3.796 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=132 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(-35*d*x*Sqrt[d^2 - e^2*x^2])/2 - (35*(d^2 - e^2*x^2)^(3/2))/(3*e) - (14*(d^2 -
e^2*x^2)^(5/2))/(e*(d + e*x)^2) - (2*(d^2 - e^2*x^2)^(7/2))/(e*(d + e*x)^4) - (3
5*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi [A]  time = 0.151591, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^5,x]

[Out]

(-35*d*x*Sqrt[d^2 - e^2*x^2])/2 - (35*(d^2 - e^2*x^2)^(3/2))/(3*e) - (14*(d^2 -
e^2*x^2)^(5/2))/(e*(d + e*x)^2) - (2*(d^2 - e^2*x^2)^(7/2))/(e*(d + e*x)^4) - (3
5*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi in Sympy [A]  time = 23.9581, size = 114, normalized size = 0.86 \[ - \frac{35 d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{35 d x \sqrt{d^{2} - e^{2} x^{2}}}{2} - \frac{35 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e} - \frac{14 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{2}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{e \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**5,x)

[Out]

-35*d**3*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) - 35*d*x*sqrt(d**2 - e**2*x**2)/
2 - 35*(d**2 - e**2*x**2)**(3/2)/(3*e) - 14*(d**2 - e**2*x**2)**(5/2)/(e*(d + e*
x)**2) - 2*(d**2 - e**2*x**2)**(7/2)/(e*(d + e*x)**4)

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Mathematica [A]  time = 0.168079, size = 85, normalized size = 0.64 \[ \frac{1}{6} \sqrt{d^2-e^2 x^2} \left (-\frac{96 d^3}{e (d+e x)}-\frac{70 d^2}{e}+15 d x-2 e x^2\right )-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^5,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*((-70*d^2)/e + 15*d*x - 2*e*x^2 - (96*d^3)/(e*(d + e*x))))/
6 - (35*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Maple [B]  time = 0.018, size = 364, normalized size = 2.8 \[ -{\frac{1}{{e}^{6}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-5}}-4\,{\frac{1}{{e}^{5}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{20}{3\,{e}^{4}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-8\,{\frac{1}{{e}^{3}{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}-8\,{\frac{1}{e{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2}}-{\frac{28\,x}{3\,{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{35\,x}{3\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,dx}{2}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{35\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240807, size = 471, normalized size = 3.57 \[ \frac{2 \, e^{7} x^{7} - 7 \, d e^{6} x^{6} + 261 \, d^{3} e^{4} x^{4} + 624 \, d^{4} e^{3} x^{3} - 324 \, d^{5} e^{2} x^{2} - 888 \, d^{6} e x + 210 \,{\left (d^{3} e^{4} x^{4} - 3 \, d^{4} e^{3} x^{3} - 8 \, d^{5} e^{2} x^{2} + 4 \, d^{6} e x + 8 \, d^{7} +{\left (d^{3} e^{3} x^{3} + 4 \, d^{4} e^{2} x^{2} - 4 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (2 \, e^{6} x^{6} - 21 \, d e^{5} x^{5} + 99 \, d^{2} e^{4} x^{4} + 180 \, d^{3} e^{3} x^{3} - 324 \, d^{4} e^{2} x^{2} - 888 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{5} x^{4} - 3 \, d e^{4} x^{3} - 8 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + 8 \, d^{4} e +{\left (e^{4} x^{3} + 4 \, d e^{3} x^{2} - 4 \, d^{2} e^{2} x - 8 \, d^{3} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/6*(2*e^7*x^7 - 7*d*e^6*x^6 + 261*d^3*e^4*x^4 + 624*d^4*e^3*x^3 - 324*d^5*e^2*x
^2 - 888*d^6*e*x + 210*(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)) - (2*e^6*x^6 - 21*d*e^5*x^5 + 99*d^2
*e^4*x^4 + 180*d^3*e^3*x^3 - 324*d^4*e^2*x^2 - 888*d^5*e*x)*sqrt(-e^2*x^2 + d^2)
)/(e^5*x^4 - 3*d*e^4*x^3 - 8*d^2*e^3*x^2 + 4*d^3*e^2*x + 8*d^4*e + (e^4*x^3 + 4*
d*e^3*x^2 - 4*d^2*e^2*x - 8*d^3*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)**5,x)

[Out]

Timed out

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GIAC/XCAS [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)^5,x, algorithm="giac")

[Out]

Timed out