3.795 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=136 \[ \frac{35}{8} d^2 x \sqrt{d^2-e^2 x^2}+\frac{35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac{7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e} \]

[Out]

(35*d^2*x*Sqrt[d^2 - e^2*x^2])/8 + (35*d*(d^2 - e^2*x^2)^(3/2))/(12*e) + (7*(d -
e*x)*(d^2 - e^2*x^2)^(3/2))/(4*e) + (2*(d^2 - e^2*x^2)^(7/2))/(e*(d + e*x)^3) +
(35*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi [A]  time = 0.170982, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{35}{8} d^2 x \sqrt{d^2-e^2 x^2}+\frac{35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac{7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^4,x]

[Out]

(35*d^2*x*Sqrt[d^2 - e^2*x^2])/8 + (35*d*(d^2 - e^2*x^2)^(3/2))/(12*e) + (7*(d -
e*x)*(d^2 - e^2*x^2)^(3/2))/(4*e) + (2*(d^2 - e^2*x^2)^(7/2))/(e*(d + e*x)^3) +
(35*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi in Sympy [A]  time = 30.3516, size = 116, normalized size = 0.85 \[ \frac{35 d^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e} + \frac{35 d^{2} x \sqrt{d^{2} - e^{2} x^{2}}}{8} + \frac{35 d \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{12 e} + \frac{7 \left (d - e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{4 e} + \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{e \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**4,x)

[Out]

35*d**4*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e) + 35*d**2*x*sqrt(d**2 - e**2*x**2
)/8 + 35*d*(d**2 - e**2*x**2)**(3/2)/(12*e) + 7*(d - e*x)*(d**2 - e**2*x**2)**(3
/2)/(4*e) + 2*(d**2 - e**2*x**2)**(7/2)/(e*(d + e*x)**3)

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Mathematica [A]  time = 0.0955933, size = 80, normalized size = 0.59 \[ \frac{105 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (160 d^3-81 d^2 e x+32 d e^2 x^2-6 e^3 x^3\right )}{24 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^4,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(160*d^3 - 81*d^2*e*x + 32*d*e^2*x^2 - 6*e^3*x^3) + 105*d^4
*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e)

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Maple [B]  time = 0.017, size = 317, normalized size = 2.3 \[{\frac{1}{{e}^{5}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}+{\frac{5}{3\,{e}^{4}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+2\,{\frac{1}{{e}^{3}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}+2\,{\frac{1}{e{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2}}+{\frac{7\,x}{3\,{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{35\,x}{12} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{d}^{2}x}{8}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{35\,{d}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^4,x)

[Out]

1/e^5/d/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+5/3/e^4/d^2/(d/e+x)^3*(-(
d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+2/e^3/d^3/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e
+x))^(9/2)+2/e/d^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+7/3/d^2*(-(d/e+x)^2*e^2+
2*d*e*(d/e+x))^(5/2)*x+35/12*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+35/8*d^2*(-(
d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+35/8*d^4/(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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233488, size = 412, normalized size = 3.03 \[ \frac{24 \, d e^{7} x^{7} - 128 \, d^{2} e^{6} x^{6} + 252 \, d^{3} e^{5} x^{5} - 96 \, d^{4} e^{4} x^{4} - 924 \, d^{5} e^{3} x^{3} + 384 \, d^{6} e^{2} x^{2} + 648 \, d^{7} e x - 210 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (6 \, e^{7} x^{7} - 32 \, d e^{6} x^{6} + 33 \, d^{2} e^{5} x^{5} + 96 \, d^{3} e^{4} x^{4} - 600 \, d^{4} e^{3} x^{3} + 384 \, d^{5} e^{2} x^{2} + 648 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (e^{5} x^{4} - 8 \, d^{2} e^{3} x^{2} + 8 \, d^{4} e + 4 \,{\left (d e^{3} x^{2} - 2 \, d^{3} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/24*(24*d*e^7*x^7 - 128*d^2*e^6*x^6 + 252*d^3*e^5*x^5 - 96*d^4*e^4*x^4 - 924*d^
5*e^3*x^3 + 384*d^6*e^2*x^2 + 648*d^7*e*x - 210*(d^4*e^4*x^4 - 8*d^6*e^2*x^2 + 8
*d^8 + 4*(d^5*e^2*x^2 - 2*d^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2
+ d^2))/(e*x)) - (6*e^7*x^7 - 32*d*e^6*x^6 + 33*d^2*e^5*x^5 + 96*d^3*e^4*x^4 - 6
00*d^4*e^3*x^3 + 384*d^5*e^2*x^2 + 648*d^6*e*x)*sqrt(-e^2*x^2 + d^2))/(e^5*x^4 -
8*d^2*e^3*x^2 + 8*d^4*e + 4*(d*e^3*x^2 - 2*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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257207, 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)^4,x, algorithm="giac")

[Out]

Done