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

Optimal. Leaf size=143 \[ \frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}+\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(d^4*(d + e*x)^2)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (22*d^3*(d + e*x))/(15*e^6*(d^
2 - e^2*x^2)^(3/2)) + (2*d*(30*d + 23*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + Sqrt[
d^2 - e^2*x^2]/e^6 - (2*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

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Rubi [A]  time = 0.415825, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}+\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d^4*(d + e*x)^2)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (22*d^3*(d + e*x))/(15*e^6*(d^
2 - e^2*x^2)^(3/2)) + (2*d*(30*d + 23*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + Sqrt[
d^2 - e^2*x^2]/e^6 - (2*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

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Rubi in Sympy [A]  time = 75.2747, size = 144, normalized size = 1.01 \[ \frac{d^{4}}{5 e^{6} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{22 d^{3}}{15 e^{6} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{4 d^{2}}{e^{6} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{46 d x}{15 e^{5} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{2 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{6}} + \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**4/(5*e**6*(d - e*x)**2*sqrt(d**2 - e**2*x**2)) - 22*d**3/(15*e**6*(d - e*x)*s
qrt(d**2 - e**2*x**2)) + 4*d**2/(e**6*sqrt(d**2 - e**2*x**2)) + 46*d*x/(15*e**5*
sqrt(d**2 - e**2*x**2)) - 2*d*atan(e*x/sqrt(d**2 - e**2*x**2))/e**6 + sqrt(d**2
- e**2*x**2)/e**6

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Mathematica [A]  time = 0.137542, size = 118, normalized size = 0.83 \[ \sqrt{d^2-e^2 x^2} \left (-\frac{d^3}{10 e^6 (e x-d)^3}-\frac{41 d^2}{60 e^6 (e x-d)^2}-\frac{383 d}{120 e^6 (e x-d)}+\frac{d}{8 e^6 (d+e x)}+\frac{1}{e^6}\right )-\frac{2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d^2 - e^2*x^2]*(e^(-6) - d^3/(10*e^6*(-d + e*x)^3) - (41*d^2)/(60*e^6*(-d +
e*x)^2) - (383*d)/(120*e^6*(-d + e*x)) + d/(8*e^6*(d + e*x))) - (2*d*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/e^6

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Maple [A]  time = 0.013, size = 193, normalized size = 1.4 \[ 7\,{\frac{{d}^{2}{x}^{4}}{{e}^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{28\,{d}^{4}{x}^{2}}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{56\,{d}^{6}}{15\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{{x}^{6} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,d{x}^{5}}{5\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{2\,d{x}^{3}}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{dx}{{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-2\,{\frac{d}{{e}^{5}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*d^2*x^4/e^2/(-e^2*x^2+d^2)^(5/2)-28/3*d^4/e^4*x^2/(-e^2*x^2+d^2)^(5/2)+56/15*d
^6/e^6/(-e^2*x^2+d^2)^(5/2)-x^6/(-e^2*x^2+d^2)^(5/2)+2/5/e*d*x^5/(-e^2*x^2+d^2)^
(5/2)-2/3/e^3*d*x^3/(-e^2*x^2+d^2)^(3/2)+2/e^5*d*x/(-e^2*x^2+d^2)^(1/2)-2/e^5*d/
(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

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Maxima [A]  time = 0.794713, size = 390, normalized size = 2.73 \[ \frac{2}{15} \, d e x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{2 \, d x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )}}{3 \, e} + \frac{7 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{28 \, d^{4} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{56 \, d^{6}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{8 \, d^{3} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{5}} - \frac{14 \, d x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{5}} - \frac{2 \, d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*d*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)) - x^6/(-e^2*x^2 + d^2)^(5/2) - 2
/3*d*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 + 7*d^2*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 28/3*d^4*x^2/((-e^2*x^2 + d^2)^(5/
2)*e^4) + 56/15*d^6/((-e^2*x^2 + d^2)^(5/2)*e^6) + 8/15*d^3*x/((-e^2*x^2 + d^2)^
(3/2)*e^5) - 14/15*d*x/(sqrt(-e^2*x^2 + d^2)*e^5) - 2*d*arcsin(e^2*x/sqrt(d^2*e^
2))/(sqrt(e^2)*e^5)

