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

Optimal. Leaf size=138 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}-\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

(-7*Sqrt[d^2 - e^2*x^2])/e - (14*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^2) + (14*
(d^2 - e^2*x^2)^(5/2))/(15*e*(d + e*x)^4) - (2*(d^2 - e^2*x^2)^(7/2))/(5*e*(d +
e*x)^6) - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Rubi [A]  time = 0.174546, antiderivative size = 138, 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 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}-\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(-7*Sqrt[d^2 - e^2*x^2])/e - (14*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^2) + (14*
(d^2 - e^2*x^2)^(5/2))/(15*e*(d + e*x)^4) - (2*(d^2 - e^2*x^2)^(7/2))/(5*e*(d +
e*x)^6) - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Rubi in Sympy [A]  time = 26.3601, size = 116, normalized size = 0.84 \[ - \frac{7 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} - \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{e} - \frac{14 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{2}} + \frac{14 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{15 e \left (d + e x\right )^{4}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{5 e \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*d*atan(e*x/sqrt(d**2 - e**2*x**2))/e - 7*sqrt(d**2 - e**2*x**2)/e - 14*(d**2
- e**2*x**2)**(3/2)/(3*e*(d + e*x)**2) + 14*(d**2 - e**2*x**2)**(5/2)/(15*e*(d +
e*x)**4) - 2*(d**2 - e**2*x**2)**(7/2)/(5*e*(d + e*x)**6)

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Mathematica [A]  time = 0.106456, size = 87, normalized size = 0.63 \[ -\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2} \left (167 d^3+381 d^2 e x+277 d e^2 x^2+15 e^3 x^3\right )}{15 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(167*d^3 + 381*d^2*e*x + 277*d*e^2*x^2 + 15*e^3*x^3))/(15*
e*(d + e*x)^3) - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Maple [B]  time = 0.014, size = 454, normalized size = 3.3 \[ -{\frac{1}{5\,{e}^{8}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-7}}+{\frac{2}{15\,{e}^{7}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-6}}-{\frac{2}{5\,{e}^{6}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-5}}-{\frac{8}{5\,{e}^{5}{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{8}{3\,{e}^{4}{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{16}{5\,{e}^{3}{d}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{16}{5\,e{d}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{56\,x}{15\,{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{14\,x}{3\,{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-7\,{\frac{x}{d}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-7\,{\frac{d}{\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/e^8/d/(d/e+x)^7*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+2/15/e^7/d^2/(d/e+x)^6
*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)-2/5/e^6/d^3/(d/e+x)^5*(-(d/e+x)^2*e^2+2*d*
e*(d/e+x))^(9/2)-8/5/e^5/d^4/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)-8/3/
e^4/d^5/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)-16/5/e^3/d^6/(d/e+x)^2*(-
(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)-16/5/e/d^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/
2)-56/15/d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x-14/3/d^3*(-(d/e+x)^2*e^2+2*d
*e*(d/e+x))^(3/2)*x-7/d*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-7*d/(e^2)^(1/2)*a
rctan((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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236021, size = 599, normalized size = 4.34 \[ \frac{15 \, e^{7} x^{7} + 155 \, d e^{6} x^{6} + 1259 \, d^{2} e^{5} x^{5} + 1205 \, d^{3} e^{4} x^{4} - 1030 \, d^{4} e^{3} x^{3} - 1980 \, d^{5} e^{2} x^{2} - 960 \, d^{6} e x + 210 \,{\left (d e^{6} x^{6} - d^{2} e^{5} x^{5} - 13 \, d^{3} e^{4} x^{4} - 15 \, d^{4} e^{3} x^{3} + 8 \, d^{5} e^{2} x^{2} + 20 \, d^{6} e x + 8 \, d^{7} +{\left (d e^{5} x^{5} + 6 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} - 12 \, d^{4} e^{2} x^{2} - 20 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{6} x^{6} + 384 \, d e^{5} x^{5} + 215 \, d^{2} e^{4} x^{4} - 1510 \, d^{3} e^{3} x^{3} - 1980 \, d^{4} e^{2} x^{2} - 960 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{7} x^{6} - d e^{6} x^{5} - 13 \, d^{2} e^{5} x^{4} - 15 \, d^{3} e^{4} x^{3} + 8 \, d^{4} e^{3} x^{2} + 20 \, d^{5} e^{2} x + 8 \, d^{6} e +{\left (e^{6} x^{5} + 6 \, d e^{5} x^{4} + 5 \, d^{2} e^{4} x^{3} - 12 \, d^{3} e^{3} x^{2} - 20 \, d^{4} e^{2} x - 8 \, d^{5} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(15*e^7*x^7 + 155*d*e^6*x^6 + 1259*d^2*e^5*x^5 + 1205*d^3*e^4*x^4 - 1030*d^
4*e^3*x^3 - 1980*d^5*e^2*x^2 - 960*d^6*e*x + 210*(d*e^6*x^6 - d^2*e^5*x^5 - 13*d
^3*e^4*x^4 - 15*d^4*e^3*x^3 + 8*d^5*e^2*x^2 + 20*d^6*e*x + 8*d^7 + (d*e^5*x^5 +
6*d^2*e^4*x^4 + 5*d^3*e^3*x^3 - 12*d^4*e^2*x^2 - 20*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)) - (15*e^6*x^6 + 384*d*e^5*x
^5 + 215*d^2*e^4*x^4 - 1510*d^3*e^3*x^3 - 1980*d^4*e^2*x^2 - 960*d^5*e*x)*sqrt(-
e^2*x^2 + d^2))/(e^7*x^6 - d*e^6*x^5 - 13*d^2*e^5*x^4 - 15*d^3*e^4*x^3 + 8*d^4*e
^3*x^2 + 20*d^5*e^2*x + 8*d^6*e + (e^6*x^5 + 6*d*e^5*x^4 + 5*d^2*e^4*x^3 - 12*d^
3*e^3*x^2 - 20*d^4*e^2*x - 8*d^5*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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.455794, 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)^7,x, algorithm="giac")

[Out]

Done