x^{10} + 210273 \, d^{8} e^{9} x^{9} - 78848 \, d^{9} e^{8} x^{8} - 100632 \, d^{10} e^{7} x^{7} + 57344 \, d^{11} e^{6} x^{6} + 19376 \, d^{12} e^{5} x^{5} - 42560 \, d^{14} e^{3} x^{3} + 26880 \, d^{16} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4032 \,{\left (9 \, d e^{14} x^{8} - 120 \, d^{3} e^{12} x^{6} + 432 \, d^{5} e^{10} x^{4} - 576 \, d^{7} e^{8} x^{2} + 256 \, d^{9} e^{6} -{\left (e^{14} x^{8} - 40 \, d^{2} e^{12} x^{6} + 240 \, d^{4} e^{10} x^{4} - 448 \, d^{6} e^{8} x^{2} + 256 \, d^{8} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4032*(448*e^18*x^18 - 1008*d*e^17*x^17 - 17856*d^2*e^16*x^16 + 41496*d^3*e^15*
x^15 + 104256*d^4*e^14*x^14 - 288918*d^5*e^13*x^13 - 157248*d^6*e^12*x^12 + 7322
49*d^7*e^11*x^11 - 80640*d^8*e^10*x^10 - 779331*d^9*e^9*x^9 + 322560*d^10*e^8*x^
8 + 320040*d^11*e^7*x^7 - 172032*d^12*e^6*x^6 - 111888*d^13*e^5*x^5 + 168000*d^1
5*e^3*x^3 - 80640*d^17*e*x + 630*(9*d^10*e^8*x^8 - 120*d^12*e^6*x^6 + 432*d^14*e
^4*x^4 - 576*d^16*e^2*x^2 + 256*d^18 - (d^9*e^8*x^8 - 40*d^11*e^6*x^6 + 240*d^13
*e^4*x^4 - 448*d^15*e^2*x^2 + 256*d^17)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(
-e^2*x^2 + d^2))/(e*x)) + 3*(1344*d*e^16*x^16 - 3024*d^2*e^15*x^15 - 16384*d^3*e
^14*x^14 + 40824*d^4*e^13*x^13 + 43456*d^5*e^12*x^12 - 151242*d^6*e^11*x^11 - 53
76*d^7*e^10*x^10 + 210273*d^8*e^9*x^9 - 78848*d^9*e^8*x^8 - 100632*d^10*e^7*x^7
+ 57344*d^11*e^6*x^6 + 19376*d^12*e^5*x^5 - 42560*d^14*e^3*x^3 + 26880*d^16*e*x)
*sqrt(-e^2*x^2 + d^2))/(9*d*e^14*x^8 - 120*d^3*e^12*x^6 + 432*d^5*e^10*x^4 - 576
*d^7*e^8*x^2 + 256*d^9*e^6 - (e^14*x^8 - 40*d^2*e^12*x^6 + 240*d^4*e^10*x^4 - 44
8*d^6*e^8*x^2 + 256*d^8*e^6)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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^5/(e*x + d)^2,x, algorithm="giac")

[Out]

Timed out