3.827 \(\int \frac{(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0833191, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.8458, size = 68, normalized size = 0.84 \[ \frac{2 \left (d + e x\right )^{3}}{3 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 \left (d + e x\right )}{e \sqrt{d^{2} - e^{2} x^{2}}} + \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(-e**2*x**2+d**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0931807, size = 62, normalized size = 0.77 \[ \frac{3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{4 (d-2 e x) \sqrt{d^2-e^2 x^2}}{(d-e x)^2}}{3 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 132, normalized size = 1.6 \[{\frac{7\,{d}^{2}x}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,x}{3}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{1\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{4\,{d}^{3}}{3\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{de{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x)

[Out]

7/3*d^2*x/(-e^2*x^2+d^2)^(3/2)-7/3*x/(-e^2*x^2+d^2)^(1/2)+1/3*e^2*x^3/(-e^2*x^2+
d^2)^(3/2)+1/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-4/3*d^3/e/(-
e^2*x^2+d^2)^(3/2)+4*e*d*x^2/(-e^2*x^2+d^2)^(3/2)

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Maxima [A]  time = 0.795555, size = 207, normalized size = 2.56 \[ \frac{1}{3} \, e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{4 \, d e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{7 \, d^{2} x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{4 \, d^{3}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{5 \, x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}}} + \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(-e^2*x^2 + d^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22511, size = 252, normalized size = 3.11 \[ \frac{2 \,{\left (2 \, e^{3} x^{3} - 6 \, d e^{2} x^{2} + 6 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{2} x^{2} - 3 \,{\left (e^{3} x^{3} - 3 \, d^{2} e x + 2 \, d^{3} -{\left (e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right )\right )}}{3 \,{\left (e^{4} x^{3} - 3 \, d^{2} e^{2} x + 2 \, d^{3} e -{\left (e^{3} x^{2} - 3 \, d e^{2} x + 2 \, d^{2} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(-e^2*x^2 + d^2)^(5/2),x, algorithm="fricas")

[Out]

2/3*(2*e^3*x^3 - 6*d*e^2*x^2 + 6*sqrt(-e^2*x^2 + d^2)*e^2*x^2 - 3*(e^3*x^3 - 3*d
^2*e*x + 2*d^3 - (e^2*x^2 - 3*d*e*x + 2*d^2)*sqrt(-e^2*x^2 + d^2))*arctan(-(d -
sqrt(-e^2*x^2 + d^2))/(e*x)))/(e^4*x^3 - 3*d^2*e^2*x + 2*d^3*e - (e^3*x^2 - 3*d*
e^2*x + 2*d^2*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral((d + e*x)**4/(-(-d + e*x)*(d + e*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.229763, size = 89, normalized size = 1.1 \[ \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{4 \,{\left (d^{3} e^{\left (-1\right )} -{\left (2 \, x e^{2} + 3 \, d e\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(-e^2*x^2 + d^2)^(5/2),x, algorithm="giac")

[Out]

arcsin(x*e/d)*e^(-1)*sign(d) - 4/3*(d^3*e^(-1) - (2*x*e^2 + 3*d*e)*x^2)*sqrt(-x^
2*e^2 + d^2)/(x^2*e^2 - d^2)^2