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

Optimal. Leaf size=132 \[ \frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{7 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e}+\frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2} \]

[Out]

(7*d^4*x*Sqrt[d^2 - e^2*x^2])/16 + (7*d^2*x*(d^2 - e^2*x^2)^(3/2))/24 + (7*d*(d^
2 - e^2*x^2)^(5/2))/(30*e) + ((d - e*x)*(d^2 - e^2*x^2)^(5/2))/(6*e) + (7*d^6*Ar
cTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e)

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Rubi [A]  time = 0.152953, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{7 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e}+\frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(7*d^4*x*Sqrt[d^2 - e^2*x^2])/16 + (7*d^2*x*(d^2 - e^2*x^2)^(3/2))/24 + (7*d*(d^
2 - e^2*x^2)^(5/2))/(30*e) + ((d - e*x)*(d^2 - e^2*x^2)^(5/2))/(6*e) + (7*d^6*Ar
cTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e)

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Rubi in Sympy [A]  time = 26.6619, size = 112, normalized size = 0.85 \[ \frac{7 d^{6} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{16 e} + \frac{7 d^{4} x \sqrt{d^{2} - e^{2} x^{2}}}{16} + \frac{7 d^{2} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24} + \frac{7 d \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{30 e} + \frac{\left (d - e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*d**6*atan(e*x/sqrt(d**2 - e**2*x**2))/(16*e) + 7*d**4*x*sqrt(d**2 - e**2*x**2)
/16 + 7*d**2*x*(d**2 - e**2*x**2)**(3/2)/24 + 7*d*(d**2 - e**2*x**2)**(5/2)/(30*
e) + (d - e*x)*(d**2 - e**2*x**2)**(5/2)/(6*e)

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Mathematica [A]  time = 0.108227, size = 102, normalized size = 0.77 \[ \frac{105 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (96 d^5+135 d^4 e x-192 d^3 e^2 x^2+10 d^2 e^3 x^3+96 d e^4 x^4-40 e^5 x^5\right )}{240 e} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(96*d^5 + 135*d^4*e*x - 192*d^3*e^2*x^2 + 10*d^2*e^3*x^3 +
96*d*e^4*x^4 - 40*e^5*x^5) + 105*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e)

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Maple [B]  time = 0.016, size = 228, normalized size = 1.7 \[{\frac{1}{5\,{e}^{3}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}+{\frac{1}{5\,de} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{7\,x}{30} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{d}^{2}x}{24} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{d}^{4}x}{16}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{7\,{d}^{6}}{16}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5/e^3/d/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+1/5/e/d*(-(d/e+x)^2*e^2
+2*d*e*(d/e+x))^(7/2)+7/30*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+7/24*d^2*(-(d/
e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+7/16*d^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*
x+7/16*d^6/(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 [A]  time = 0.793836, size = 188, normalized size = 1.42 \[ -\frac{7 i \, d^{6} \arcsin \left (\frac{e x}{d} + 2\right )}{16 \, e} + \frac{7}{16} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x + \frac{7 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{8 \, e} + \frac{7}{24} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} x + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}{6 \,{\left (e^{2} x + d e\right )}} + \frac{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d}{30 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/16*I*d^6*arcsin(e*x/d + 2)/e + 7/16*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4*x + 7
/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^5/e + 7/24*(-e^2*x^2 + d^2)^(3/2)*d^2*x + 1
/6*(-e^2*x^2 + d^2)^(7/2)/(e^2*x + d*e) + 7/30*(-e^2*x^2 + d^2)^(5/2)*d/e

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Fricas [A]  time = 0.232924, size = 593, normalized size = 4.49 \[ \frac{240 \, d e^{11} x^{11} - 576 \, d^{2} e^{10} x^{10} - 1580 \, d^{3} e^{9} x^{9} + 4800 \, d^{4} e^{8} x^{8} + 2130 \, d^{5} e^{7} x^{7} - 13920 \, d^{6} e^{6} x^{6} + 3210 \, d^{7} e^{5} x^{5} + 17280 \, d^{8} e^{4} x^{4} - 8320 \, d^{9} e^{3} x^{3} - 7680 \, d^{10} e^{2} x^{2} + 4320 \, d^{11} e x - 210 \,{\left (d^{6} e^{6} x^{6} - 18 \, d^{8} e^{4} x^{4} + 48 \, d^{10} e^{2} x^{2} - 32 \, d^{12} + 2 \,{\left (3 \, d^{7} e^{4} x^{4} - 16 \, d^{9} e^{2} x^{2} + 16 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (40 \, e^{11} x^{11} - 96 \, d e^{10} x^{10} - 730 \, d^{2} e^{9} x^{9} + 1920 \, d^{3} e^{8} x^{8} + 1965 \, d^{4} e^{7} x^{7} - 8160 \, d^{5} e^{6} x^{6} + 670 \, d^{6} e^{5} x^{5} + 13440 \, d^{7} e^{4} x^{4} - 6160 \, d^{8} e^{3} x^{3} - 7680 \, d^{9} e^{2} x^{2} + 4320 \, d^{10} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \,{\left (e^{7} x^{6} - 18 \, d^{2} e^{5} x^{4} + 48 \, d^{4} e^{3} x^{2} - 32 \, d^{6} e + 2 \,{\left (3 \, d e^{5} x^{4} - 16 \, d^{3} e^{3} x^{2} + 16 \, d^{5} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(240*d*e^11*x^11 - 576*d^2*e^10*x^10 - 1580*d^3*e^9*x^9 + 4800*d^4*e^8*x^8
+ 2130*d^5*e^7*x^7 - 13920*d^6*e^6*x^6 + 3210*d^7*e^5*x^5 + 17280*d^8*e^4*x^4 -
8320*d^9*e^3*x^3 - 7680*d^10*e^2*x^2 + 4320*d^11*e*x - 210*(d^6*e^6*x^6 - 18*d^
8*e^4*x^4 + 48*d^10*e^2*x^2 - 32*d^12 + 2*(3*d^7*e^4*x^4 - 16*d^9*e^2*x^2 + 16*d
^11)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (40*e^11*
x^11 - 96*d*e^10*x^10 - 730*d^2*e^9*x^9 + 1920*d^3*e^8*x^8 + 1965*d^4*e^7*x^7 -
8160*d^5*e^6*x^6 + 670*d^6*e^5*x^5 + 13440*d^7*e^4*x^4 - 6160*d^8*e^3*x^3 - 7680
*d^9*e^2*x^2 + 4320*d^10*e*x)*sqrt(-e^2*x^2 + d^2))/(e^7*x^6 - 18*d^2*e^5*x^4 +
48*d^4*e^3*x^2 - 32*d^6*e + 2*(3*d*e^5*x^4 - 16*d^3*e^3*x^2 + 16*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)**2,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)^2,x, algorithm="giac")

[Out]

Timed out