3.83 \(\int \frac{x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}-\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(d^4*(d + e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (23*d^3*(d + e*x)^2)/(15*e^6*(
d^2 - e^2*x^2)^(3/2)) + (127*d^2*(d + e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + (3*d*
Sqrt[d^2 - e^2*x^2])/e^6 + (x*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (13*d^2*ArcTan[(e*x
)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

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Rubi [A]  time = 0.595229, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}-\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d^4*(d + e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (23*d^3*(d + e*x)^2)/(15*e^6*(
d^2 - e^2*x^2)^(3/2)) + (127*d^2*(d + e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + (3*d*
Sqrt[d^2 - e^2*x^2])/e^6 + (x*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (13*d^2*ArcTan[(e*x
)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

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Rubi in Sympy [A]  time = 68.4353, size = 156, normalized size = 0.9 \[ \frac{d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{6} \left (d - e x\right )^{3}} - \frac{23 d^{3} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6} \left (d - e x\right )^{2}} - \frac{13 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{6}} + \frac{127 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6} \left (d - e x\right )} + \frac{3 d \sqrt{d^{2} - e^{2} x^{2}}}{e^{6}} + \frac{x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d**4*sqrt(d**2 - e**2*x**2)/(5*e**6*(d - e*x)**3) - 23*d**3*sqrt(d**2 - e**2*x**
2)/(15*e**6*(d - e*x)**2) - 13*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e**6) +
127*d**2*sqrt(d**2 - e**2*x**2)/(15*e**6*(d - e*x)) + 3*d*sqrt(d**2 - e**2*x**2)
/e**6 + x*sqrt(d**2 - e**2*x**2)/(2*e**5)

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Mathematica [A]  time = 0.134595, size = 108, normalized size = 0.62 \[ \frac{195 d^2 (d-e x)^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-304 d^4+717 d^3 e x-479 d^2 e^2 x^2+45 d e^3 x^3+15 e^4 x^4\right )}{30 e^6 (e x-d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-304*d^4 + 717*d^3*e*x - 479*d^2*e^2*x^2 + 45*d*e^3*x^3 +
15*e^4*x^4) + 195*d^2*(d - e*x)^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^6*(-d
+ e*x)^3)

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Maple [A]  time = 0.014, size = 222, normalized size = 1.3 \[ 19\,{\frac{{d}^{3}{x}^{4}}{{e}^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{76\,{d}^{5}{x}^{2}}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{152\,{d}^{7}}{15\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{e{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{13\,{d}^{2}{x}^{5}}{10\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{13\,{d}^{2}{x}^{3}}{6\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{13\,{d}^{2}x}{2\,{e}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{13\,{d}^{2}}{2\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-3\,{\frac{d{x}^{6}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

19*d^3*x^4/e^2/(-e^2*x^2+d^2)^(5/2)-76/3*d^5/e^4*x^2/(-e^2*x^2+d^2)^(5/2)+152/15
*d^7/e^6/(-e^2*x^2+d^2)^(5/2)-1/2*e*x^7/(-e^2*x^2+d^2)^(5/2)+13/10/e*d^2*x^5/(-e
^2*x^2+d^2)^(5/2)-13/6/e^3*d^2*x^3/(-e^2*x^2+d^2)^(3/2)+13/2/e^5*d^2*x/(-e^2*x^2
+d^2)^(1/2)-13/2/e^5*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-
3*d*x^6/(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.804762, size = 429, normalized size = 2.47 \[ -\frac{e x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{13}{30} \, d^{2} e x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{3 \, d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, d^{2} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )}}{6 \, e} + \frac{19 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{76 \, d^{5} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{152 \, d^{7}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{26 \, d^{4} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{5}} - \frac{91 \, d^{2} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{5}} - \frac{13 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*x^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*e*x^7/(-e^2*x^2 + d^2)^(5/2) + 13/30*d^2*e*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2
)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)
*e^6)) - 3*d*x^6/(-e^2*x^2 + d^2)^(5/2) - 13/6*d^2*x*(3*x^2/((-e^2*x^2 + d^2)^(3
/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e + 19*d^3*x^4/((-e^2*x^2 + d^2)^
(5/2)*e^2) - 76/3*d^5*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 152/15*d^7/((-e^2*x^2 +
d^2)^(5/2)*e^6) + 26/15*d^4*x/((-e^2*x^2 + d^2)^(3/2)*e^5) - 91/30*d^2*x/(sqrt(
-e^2*x^2 + d^2)*e^5) - 13/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^5)

