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

Optimal. Leaf size=122 \[ \frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(x^4*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^2*(4*d + 5*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (8*d + 15*e*x)/(15*e^6*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/
Sqrt[d^2 - e^2*x^2]]/e^6

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Rubi [A]  time = 0.265515, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^2*(4*d + 5*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (8*d + 15*e*x)/(15*e^6*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/
Sqrt[d^2 - e^2*x^2]]/e^6

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Rubi in Sympy [A]  time = 34.2947, size = 109, normalized size = 0.89 \[ \frac{x^{4} \left (2 d + 2 e x\right )}{10 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{x^{2} \left (16 d + 20 e x\right )}{60 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{32 d + 60 e x}{60 e^{6} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*(2*d + 2*e*x)/(10*e**2*(d**2 - e**2*x**2)**(5/2)) - x**2*(16*d + 20*e*x)/(6
0*e**4*(d**2 - e**2*x**2)**(3/2)) + (32*d + 60*e*x)/(60*e**6*sqrt(d**2 - e**2*x*
*2)) - atan(e*x/sqrt(d**2 - e**2*x**2))/e**6

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Mathematica [A]  time = 0.150678, size = 103, normalized size = 0.84 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (8 d^4+7 d^3 e x-27 d^2 e^2 x^2-8 d e^3 x^3+23 e^4 x^4\right )}{(d-e x)^3 (d+e x)^2}-15 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^6} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(8*d^4 + 7*d^3*e*x - 27*d^2*e^2*x^2 - 8*d*e^3*x^3 + 23*e^4
*x^4))/((d - e*x)^3*(d + e*x)^2) - 15*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6
)

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Maple [A]  time = 0.03, size = 166, normalized size = 1.4 \[{\frac{d{x}^{4}}{{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,{d}^{3}{x}^{2}}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{d}^{5}}{15\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{{x}^{5}}{5\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{x}^{3}}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{x}{{e}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x^4/e^2/(-e^2*x^2+d^2)^(5/2)-4/3*d^3/e^4*x^2/(-e^2*x^2+d^2)^(5/2)+8/15*d^5/e^6
/(-e^2*x^2+d^2)^(5/2)+1/5*x^5/e/(-e^2*x^2+d^2)^(5/2)-1/3/e^3*x^3/(-e^2*x^2+d^2)^
(3/2)+1/e^5*x/(-e^2*x^2+d^2)^(1/2)-1/e^5/(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 = 0.798007, size = 355, normalized size = 2.91 \[ \frac{1}{15} \, e x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )}}{3 \, e} + \frac{d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{4 \, d^{3} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{5}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{4 \, d^{2} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{5}} - \frac{7 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{5}} - \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*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)) - 1/3*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 + d*x^4/((-e^2*x^2 + d^2)^(5/2
)*e^2) - 4/3*d^3*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8/15*d^5/((-e^2*x^2 + d^2)^(
5/2)*e^6) + 4/15*d^2*x/((-e^2*x^2 + d^2)^(3/2)*e^5) - 7/15*x/(sqrt(-e^2*x^2 + d^
2)*e^5) - arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^5)

