3.159 \(\int \frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=171 \[ -\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4}-\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4} \]

[Out]

-(d^5*x*Sqrt[d^2 - e^2*x^2])/(8*e^3) - (11*d^2*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^
2) + (d*x^3*(d^2 - e^2*x^2)^(3/2))/(3*e) - (x^4*(d^2 - e^2*x^2)^(3/2))/7 - (d^3*
(88*d - 105*e*x)*(d^2 - e^2*x^2)^(3/2))/(420*e^4) - (d^7*ArcTan[(e*x)/Sqrt[d^2 -
e^2*x^2]])/(8*e^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.532324, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4}-\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]

[Out]

-(d^5*x*Sqrt[d^2 - e^2*x^2])/(8*e^3) - (11*d^2*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^
2) + (d*x^3*(d^2 - e^2*x^2)^(3/2))/(3*e) - (x^4*(d^2 - e^2*x^2)^(3/2))/7 - (d^3*
(88*d - 105*e*x)*(d^2 - e^2*x^2)^(3/2))/(420*e^4) - (d^7*ArcTan[(e*x)/Sqrt[d^2 -
e^2*x^2]])/(8*e^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 69.4826, size = 162, normalized size = 0.95 \[ - \frac{d^{7} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e^{4}} + \frac{d^{5} x \sqrt{d^{2} - e^{2} x^{2}}}{8 e^{3}} - \frac{2 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{4}} + \frac{d^{3} x^{3} \sqrt{d^{2} - e^{2} x^{2}}}{12 e} + \frac{3 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{4}} - \frac{d e x^{5} \sqrt{d^{2} - e^{2} x^{2}}}{3} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{7 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**2,x)

[Out]

-d**7*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e**4) + d**5*x*sqrt(d**2 - e**2*x**2)/
(8*e**3) - 2*d**4*(d**2 - e**2*x**2)**(3/2)/(3*e**4) + d**3*x**3*sqrt(d**2 - e**
2*x**2)/(12*e) + 3*d**2*(d**2 - e**2*x**2)**(5/2)/(5*e**4) - d*e*x**5*sqrt(d**2
- e**2*x**2)/3 - (d**2 - e**2*x**2)**(7/2)/(7*e**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.105648, size = 113, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-176 d^6+105 d^5 e x-88 d^4 e^2 x^2+70 d^3 e^3 x^3+144 d^2 e^4 x^4-280 d e^5 x^5+120 e^6 x^6\right )-105 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{840 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-176*d^6 + 105*d^5*e*x - 88*d^4*e^2*x^2 + 70*d^3*e^3*x^3 +
144*d^2*e^4*x^4 - 280*d*e^5*x^5 + 120*e^6*x^6) - 105*d^7*ArcTan[(e*x)/Sqrt[d^2
- e^2*x^2]])/(840*e^4)

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 327, normalized size = 1.9 \[ -{\frac{1}{7\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{dx}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{3}x}{12\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{5}x}{8\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{5\,{d}^{7}}{8\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{4\,{d}^{2}}{15\,{e}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}x}{3\,{e}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}x}{2\,{e}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{d}^{7}}{2\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{2}}{3\,{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^2,x)

[Out]

-1/7/e^4*(-e^2*x^2+d^2)^(7/2)-1/3*d/e^3*x*(-e^2*x^2+d^2)^(5/2)-5/12/e^3*d^3*x*(-
e^2*x^2+d^2)^(3/2)-5/8*d^5*x*(-e^2*x^2+d^2)^(1/2)/e^3-5/8/e^3*d^7/(e^2)^(1/2)*ar
ctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+4/15/e^4*d^2*(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(5/2)+1/3/e^3*d^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+1/2/e^3*d^5*(-(x+d
/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x+1/2/e^3*d^7/(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^2/e^6/(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^3/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.289817, size = 657, normalized size = 3.84 \[ \frac{120 \, e^{14} x^{14} - 280 \, d e^{13} x^{13} - 2856 \, d^{2} e^{12} x^{12} + 7070 \, d^{3} e^{11} x^{11} + 8792 \, d^{4} e^{10} x^{10} - 30765 \, d^{5} e^{9} x^{9} - 280 \, d^{6} e^{8} x^{8} + 44975 \, d^{7} e^{7} x^{7} - 19040 \, d^{8} e^{6} x^{6} - 17080 \, d^{9} e^{5} x^{5} + 13440 \, d^{10} e^{4} x^{4} - 10640 \, d^{11} e^{3} x^{3} + 6720 \, d^{13} e x + 210 \,{\left (7 \, d^{8} e^{6} x^{6} - 56 \, d^{10} e^{4} x^{4} + 112 \, d^{12} e^{2} x^{2} - 64 \, d^{14} -{\left (d^{7} e^{6} x^{6} - 24 \, d^{9} e^{4} x^{4} + 80 \, d^{11} e^{2} x^{2} - 64 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 7 \,{\left (120 \, d e^{12} x^{12} - 280 \, d^{2} e^{11} x^{11} - 816 \, d^{3} e^{10} x^{10} + 2310 \, d^{4} e^{9} x^{9} + 680 \, d^{5} e^{8} x^{8} - 4935 \, d^{6} e^{7} x^{7} + 1760 \, d^{7} e^{6} x^{6} + 2840 \, d^{8} e^{5} x^{5} - 1920 \, d^{9} e^{4} x^{4} + 1040 \, d^{10} e^{3} x^{3} - 960 \, d^{12} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{840 \,{\left (7 \, d e^{10} x^{6} - 56 \, d^{3} e^{8} x^{4} + 112 \, d^{5} e^{6} x^{2} - 64 \, d^{7} e^{4} -{\left (e^{10} x^{6} - 24 \, d^{2} e^{8} x^{4} + 80 \, d^{4} e^{6} x^{2} - 64 \, d^{6} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*x^3/(e*x + d)^2,x, algorithm="fricas")

[Out]

1/840*(120*e^14*x^14 - 280*d*e^13*x^13 - 2856*d^2*e^12*x^12 + 7070*d^3*e^11*x^11
+ 8792*d^4*e^10*x^10 - 30765*d^5*e^9*x^9 - 280*d^6*e^8*x^8 + 44975*d^7*e^7*x^7
- 19040*d^8*e^6*x^6 - 17080*d^9*e^5*x^5 + 13440*d^10*e^4*x^4 - 10640*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*(120*d*e^12*x^12 - 280*d^
2*e^11*x^11 - 816*d^3*e^10*x^10 + 2310*d^4*e^9*x^9 + 680*d^5*e^8*x^8 - 4935*d^6*
e^7*x^7 + 1760*d^7*e^6*x^6 + 2840*d^8*e^5*x^5 - 1920*d^9*e^4*x^4 + 1040*d^10*e^3
*x^3 - 960*d^12*e*x)*sqrt(-e^2*x^2 + d^2))/(7*d*e^10*x^6 - 56*d^3*e^8*x^4 + 112*
d^5*e^6*x^2 - 64*d^7*e^4 - (e^10*x^6 - 24*d^2*e^8*x^4 + 80*d^4*e^6*x^2 - 64*d^6*
e^4)*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**3*(-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^3/(e*x + d)^2,x, algorithm="giac")

[Out]

Timed out