3.137 \(\int \frac{x^6}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[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 = 0.454123, antiderivative size = 148, 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 \[ \frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),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 = 77.2055, size = 182, normalized size = 1.23 \[ - \frac{d^{5}}{5 e^{7} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{d^{4}}{e^{7} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{d^{3} x}{15 e^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{3 d^{2}}{e^{7} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{2 d x^{3}}{3 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{13 d x}{15 e^{6} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{7}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

-d**5/(5*e**7*(d + e*x)*(d**2 - e**2*x**2)**(3/2)) + d**4/(e**7*(d**2 - e**2*x**
2)**(3/2)) - d**3*x/(15*e**6*(d**2 - e**2*x**2)**(3/2)) - 3*d**2/(e**7*sqrt(d**2
- e**2*x**2)) - 2*d*x**3/(3*e**4*(d**2 - e**2*x**2)**(3/2)) + 13*d*x/(15*e**6*s
qrt(d**2 - e**2*x**2)) - d*atan(e*x/sqrt(d**2 - e**2*x**2))/e**7 - sqrt(d**2 - e
**2*x**2)/e**7

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Mathematica [A]  time = 0.161397, size = 115, normalized size = 0.78 \[ -\frac{15 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (48 d^5+33 d^4 e x-87 d^3 e^2 x^2-52 d^2 e^3 x^3+38 d e^4 x^4+15 e^5 x^5\right )}{(d-e x)^2 (d+e x)^3}}{15 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),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 +
38*d*e^4*x^4 + 15*e^5*x^5))/((d - e*x)^2*(d + e*x)^3) + 15*d*ArcTan[(e*x)/Sqrt[d
^2 - e^2*x^2]])/(15*e^7)

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Maple [B]  time = 0.045, size = 288, normalized size = 2. \[ -{\frac{{x}^{4}}{{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{{d}^{2}{x}^{2}}{{e}^{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-3\,{\frac{{d}^{4}}{{e}^{7} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{{d}^{5}}{5\,{e}^{8}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}x}{15\,{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,dx}{15\,{e}^{6}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{2\,{d}^{3}x}{3\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,dx}{3\,{e}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{d{x}^{3}}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{d}{{e}^{6}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

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

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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(x^6/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306824, size = 838, normalized size = 5.66 \[ \frac{27 \, d e^{9} x^{9} + 142 \, d^{2} e^{8} x^{8} + 112 \, d^{3} e^{7} x^{7} - 523 \, d^{4} e^{6} x^{6} - 523 \, d^{5} e^{5} x^{5} + 620 \, d^{6} e^{4} x^{4} + 620 \, d^{7} e^{3} x^{3} - 240 \, d^{8} e^{2} x^{2} - 240 \, d^{9} e x + 30 \,{\left (d e^{9} x^{9} + d^{2} e^{8} x^{8} - 14 \, d^{3} e^{7} x^{7} - 14 \, d^{4} e^{6} x^{6} + 41 \, d^{5} e^{5} x^{5} + 41 \, d^{6} e^{4} x^{4} - 44 \, d^{7} e^{3} x^{3} - 44 \, d^{8} e^{2} x^{2} + 16 \, d^{9} e x + 16 \, d^{10} +{\left (5 \, d^{2} e^{7} x^{7} + 5 \, d^{3} e^{6} x^{6} - 25 \, d^{4} e^{5} x^{5} - 25 \, d^{5} e^{4} x^{4} + 36 \, d^{6} e^{3} x^{3} + 36 \, d^{7} e^{2} x^{2} - 16 \, d^{8} e x - 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{9} x^{9} + 38 \, d e^{8} x^{8} + 8 \, d^{2} e^{7} x^{7} - 303 \, d^{3} e^{6} x^{6} - 303 \, d^{4} e^{5} x^{5} + 500 \, d^{5} e^{4} x^{4} + 500 \, d^{6} e^{3} x^{3} - 240 \, d^{7} e^{2} x^{2} - 240 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{16} x^{9} + d e^{15} x^{8} - 14 \, d^{2} e^{14} x^{7} - 14 \, d^{3} e^{13} x^{6} + 41 \, d^{4} e^{12} x^{5} + 41 \, d^{5} e^{11} x^{4} - 44 \, d^{6} e^{10} x^{3} - 44 \, d^{7} e^{9} x^{2} + 16 \, d^{8} e^{8} x + 16 \, d^{9} e^{7} +{\left (5 \, d e^{14} x^{7} + 5 \, d^{2} e^{13} x^{6} - 25 \, d^{3} e^{12} x^{5} - 25 \, d^{4} e^{11} x^{4} + 36 \, d^{5} e^{10} x^{3} + 36 \, d^{6} e^{9} x^{2} - 16 \, d^{7} e^{8} x - 16 \, d^{8} e^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),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 - 2
40*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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**6/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, 1]