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Fricas [A]  time = 0.294262, size = 729, normalized size = 5.98 \[ -\frac{23 \, e^{8} x^{8} - 40 \, d e^{7} x^{7} - 179 \, d^{2} e^{6} x^{6} + 199 \, d^{3} e^{5} x^{5} + 280 \, d^{4} e^{4} x^{4} - 280 \, d^{5} e^{3} x^{3} - 120 \, d^{6} e^{2} x^{2} + 120 \, d^{7} e x - 30 \,{\left (4 \, d e^{7} x^{7} - 4 \, d^{2} e^{6} x^{6} - 16 \, d^{3} e^{5} x^{5} + 16 \, d^{4} e^{4} x^{4} + 20 \, d^{5} e^{3} x^{3} - 20 \, d^{6} e^{2} x^{2} - 8 \, d^{7} e x + 8 \, d^{8} -{\left (e^{7} x^{7} - d e^{6} x^{6} - 9 \, d^{2} e^{5} x^{5} + 9 \, d^{3} e^{4} x^{4} + 16 \, d^{4} e^{3} x^{3} - 16 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x + 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \,{\left (2 \, e^{7} x^{7} + 21 \, d e^{6} x^{6} - 26 \, d^{2} e^{5} x^{5} - 55 \, d^{3} e^{4} x^{4} + 55 \, d^{4} e^{3} x^{3} + 30 \, d^{5} e^{2} x^{2} - 30 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d e^{13} x^{7} - 4 \, d^{2} e^{12} x^{6} - 16 \, d^{3} e^{11} x^{5} + 16 \, d^{4} e^{10} x^{4} + 20 \, d^{5} e^{9} x^{3} - 20 \, d^{6} e^{8} x^{2} - 8 \, d^{7} e^{7} x + 8 \, d^{8} e^{6} -{\left (e^{13} x^{7} - d e^{12} x^{6} - 9 \, d^{2} e^{11} x^{5} + 9 \, d^{3} e^{10} x^{4} + 16 \, d^{4} e^{9} x^{3} - 16 \, d^{5} e^{8} x^{2} - 8 \, d^{6} e^{7} x + 8 \, d^{7} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(23*e^8*x^8 - 40*d*e^7*x^7 - 179*d^2*e^6*x^6 + 199*d^3*e^5*x^5 + 280*d^4*e
^4*x^4 - 280*d^5*e^3*x^3 - 120*d^6*e^2*x^2 + 120*d^7*e*x - 30*(4*d*e^7*x^7 - 4*d
^2*e^6*x^6 - 16*d^3*e^5*x^5 + 16*d^4*e^4*x^4 + 20*d^5*e^3*x^3 - 20*d^6*e^2*x^2 -
8*d^7*e*x + 8*d^8 - (e^7*x^7 - d*e^6*x^6 - 9*d^2*e^5*x^5 + 9*d^3*e^4*x^4 + 16*d
^4*e^3*x^3 - 16*d^5*e^2*x^2 - 8*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*(2*e^7*x^7 + 21*d*e^6*x^6 - 26*d^2*e^5*x^5
- 55*d^3*e^4*x^4 + 55*d^4*e^3*x^3 + 30*d^5*e^2*x^2 - 30*d^6*e*x)*sqrt(-e^2*x^2 +
d^2))/(4*d*e^13*x^7 - 4*d^2*e^12*x^6 - 16*d^3*e^11*x^5 + 16*d^4*e^10*x^4 + 20*d
^5*e^9*x^3 - 20*d^6*e^8*x^2 - 8*d^7*e^7*x + 8*d^8*e^6 - (e^13*x^7 - d*e^12*x^6 -
9*d^2*e^11*x^5 + 9*d^3*e^10*x^4 + 16*d^4*e^9*x^3 - 16*d^5*e^8*x^2 - 8*d^6*e^7*x
+ 8*d^7*e^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 39.8303, size = 1739, normalized size = 14.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((8*d**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqr
t(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)) - 20*d**2*e**2*x**2/
(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2)
+ 15*e**10*x**4*sqrt(d**2 - e**2*x**2)) + 15*e**4*x**4/(15*d**4*e**6*sqrt(d**2 -
e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2
- e**2*x**2)), Ne(e, 0)), (x**6/(6*(d**2)**(7/2)), True)) + e*Piecewise((30*I*d
**5*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**
2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**
2*x**2/d**2)) - 15*pi*d**5*sqrt(-1 + e**2*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**
2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqr
t(-1 + e**2*x**2/d**2)) - 30*I*d**4*e*x/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2)
- 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x
**2/d**2)) - 60*I*d**3*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5
*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) +
30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*pi*d**3*e**2*x**2*sqrt(-1 + e**2
*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1
+ e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 70*I*d**2*e**3*
x**3/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*
x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*I*d*e**4*x**4*sqrt(
-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d
**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d*
*2)) - 15*pi*d*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*
x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(
-1 + e**2*x**2/d**2)) - 46*I*e**5*x**5/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) -
60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x*
*2/d**2)), Abs(e**2*x**2/d**2) > 1), (-15*d**5*sqrt(1 - e**2*x**2/d**2)*asin(e*x
/d)/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**
2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) + 15*d**4*e*x/(15*d**5*e**7*
sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**
11*x**4*sqrt(1 - e**2*x**2/d**2)) + 30*d**3*e**2*x**2*sqrt(1 - e**2*x**2/d**2)*a
sin(e*x/d)/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e
**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) - 35*d**2*e**3*x**3/(
15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**
2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) - 15*d*e**4*x**4*sqrt(1 - e**2*x*
*2/d**2)*asin(e*x/d)/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*
sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) + 23*e**5*x
**5/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**
2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)), True))

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GIAC/XCAS [A]  time = 0.296982, size = 131, normalized size = 1.07 \[ -\arcsin \left (\frac{x e}{d}\right ) e^{\left (-6\right )}{\rm sign}\left (d\right ) - \frac{{\left (8 \, d^{5} e^{\left (-6\right )} +{\left (15 \, d^{4} e^{\left (-5\right )} -{\left (20 \, d^{3} e^{\left (-4\right )} +{\left (35 \, d^{2} e^{\left (-3\right )} -{\left (23 \, x e^{\left (-1\right )} + 15 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(x*e/d)*e^(-6)*sign(d) - 1/15*(8*d^5*e^(-6) + (15*d^4*e^(-5) - (20*d^3*e^
(-4) + (35*d^2*e^(-3) - (23*x*e^(-1) + 15*d*e^(-2))*x)*x)*x)*x)*sqrt(-x^2*e^2 +
d^2)/(x^2*e^2 - d^2)^3