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Fricas [A]  time = 0.280377, size = 635, normalized size = 4.44 \[ \frac{15 \, e^{8} x^{8} - 76 \, d e^{7} x^{7} + 136 \, d^{2} e^{6} x^{6} + 242 \, d^{3} e^{5} x^{5} - 640 \, d^{4} e^{4} x^{4} + 80 \, d^{5} e^{3} x^{3} + 480 \, d^{6} e^{2} x^{2} - 240 \, d^{7} e x + 60 \,{\left (4 \, d^{2} e^{6} x^{6} - 8 \, d^{3} e^{5} x^{5} - 8 \, d^{4} e^{4} x^{4} + 24 \, d^{5} e^{3} x^{3} - 4 \, d^{6} e^{2} x^{2} - 16 \, d^{7} e x + 8 \, d^{8} -{\left (d e^{6} x^{6} - 2 \, d^{2} e^{5} x^{5} - 7 \, d^{3} e^{4} x^{4} + 16 \, d^{4} e^{3} x^{3} - 16 \, d^{6} e x + 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \,{\left (d e^{6} x^{6} - 48 \, d^{2} e^{5} x^{5} + 100 \, d^{3} e^{4} x^{4} + 10 \, d^{4} e^{3} x^{3} - 120 \, d^{5} e^{2} x^{2} + 60 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d e^{12} x^{6} - 8 \, d^{2} e^{11} x^{5} - 8 \, d^{3} e^{10} x^{4} + 24 \, d^{4} e^{9} x^{3} - 4 \, d^{5} e^{8} x^{2} - 16 \, d^{6} e^{7} x + 8 \, d^{7} e^{6} -{\left (e^{12} x^{6} - 2 \, d e^{11} x^{5} - 7 \, d^{2} e^{10} x^{4} + 16 \, d^{3} e^{9} x^{3} - 16 \, d^{5} e^{7} x + 8 \, d^{6} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(15*e^8*x^8 - 76*d*e^7*x^7 + 136*d^2*e^6*x^6 + 242*d^3*e^5*x^5 - 640*d^4*e^
4*x^4 + 80*d^5*e^3*x^3 + 480*d^6*e^2*x^2 - 240*d^7*e*x + 60*(4*d^2*e^6*x^6 - 8*d
^3*e^5*x^5 - 8*d^4*e^4*x^4 + 24*d^5*e^3*x^3 - 4*d^6*e^2*x^2 - 16*d^7*e*x + 8*d^8
- (d*e^6*x^6 - 2*d^2*e^5*x^5 - 7*d^3*e^4*x^4 + 16*d^4*e^3*x^3 - 16*d^6*e*x + 8*
d^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 4*(d*e^6*
x^6 - 48*d^2*e^5*x^5 + 100*d^3*e^4*x^4 + 10*d^4*e^3*x^3 - 120*d^5*e^2*x^2 + 60*d
^6*e*x)*sqrt(-e^2*x^2 + d^2))/(4*d*e^12*x^6 - 8*d^2*e^11*x^5 - 8*d^3*e^10*x^4 +
24*d^4*e^9*x^3 - 4*d^5*e^8*x^2 - 16*d^6*e^7*x + 8*d^7*e^6 - (e^12*x^6 - 2*d*e^11
*x^5 - 7*d^2*e^10*x^4 + 16*d^3*e^9*x^3 - 16*d^5*e^7*x + 8*d^6*e^6)*sqrt(-e^2*x^2
+ d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.300413, size = 143, normalized size = 1. \[ -2 \, d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-6\right )}{\rm sign}\left (d\right ) - \frac{{\left (56 \, d^{6} e^{\left (-6\right )} +{\left (30 \, d^{5} e^{\left (-5\right )} -{\left (140 \, d^{4} e^{\left (-4\right )} +{\left (70 \, d^{3} e^{\left (-3\right )} -{\left (105 \, d^{2} e^{\left (-2\right )} +{\left (46 \, d e^{\left (-1\right )} - 15 \, x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d*arcsin(x*e/d)*e^(-6)*sign(d) - 1/15*(56*d^6*e^(-6) + (30*d^5*e^(-5) - (140*
d^4*e^(-4) + (70*d^3*e^(-3) - (105*d^2*e^(-2) + (46*d*e^(-1) - 15*x)*x)*x)*x)*x)
*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3