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Fricas [A]  time = 0.30001, size = 729, normalized size = 4.19 \[ -\frac{15 \, e^{9} x^{9} + 120 \, d e^{8} x^{8} - 678 \, d^{2} e^{7} x^{7} - 210 \, d^{3} e^{6} x^{6} + 5421 \, d^{4} e^{5} x^{5} - 6500 \, d^{5} e^{4} x^{4} - 2860 \, d^{6} e^{3} x^{3} + 7800 \, d^{7} e^{2} x^{2} - 3120 \, d^{8} e x - 390 \,{\left (d^{2} e^{7} x^{7} - 7 \, d^{3} e^{6} x^{6} + 3 \, d^{4} e^{5} x^{5} + 31 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} - 12 \, d^{7} e^{2} x^{2} + 40 \, d^{8} e x - 16 \, d^{9} +{\left (d^{2} e^{6} x^{6} + 2 \, d^{3} e^{5} x^{5} - 19 \, d^{4} e^{4} x^{4} + 20 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 40 \, d^{7} e x + 16 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{8} x^{8} - 15 \, d e^{7} x^{7} - 535 \, d^{2} e^{6} x^{6} + 2821 \, d^{3} e^{5} x^{5} - 2600 \, d^{4} e^{4} x^{4} - 4420 \, d^{5} e^{3} x^{3} + 7800 \, d^{6} e^{2} x^{2} - 3120 \, d^{7} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{13} x^{7} - 7 \, d e^{12} x^{6} + 3 \, d^{2} e^{11} x^{5} + 31 \, d^{3} e^{10} x^{4} - 40 \, d^{4} e^{9} x^{3} - 12 \, d^{5} e^{8} x^{2} + 40 \, d^{6} e^{7} x - 16 \, d^{7} e^{6} +{\left (e^{12} x^{6} + 2 \, d e^{11} x^{5} - 19 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 20 \, d^{4} e^{8} x^{2} - 40 \, d^{5} e^{7} x + 16 \, d^{6} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*x^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/30*(15*e^9*x^9 + 120*d*e^8*x^8 - 678*d^2*e^7*x^7 - 210*d^3*e^6*x^6 + 5421*d^4
*e^5*x^5 - 6500*d^5*e^4*x^4 - 2860*d^6*e^3*x^3 + 7800*d^7*e^2*x^2 - 3120*d^8*e*x
- 390*(d^2*e^7*x^7 - 7*d^3*e^6*x^6 + 3*d^4*e^5*x^5 + 31*d^5*e^4*x^4 - 40*d^6*e^
3*x^3 - 12*d^7*e^2*x^2 + 40*d^8*e*x - 16*d^9 + (d^2*e^6*x^6 + 2*d^3*e^5*x^5 - 19
*d^4*e^4*x^4 + 20*d^5*e^3*x^3 + 20*d^6*e^2*x^2 - 40*d^7*e*x + 16*d^8)*sqrt(-e^2*
x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (15*e^8*x^8 - 15*d*e^7*x
^7 - 535*d^2*e^6*x^6 + 2821*d^3*e^5*x^5 - 2600*d^4*e^4*x^4 - 4420*d^5*e^3*x^3 +
7800*d^6*e^2*x^2 - 3120*d^7*e*x)*sqrt(-e^2*x^2 + d^2))/(e^13*x^7 - 7*d*e^12*x^6
+ 3*d^2*e^11*x^5 + 31*d^3*e^10*x^4 - 40*d^4*e^9*x^3 - 12*d^5*e^8*x^2 + 40*d^6*e^
7*x - 16*d^7*e^6 + (e^12*x^6 + 2*d*e^11*x^5 - 19*d^2*e^10*x^4 + 20*d^3*e^9*x^3 +
20*d^4*e^8*x^2 - 40*d^5*e^7*x + 16*d^6*e^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x**5*(d + e*x)**3/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.30419, size = 159, normalized size = 0.91 \[ -\frac{13}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-6\right )}{\rm sign}\left (d\right ) - \frac{{\left (304 \, d^{7} e^{\left (-6\right )} +{\left (195 \, d^{6} e^{\left (-5\right )} -{\left (760 \, d^{5} e^{\left (-4\right )} +{\left (455 \, d^{4} e^{\left (-3\right )} -{\left (570 \, d^{3} e^{\left (-2\right )} +{\left (299 \, d^{2} e^{\left (-1\right )} - 15 \,{\left (x e + 6 \, d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{30 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*x^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-13/2*d^2*arcsin(x*e/d)*e^(-6)*sign(d) - 1/30*(304*d^7*e^(-6) + (195*d^6*e^(-5)
- (760*d^5*e^(-4) + (455*d^4*e^(-3) - (570*d^3*e^(-2) + (299*d^2*e^(-1) - 15*(x*
e + 6*d)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3