3.20 \(\int \frac{x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}+\frac{x (5 d+8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(x^5*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^3*(5*d + 6*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (x*(5*d + 8*e*x))/(5*e^6*Sqrt[d^2 - e^2*x^2]) + (16*Sqrt[d^
2 - e^2*x^2])/(5*e^7) - (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^7

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Rubi [A]  time = 0.362028, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}+\frac{x (5 d+8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(x^5*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^3*(5*d + 6*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (x*(5*d + 8*e*x))/(5*e^6*Sqrt[d^2 - e^2*x^2]) + (16*Sqrt[d^
2 - e^2*x^2])/(5*e^7) - (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^7

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Rubi in Sympy [A]  time = 45.8304, size = 133, normalized size = 0.9 \[ - \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{7}} + \frac{x^{5} \left (2 d + 2 e x\right )}{10 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{x^{3} \left (20 d + 24 e x\right )}{60 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{x \left (120 d + 192 e x\right )}{120 e^{6} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{16 \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-d*atan(e*x/sqrt(d**2 - e**2*x**2))/e**7 + x**5*(2*d + 2*e*x)/(10*e**2*(d**2 - e
**2*x**2)**(5/2)) - x**3*(20*d + 24*e*x)/(60*e**4*(d**2 - e**2*x**2)**(3/2)) + x
*(120*d + 192*e*x)/(120*e**6*sqrt(d**2 - e**2*x**2)) + 16*sqrt(d**2 - e**2*x**2)
/(5*e**7)

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Mathematica [A]  time = 0.172396, size = 115, normalized size = 0.78 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (48 d^5-33 d^4 e x-87 d^3 e^2 x^2+52 d^2 e^3 x^3+38 d e^4 x^4-15 e^5 x^5\right )}{(d-e x)^3 (d+e x)^2}-15 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(48*d^5 - 33*d^4*e*x - 87*d^3*e^2*x^2 + 52*d^2*e^3*x^3 + 3
8*d*e^4*x^4 - 15*e^5*x^5))/((d - e*x)^3*(d + e*x)^2) - 15*d*ArcTan[(e*x)/Sqrt[d^
2 - e^2*x^2]])/(15*e^7)

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Maple [A]  time = 0.034, size = 195, normalized size = 1.3 \[{\frac{d{x}^{5}}{5\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{d{x}^{3}}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{dx}{{e}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{d}{{e}^{6}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{x}^{6}}{e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+6\,{\frac{{d}^{2}{x}^{4}}{{e}^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-8\,{\frac{{d}^{4}{x}^{2}}{{e}^{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{16\,{d}^{6}}{5\,{e}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/5*d*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/3*d/e^4*x^3/(-e^2*x^2+d^2)^(3/2)+d/e^6*x/(-
e^2*x^2+d^2)^(1/2)-d/e^6/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-
x^6/e/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^3*x^4/(-e^2*x^2+d^2)^(5/2)-8*d^4/e^5*x^2/(-e^
2*x^2+d^2)^(5/2)+16/5*d^6/e^7/(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.804294, size = 393, normalized size = 2.67 \[ \frac{1}{15} \, d x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} - \frac{d x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )}}{3 \, e^{2}} + \frac{6 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{8 \, d^{4} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{5}} + \frac{16 \, d^{6}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{7}} + \frac{4 \, d^{3} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{6}} - \frac{7 \, d x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{6}} - \frac{d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^6/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*d*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)*e) -
1/3*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^2 + 6*d^2*x^4/((-e^2*x^2 + d^2)^(5/2)*e^3) - 8*d^4*x^2/((-e^2*x^2 + d^2)^(5
/2)*e^5) + 16/5*d^6/((-e^2*x^2 + d^2)^(5/2)*e^7) + 4/15*d^3*x/((-e^2*x^2 + d^2)^
(3/2)*e^6) - 7/15*d*x/(sqrt(-e^2*x^2 + d^2)*e^6) - d*arcsin(e^2*x/sqrt(d^2*e^2))
/(sqrt(e^2)*e^6)

