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

Optimal. Leaf size=145 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(35*d*Sqrt[d^2 - e^2*x^2])/(2*e) + (35*(d^2 - e^2*x^2)^(3/2))/(6*e*(d + e*x)) +
(14*(d^2 - e^2*x^2)^(5/2))/(3*e*(d + e*x)^3) - (2*(d^2 - e^2*x^2)^(7/2))/(3*e*(d
+ e*x)^5) + (35*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi [A]  time = 0.177385, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

(35*d*Sqrt[d^2 - e^2*x^2])/(2*e) + (35*(d^2 - e^2*x^2)^(3/2))/(6*e*(d + e*x)) +
(14*(d^2 - e^2*x^2)^(5/2))/(3*e*(d + e*x)^3) - (2*(d^2 - e^2*x^2)^(7/2))/(3*e*(d
+ e*x)^5) + (35*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi in Sympy [A]  time = 27.3264, size = 121, normalized size = 0.83 \[ \frac{35 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} + \frac{35 d \sqrt{d^{2} - e^{2} x^{2}}}{2 e} + \frac{35 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{6 e \left (d + e x\right )} + \frac{14 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{3}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{3 e \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) + 35*d*sqrt(d**2 - e**2*x**2)/(2*
e) + 35*(d**2 - e**2*x**2)**(3/2)/(6*e*(d + e*x)) + 14*(d**2 - e**2*x**2)**(5/2)
/(3*e*(d + e*x)**3) - 2*(d**2 - e**2*x**2)**(7/2)/(3*e*(d + e*x)**5)

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Mathematica [A]  time = 0.131661, size = 87, normalized size = 0.6 \[ \frac{105 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (164 d^3+229 d^2 e x+30 d e^2 x^2-3 e^3 x^3\right )}{(d+e x)^2}}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(164*d^3 + 229*d^2*e*x + 30*d*e^2*x^2 - 3*e^3*x^3))/(d + e
*x)^2 + 105*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e)

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Maple [B]  time = 0.017, size = 407, normalized size = 2.8 \[ -{\frac{1}{3\,{e}^{7}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-6}}+{\frac{1}{{e}^{6}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-5}}+4\,{\frac{1}{{e}^{5}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-4}}+{\frac{20}{3\,{e}^{4}{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+8\,{\frac{1}{{e}^{3}{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}+8\,{\frac{1}{e{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2}}+{\frac{28\,x}{3\,{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{35\,x}{3\,{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{35\,x}{2}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{35\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/e^7/d/(d/e+x)^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+1/e^6/d^2/(d/e+x)^5*(-
(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+4/e^5/d^3/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/
e+x))^(9/2)+20/3/e^4/d^4/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+8/e^3/d^
5/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+8/e/d^5*(-(d/e+x)^2*e^2+2*d*e*(
d/e+x))^(7/2)+28/3/d^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+35/3/d^2*(-(d/e+x)
^2*e^2+2*d*e*(d/e+x))^(3/2)*x+35/2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+35/2*d
^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(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((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238298, size = 516, normalized size = 3.56 \[ \frac{3 \, e^{7} x^{7} - 21 \, d e^{6} x^{6} - 179 \, d^{2} e^{5} x^{5} - 951 \, d^{3} e^{4} x^{4} - 320 \, d^{4} e^{3} x^{3} + 1332 \, d^{5} e^{2} x^{2} + 792 \, d^{6} e x - 210 \,{\left (d^{2} e^{5} x^{5} - 2 \, d^{3} e^{4} x^{4} - 11 \, d^{4} e^{3} x^{3} - 4 \, d^{5} e^{2} x^{2} + 12 \, d^{6} e x + 8 \, d^{7} +{\left (d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} - 12 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (3 \, e^{6} x^{6} - 42 \, d e^{5} x^{5} - 285 \, d^{2} e^{4} x^{4} + 76 \, d^{3} e^{3} x^{3} + 1332 \, d^{4} e^{2} x^{2} + 792 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{6} x^{5} - 2 \, d e^{5} x^{4} - 11 \, d^{2} e^{4} x^{3} - 4 \, d^{3} e^{3} x^{2} + 12 \, d^{4} e^{2} x + 8 \, d^{5} e +{\left (e^{5} x^{4} + 5 \, d e^{4} x^{3} - 12 \, d^{3} e^{2} x - 8 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(3*e^7*x^7 - 21*d*e^6*x^6 - 179*d^2*e^5*x^5 - 951*d^3*e^4*x^4 - 320*d^4*e^3*
x^3 + 1332*d^5*e^2*x^2 + 792*d^6*e*x - 210*(d^2*e^5*x^5 - 2*d^3*e^4*x^4 - 11*d^4
*e^3*x^3 - 4*d^5*e^2*x^2 + 12*d^6*e*x + 8*d^7 + (d^2*e^4*x^4 + 5*d^3*e^3*x^3 - 1
2*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
)) - (3*e^6*x^6 - 42*d*e^5*x^5 - 285*d^2*e^4*x^4 + 76*d^3*e^3*x^3 + 1332*d^4*e^2
*x^2 + 792*d^5*e*x)*sqrt(-e^2*x^2 + d^2))/(e^6*x^5 - 2*d*e^5*x^4 - 11*d^2*e^4*x^
3 - 4*d^3*e^3*x^2 + 12*d^4*e^2*x + 8*d^5*e + (e^5*x^4 + 5*d*e^4*x^3 - 12*d^3*e^2
*x - 8*d^4*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.64258, 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)^(7/2)/(e*x + d)^6,x, algorithm="giac")

[Out]

Done