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

Optimal. Leaf size=108 \[ \frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+\frac{5 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

(2*(d + e*x)^4)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (10*(d + e*x)^2)/(3*e*Sqrt[d^2 - e
^2*x^2]) - (5*Sqrt[d^2 - e^2*x^2])/e + (5*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Rubi [A]  time = 0.13153, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+\frac{5 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^4)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (10*(d + e*x)^2)/(3*e*Sqrt[d^2 - e
^2*x^2]) - (5*Sqrt[d^2 - e^2*x^2])/e + (5*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Rubi in Sympy [A]  time = 22.0805, size = 90, normalized size = 0.83 \[ \frac{5 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} + \frac{2 \left (d + e x\right )^{4}}{3 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{10 \left (d + e x\right )^{2}}{3 e \sqrt{d^{2} - e^{2} x^{2}}} - \frac{5 \sqrt{d^{2} - e^{2} x^{2}}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*d*atan(e*x/sqrt(d**2 - e**2*x**2))/e + 2*(d + e*x)**4/(3*e*(d**2 - e**2*x**2)*
*(3/2)) - 10*(d + e*x)**2/(3*e*sqrt(d**2 - e**2*x**2)) - 5*sqrt(d**2 - e**2*x**2
)/e

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Mathematica [A]  time = 0.124812, size = 75, normalized size = 0.69 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-23 d^2+34 d e x-3 e^2 x^2\right )}{(d-e x)^2}+15 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(((-23*d^2 + 34*d*e*x - 3*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(d - e*x)^2 + 15*d*ArcTa
n[(e*x)/Sqrt[d^2 - e^2*x^2]])/(3*e)

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Maple [A]  time = 0.009, size = 160, normalized size = 1.5 \[{\frac{11\,{d}^{3}x}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{23\,dx}{3}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{{e}^{3}{x}^{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+14\,{\frac{e{d}^{2}{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{23\,{d}^{4}}{3\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,d{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{d}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/3*d^3*x/(-e^2*x^2+d^2)^(3/2)-23/3*d*x/(-e^2*x^2+d^2)^(1/2)-e^3*x^4/(-e^2*x^2+
d^2)^(3/2)+14*e*d^2*x^2/(-e^2*x^2+d^2)^(3/2)-23/3*d^4/e/(-e^2*x^2+d^2)^(3/2)+5/3
*d*e^2*x^3/(-e^2*x^2+d^2)^(3/2)+5*d/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d
^2)^(1/2))

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Maxima [A]  time = 0.793233, size = 244, normalized size = 2.26 \[ \frac{5}{3} \, d e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} - \frac{e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{14 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{11 \, d^{3} x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{23 \, d^{4}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{13 \, d x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}}} + \frac{5 \, d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/3*d*e^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*
e^4)) - e^3*x^4/(-e^2*x^2 + d^2)^(3/2) + 14*d^2*e*x^2/(-e^2*x^2 + d^2)^(3/2) + 1
1/3*d^3*x/(-e^2*x^2 + d^2)^(3/2) - 23/3*d^4/((-e^2*x^2 + d^2)^(3/2)*e) - 13/3*d*
x/sqrt(-e^2*x^2 + d^2) + 5*d*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2)

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Fricas [A]  time = 0.232517, size = 416, normalized size = 3.85 \[ \frac{3 \, e^{5} x^{5} - 48 \, d e^{4} x^{4} + 7 \, d^{2} e^{3} x^{3} + 102 \, d^{3} e^{2} x^{2} - 48 \, d^{4} e x - 30 \,{\left (d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} + d^{3} e^{2} x^{2} + 6 \, d^{4} e x - 4 \, d^{5} +{\left (d e^{3} x^{3} + d^{2} e^{2} x^{2} - 6 \, d^{3} e x + 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (3 \, e^{4} x^{4} - 17 \, d e^{3} x^{3} + 102 \, d^{2} e^{2} x^{2} - 48 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (e^{5} x^{4} - 4 \, d e^{4} x^{3} + d^{2} e^{3} x^{2} + 6 \, d^{3} e^{2} x - 4 \, d^{4} e +{\left (e^{4} x^{3} + d e^{3} x^{2} - 6 \, d^{2} e^{2} x + 4 \, d^{3} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*e^5*x^5 - 48*d*e^4*x^4 + 7*d^2*e^3*x^3 + 102*d^3*e^2*x^2 - 48*d^4*e*x - 3
0*(d*e^4*x^4 - 4*d^2*e^3*x^3 + d^3*e^2*x^2 + 6*d^4*e*x - 4*d^5 + (d*e^3*x^3 + d^
2*e^2*x^2 - 6*d^3*e*x + 4*d^4)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2
+ d^2))/(e*x)) - (3*e^4*x^4 - 17*d*e^3*x^3 + 102*d^2*e^2*x^2 - 48*d^3*e*x)*sqrt(
-e^2*x^2 + d^2))/(e^5*x^4 - 4*d*e^4*x^3 + d^2*e^3*x^2 + 6*d^3*e^2*x - 4*d^4*e +
(e^4*x^3 + d*e^3*x^2 - 6*d^2*e^2*x + 4*d^3*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**5/(-(-d + e*x)*(d + e*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.228999, size = 116, normalized size = 1.07 \[ 5 \, d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{{\left (23 \, d^{4} e^{\left (-1\right )} +{\left (12 \, d^{3} -{\left (42 \, d^{2} e -{\left (3 \, x e^{3} - 28 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*d*arcsin(x*e/d)*e^(-1)*sign(d) - 1/3*(23*d^4*e^(-1) + (12*d^3 - (42*d^2*e - (3
*x*e^3 - 28*d*e^2)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^2