3.794 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{7 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}+\frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2} \]

[Out]

(7*d^3*x*Sqrt[d^2 - e^2*x^2])/8 + (7*d^2*(d^2 - e^2*x^2)^(3/2))/(12*e) + (7*d*(d
- e*x)*(d^2 - e^2*x^2)^(3/2))/(20*e) + ((d - e*x)^2*(d^2 - e^2*x^2)^(3/2))/(5*e
) + (7*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi [A]  time = 0.181704, antiderivative size = 142, 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 \[ \frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{7 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}+\frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^3,x]

[Out]

(7*d^3*x*Sqrt[d^2 - e^2*x^2])/8 + (7*d^2*(d^2 - e^2*x^2)^(3/2))/(12*e) + (7*d*(d
- e*x)*(d^2 - e^2*x^2)^(3/2))/(20*e) + ((d - e*x)^2*(d^2 - e^2*x^2)^(3/2))/(5*e
) + (7*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi in Sympy [A]  time = 31.4287, size = 119, normalized size = 0.84 \[ \frac{7 d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e} + \frac{7 d^{3} x \sqrt{d^{2} - e^{2} x^{2}}}{8} + \frac{7 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{12 e} + \frac{7 d \left (d - e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{20 e} + \frac{\left (d - e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**3,x)

[Out]

7*d**5*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e) + 7*d**3*x*sqrt(d**2 - e**2*x**2)/
8 + 7*d**2*(d**2 - e**2*x**2)**(3/2)/(12*e) + 7*d*(d - e*x)*(d**2 - e**2*x**2)**
(3/2)/(20*e) + (d - e*x)**2*(d**2 - e**2*x**2)**(3/2)/(5*e)

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Mathematica [A]  time = 0.0968285, size = 91, normalized size = 0.64 \[ \frac{105 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (136 d^4+15 d^3 e x-112 d^2 e^2 x^2+90 d e^3 x^3-24 e^4 x^4\right )}{120 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^3,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(136*d^4 + 15*d^3*e*x - 112*d^2*e^2*x^2 + 90*d*e^3*x^3 - 24
*e^4*x^4) + 105*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(120*e)

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Maple [B]  time = 0.018, size = 274, normalized size = 1.9 \[{\frac{1}{3\,{e}^{4}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+{\frac{2}{5\,{e}^{3}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}+{\frac{2}{5\,e{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{7\,x}{15\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{7\,dx}{12} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{d}^{3}x}{8}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{7\,{d}^{5}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^3,x)

[Out]

1/3/e^4/d/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+2/5/e^3/d^2/(d/e+x)^2*(
-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+2/5/e/d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/
2)+7/15/d*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+7/12*d*(-(d/e+x)^2*e^2+2*d*e*(d
/e+x))^(3/2)*x+7/8*d^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+7/8*d^5/(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 = 0.806919, size = 216, normalized size = 1.52 \[ -\frac{7 i \, d^{5} \arcsin \left (\frac{e x}{d} + 2\right )}{8 \, e} + \frac{7}{8} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x + \frac{7 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{4 \, e} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}{5 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d}{20 \,{\left (e^{2} x + d e\right )}} + \frac{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2}}{12 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^3,x, algorithm="maxima")

[Out]

-7/8*I*d^5*arcsin(e*x/d + 2)/e + 7/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^3*x + 7/4
*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4/e + 1/5*(-e^2*x^2 + d^2)^(7/2)/(e^3*x^2 + 2
*d*e^2*x + d^2*e) + 7/20*(-e^2*x^2 + d^2)^(5/2)*d/(e^2*x + d*e) + 7/12*(-e^2*x^2
+ d^2)^(3/2)*d^2/e

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Fricas [A]  time = 0.229675, size = 504, normalized size = 3.55 \[ -\frac{24 \, e^{10} x^{10} - 90 \, d e^{9} x^{9} - 200 \, d^{2} e^{8} x^{8} + 1155 \, d^{3} e^{7} x^{7} - 920 \, d^{4} e^{6} x^{6} - 2325 \, d^{5} e^{5} x^{5} + 3840 \, d^{6} e^{4} x^{4} + 1020 \, d^{7} e^{3} x^{3} - 2880 \, d^{8} e^{2} x^{2} + 240 \, d^{9} e x + 210 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (24 \, d e^{8} x^{8} - 90 \, d^{2} e^{7} x^{7} + 16 \, d^{3} e^{6} x^{6} + 345 \, d^{4} e^{5} x^{5} - 480 \, d^{5} e^{4} x^{4} - 228 \, d^{6} e^{3} x^{3} + 576 \, d^{7} e^{2} x^{2} - 48 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \,{\left (5 \, d e^{5} x^{4} - 20 \, d^{3} e^{3} x^{2} + 16 \, d^{5} e -{\left (e^{5} x^{4} - 12 \, d^{2} e^{3} x^{2} + 16 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^3,x, algorithm="fricas")

[Out]

-1/120*(24*e^10*x^10 - 90*d*e^9*x^9 - 200*d^2*e^8*x^8 + 1155*d^3*e^7*x^7 - 920*d
^4*e^6*x^6 - 2325*d^5*e^5*x^5 + 3840*d^6*e^4*x^4 + 1020*d^7*e^3*x^3 - 2880*d^8*e
^2*x^2 + 240*d^9*e*x + 210*(5*d^6*e^4*x^4 - 20*d^8*e^2*x^2 + 16*d^10 - (d^5*e^4*
x^4 - 12*d^7*e^2*x^2 + 16*d^9)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2
+ d^2))/(e*x)) + 5*(24*d*e^8*x^8 - 90*d^2*e^7*x^7 + 16*d^3*e^6*x^6 + 345*d^4*e^5
*x^5 - 480*d^5*e^4*x^4 - 228*d^6*e^3*x^3 + 576*d^7*e^2*x^2 - 48*d^8*e*x)*sqrt(-e
^2*x^2 + d^2))/(5*d*e^5*x^4 - 20*d^3*e^3*x^2 + 16*d^5*e - (e^5*x^4 - 12*d^2*e^3*
x^2 + 16*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269968, 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)^3,x, algorithm="giac")

[Out]

Done