3.202 \(\int \frac{x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=130 \[ \frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]

[Out]

(10*d*x*Sqrt[d^2 - e^2*x^2])/e + (20*(d^2 - e^2*x^2)^(3/2))/(3*e^2) + (8*(d^2 -
e^2*x^2)^(5/2))/(e^2*(d + e*x)^2) + (d^2 - e^2*x^2)^(7/2)/(e^2*(d + e*x)^4) + (1
0*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2

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Rubi [A]  time = 0.16634, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

[In]  Int[(x*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(10*d*x*Sqrt[d^2 - e^2*x^2])/e + (20*(d^2 - e^2*x^2)^(3/2))/(3*e^2) + (8*(d^2 -
e^2*x^2)^(5/2))/(e^2*(d + e*x)^2) + (d^2 - e^2*x^2)^(7/2)/(e^2*(d + e*x)^4) + (1
0*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2

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Rubi in Sympy [A]  time = 25.2626, size = 116, normalized size = 0.89 \[ \frac{10 d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{2}} + \frac{10 d x \sqrt{d^{2} - e^{2} x^{2}}}{e} + \frac{20 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{8 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )^{2}} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{e^{2} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)

[Out]

10*d**3*atan(e*x/sqrt(d**2 - e**2*x**2))/e**2 + 10*d*x*sqrt(d**2 - e**2*x**2)/e
+ 20*(d**2 - e**2*x**2)**(3/2)/(3*e**2) + 8*(d**2 - e**2*x**2)**(5/2)/(e**2*(d +
e*x)**2) + (d**2 - e**2*x**2)**(7/2)/(e**2*(d + e*x)**4)

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Mathematica [A]  time = 0.131566, size = 83, normalized size = 0.64 \[ \frac{1}{3} \sqrt{d^2-e^2 x^2} \left (\frac{24 d^3}{e^2 (d+e x)}+\frac{23 d^2}{e^2}-\frac{6 d x}{e}+x^2\right )+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*((23*d^2)/e^2 - (6*d*x)/e + x^2 + (24*d^3)/(e^2*(d + e*x)))
)/3 + (10*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2

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Maple [B]  time = 0.016, size = 290, normalized size = 2.2 \[ 4\,{\frac{1}{{e}^{5}d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{7/2} \left ( x+{\frac{d}{e}} \right ) ^{-3}}+{\frac{16}{3\,{e}^{4}{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{16}{3\,{e}^{2}{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{20\,x}{3\,de} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+10\,{\frac{dx}{e}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+10\,{\frac{{d}^{3}}{e\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ) }+{\frac{1}{{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x)

[Out]

4/e^5/d/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+16/3/e^4/d^2/(x+d/e)^2*(-
(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+16/3/e^2/d^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(
5/2)+20/3/e/d*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+10/e*d*(-(x+d/e)^2*e^2+2*d*
e*(x+d/e))^(1/2)*x+10/e*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d
*e*(x+d/e))^(1/2))+1/e^6/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)

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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((-e^2*x^2 + d^2)^(5/2)*x/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292027, size = 489, normalized size = 3.76 \[ -\frac{e^{7} x^{7} - 2 \, d e^{6} x^{6} - 6 \, d^{2} e^{5} x^{5} + 87 \, d^{3} e^{4} x^{4} + 174 \, d^{4} e^{3} x^{3} - 108 \, d^{5} e^{2} x^{2} - 240 \, d^{6} e x + 60 \,{\left (d^{3} e^{4} x^{4} - 3 \, d^{4} e^{3} x^{3} - 8 \, d^{5} e^{2} x^{2} + 4 \, d^{6} e x + 8 \, d^{7} +{\left (d^{3} e^{3} x^{3} + 4 \, d^{4} e^{2} x^{2} - 4 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (e^{6} x^{6} - 9 \, d e^{5} x^{5} + 33 \, d^{2} e^{4} x^{4} + 54 \, d^{3} e^{3} x^{3} - 108 \, d^{4} e^{2} x^{2} - 240 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (e^{6} x^{4} - 3 \, d e^{5} x^{3} - 8 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + 8 \, d^{4} e^{2} +{\left (e^{5} x^{3} + 4 \, d e^{4} x^{2} - 4 \, d^{2} e^{3} x - 8 \, d^{3} e^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*x/(e*x + d)^4,x, algorithm="fricas")

[Out]

-1/3*(e^7*x^7 - 2*d*e^6*x^6 - 6*d^2*e^5*x^5 + 87*d^3*e^4*x^4 + 174*d^4*e^3*x^3 -
108*d^5*e^2*x^2 - 240*d^6*e*x + 60*(d^3*e^4*x^4 - 3*d^4*e^3*x^3 - 8*d^5*e^2*x^2
+ 4*d^6*e*x + 8*d^7 + (d^3*e^3*x^3 + 4*d^4*e^2*x^2 - 4*d^5*e*x - 8*d^6)*sqrt(-e
^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (e^6*x^6 - 9*d*e^5*x^
5 + 33*d^2*e^4*x^4 + 54*d^3*e^3*x^3 - 108*d^4*e^2*x^2 - 240*d^5*e*x)*sqrt(-e^2*x
^2 + d^2))/(e^6*x^4 - 3*d*e^5*x^3 - 8*d^2*e^4*x^2 + 4*d^3*e^3*x + 8*d^4*e^2 + (e
^5*x^3 + 4*d*e^4*x^2 - 4*d^2*e^3*x - 8*d^3*e^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral(x*(-(-d + e*x)*(d + e*x))**(5/2)/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.30959, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*x/(e*x + d)^4,x, algorithm="giac")

[Out]

Done