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Fricas [A]  time = 0.301228, size = 841, normalized size = 5.72 \[ -\frac{27 \, d e^{9} x^{9} - 142 \, d^{2} e^{8} x^{8} + 112 \, d^{3} e^{7} x^{7} + 523 \, d^{4} e^{6} x^{6} - 523 \, d^{5} e^{5} x^{5} - 620 \, d^{6} e^{4} x^{4} + 620 \, d^{7} e^{3} x^{3} + 240 \, d^{8} e^{2} x^{2} - 240 \, d^{9} e x - 30 \,{\left (d e^{9} x^{9} - d^{2} e^{8} x^{8} - 14 \, d^{3} e^{7} x^{7} + 14 \, d^{4} e^{6} x^{6} + 41 \, d^{5} e^{5} x^{5} - 41 \, d^{6} e^{4} x^{4} - 44 \, d^{7} e^{3} x^{3} + 44 \, d^{8} e^{2} x^{2} + 16 \, d^{9} e x - 16 \, d^{10} +{\left (5 \, d^{2} e^{7} x^{7} - 5 \, d^{3} e^{6} x^{6} - 25 \, d^{4} e^{5} x^{5} + 25 \, d^{5} e^{4} x^{4} + 36 \, d^{6} e^{3} x^{3} - 36 \, d^{7} e^{2} x^{2} - 16 \, d^{8} e x + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{9} x^{9} - 38 \, d e^{8} x^{8} + 8 \, d^{2} e^{7} x^{7} + 303 \, d^{3} e^{6} x^{6} - 303 \, d^{4} e^{5} x^{5} - 500 \, d^{5} e^{4} x^{4} + 500 \, d^{6} e^{3} x^{3} + 240 \, d^{7} e^{2} x^{2} - 240 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{16} x^{9} - d e^{15} x^{8} - 14 \, d^{2} e^{14} x^{7} + 14 \, d^{3} e^{13} x^{6} + 41 \, d^{4} e^{12} x^{5} - 41 \, d^{5} e^{11} x^{4} - 44 \, d^{6} e^{10} x^{3} + 44 \, d^{7} e^{9} x^{2} + 16 \, d^{8} e^{8} x - 16 \, d^{9} e^{7} +{\left (5 \, d e^{14} x^{7} - 5 \, d^{2} e^{13} x^{6} - 25 \, d^{3} e^{12} x^{5} + 25 \, d^{4} e^{11} x^{4} + 36 \, d^{5} e^{10} x^{3} - 36 \, d^{6} e^{9} x^{2} - 16 \, d^{7} e^{8} x + 16 \, d^{8} e^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^6/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(27*d*e^9*x^9 - 142*d^2*e^8*x^8 + 112*d^3*e^7*x^7 + 523*d^4*e^6*x^6 - 523*
d^5*e^5*x^5 - 620*d^6*e^4*x^4 + 620*d^7*e^3*x^3 + 240*d^8*e^2*x^2 - 240*d^9*e*x
- 30*(d*e^9*x^9 - d^2*e^8*x^8 - 14*d^3*e^7*x^7 + 14*d^4*e^6*x^6 + 41*d^5*e^5*x^5
- 41*d^6*e^4*x^4 - 44*d^7*e^3*x^3 + 44*d^8*e^2*x^2 + 16*d^9*e*x - 16*d^10 + (5*
d^2*e^7*x^7 - 5*d^3*e^6*x^6 - 25*d^4*e^5*x^5 + 25*d^5*e^4*x^4 + 36*d^6*e^3*x^3 -
36*d^7*e^2*x^2 - 16*d^8*e*x + 16*d^9)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-
e^2*x^2 + d^2))/(e*x)) - (15*e^9*x^9 - 38*d*e^8*x^8 + 8*d^2*e^7*x^7 + 303*d^3*e^
6*x^6 - 303*d^4*e^5*x^5 - 500*d^5*e^4*x^4 + 500*d^6*e^3*x^3 + 240*d^7*e^2*x^2 -
240*d^8*e*x)*sqrt(-e^2*x^2 + d^2))/(e^16*x^9 - d*e^15*x^8 - 14*d^2*e^14*x^7 + 14
*d^3*e^13*x^6 + 41*d^4*e^12*x^5 - 41*d^5*e^11*x^4 - 44*d^6*e^10*x^3 + 44*d^7*e^9
*x^2 + 16*d^8*e^8*x - 16*d^9*e^7 + (5*d*e^14*x^7 - 5*d^2*e^13*x^6 - 25*d^3*e^12*
x^5 + 25*d^4*e^11*x^4 + 36*d^5*e^10*x^3 - 36*d^6*e^9*x^2 - 16*d^7*e^8*x + 16*d^8
*e^7)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 40.6592, size = 1821, normalized size = 12.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*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*sqrt(-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)*ac
osh(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*sqr
t(-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)*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)) -
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*sqr
t(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)) +
e*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*s
qrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 40*d**4*e**2*x**2
/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2)
+ 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) + 30*d**2*e**4*x**4/(5*d**4*e**8*sqrt(d*
*2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(
d**2 - e**2*x**2)) - 5*e**6*x**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e
**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0
)), (x**8/(8*(d**2)**(7/2)), True))

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GIAC/XCAS [A]  time = 0.300842, size = 147, normalized size = 1. \[ -d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-7\right )}{\rm sign}\left (d\right ) - \frac{{\left (48 \, d^{6} e^{\left (-7\right )} +{\left (15 \, d^{5} e^{\left (-6\right )} -{\left (120 \, d^{4} e^{\left (-5\right )} +{\left (35 \, d^{3} e^{\left (-4\right )} -{\left (90 \, d^{2} e^{\left (-3\right )} -{\left (15 \, x e^{\left (-1\right )} - 23 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^6/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-d*arcsin(x*e/d)*e^(-7)*sign(d) - 1/15*(48*d^6*e^(-7) + (15*d^5*e^(-6) - (120*d^
4*e^(-5) + (35*d^3*e^(-4) - (90*d^2*e^(-3) - (15*x*e^(-1) - 23*d*e^(-2))*x)*x)*x
)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3