3.817 \(\int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{5 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(-5*d^2*Sqrt[d^2 - e^2*x^2])/(2*e) - (5*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(6*e) -
((d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(3*e) + (5*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(2*e)

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Rubi [A]  time = 0.130858, antiderivative size = 116, 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 \[ -\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{5 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d^2*Sqrt[d^2 - e^2*x^2])/(2*e) - (5*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(6*e) -
((d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(3*e) + (5*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(2*e)

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Rubi in Sympy [A]  time = 21.4899, size = 97, normalized size = 0.84 \[ \frac{5 d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{5 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{2 e} - \frac{5 d \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{6 e} - \frac{\left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

5*d**3*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) - 5*d**2*sqrt(d**2 - e**2*x**2)/(2
*e) - 5*d*(d + e*x)*sqrt(d**2 - e**2*x**2)/(6*e) - (d + e*x)**2*sqrt(d**2 - e**2
*x**2)/(3*e)

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Mathematica [A]  time = 0.074686, size = 70, normalized size = 0.6 \[ \frac{15 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (22 d^2+9 d e x+2 e^2 x^2\right )}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(22*d^2 + 9*d*e*x + 2*e^2*x^2)) + 15*d^3*ArcTan[(e*x)/Sqr
t[d^2 - e^2*x^2]])/(6*e)

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Maple [A]  time = 0.011, size = 94, normalized size = 0.8 \[{\frac{5\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e{x}^{2}}{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{11\,{d}^{2}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,dx}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.788577, size = 116, normalized size = 1. \[ -\frac{1}{3} \, \sqrt{-e^{2} x^{2} + d^{2}} e x^{2} + \frac{5 \, d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} - \frac{3}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} d x - \frac{11 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(-e^2*x^2 + d^2)*e*x^2 + 5/2*d^3*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2)
- 3/2*sqrt(-e^2*x^2 + d^2)*d*x - 11/3*sqrt(-e^2*x^2 + d^2)*d^2/e

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Fricas [A]  time = 0.229524, size = 325, normalized size = 2.8 \[ -\frac{2 \, e^{6} x^{6} + 9 \, d e^{5} x^{5} + 12 \, d^{2} e^{4} x^{4} - 45 \, d^{3} e^{3} x^{3} - 36 \, d^{4} e^{2} x^{2} + 36 \, d^{5} e x + 30 \,{\left (3 \, d^{4} e^{2} x^{2} - 4 \, d^{6} -{\left (d^{3} e^{2} x^{2} - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (2 \, d e^{4} x^{4} + 9 \, d^{2} e^{3} x^{3} + 12 \, d^{3} e^{2} x^{2} - 12 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (3 \, d e^{3} x^{2} - 4 \, d^{3} e -{\left (e^{3} x^{2} - 4 \, d^{2} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*e^6*x^6 + 9*d*e^5*x^5 + 12*d^2*e^4*x^4 - 45*d^3*e^3*x^3 - 36*d^4*e^2*x^2
+ 36*d^5*e*x + 30*(3*d^4*e^2*x^2 - 4*d^6 - (d^3*e^2*x^2 - 4*d^5)*sqrt(-e^2*x^2
+ d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 3*(2*d*e^4*x^4 + 9*d^2*e^3*x
^3 + 12*d^3*e^2*x^2 - 12*d^4*e*x)*sqrt(-e^2*x^2 + d^2))/(3*d*e^3*x^2 - 4*d^3*e -
(e^3*x^2 - 4*d^2*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 11.7864, size = 342, normalized size = 2.95 \[ d^{3} \left (\begin{cases} \frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{asin}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} < 0 \\\frac{\sqrt{- \frac{d^{2}}{e^{2}}} \operatorname{asinh}{\left (x \sqrt{- \frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} > 0 \\\frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{acosh}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{- d^{2}}} & \text{for}\: - e^{2} > 0 \wedge d^{2} < 0 \end{cases}\right ) + 3 d^{2} e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{d^{2}}} & \text{for}\: e^{2} = 0 \\- \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) + 3 d e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{i d x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{d x}{2 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{3}}{2 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{2 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac{x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \sqrt{d^{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**3*Piecewise((sqrt(d**2/e**2)*asin(x*sqrt(e**2/d**2))/sqrt(d**2), (d**2 > 0) &
(-e**2 < 0)), (sqrt(-d**2/e**2)*asinh(x*sqrt(-e**2/d**2))/sqrt(d**2), (d**2 > 0
) & (-e**2 > 0)), (sqrt(d**2/e**2)*acosh(x*sqrt(e**2/d**2))/sqrt(-d**2), (d**2 <
0) & (-e**2 > 0))) + 3*d**2*e*Piecewise((x**2/(2*sqrt(d**2)), Eq(e**2, 0)), (-s
qrt(d**2 - e**2*x**2)/e**2, True)) + 3*d*e**2*Piecewise((-I*d**2*acosh(e*x/d)/(2
*e**3) - I*d*x*sqrt(-1 + e**2*x**2/d**2)/(2*e**2), Abs(e**2*x**2/d**2) > 1), (d*
*2*asin(e*x/d)/(2*e**3) - d*x/(2*e**2*sqrt(1 - e**2*x**2/d**2)) + x**3/(2*d*sqrt
(1 - e**2*x**2/d**2)), True)) + e**3*Piecewise((-2*d**2*sqrt(d**2 - e**2*x**2)/(
3*e**4) - x**2*sqrt(d**2 - e**2*x**2)/(3*e**2), Ne(e, 0)), (x**4/(4*sqrt(d**2)),
True))

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GIAC/XCAS [A]  time = 0.226701, size = 70, normalized size = 0.6 \[ \frac{5}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (22 \, d^{2} e^{\left (-1\right )} +{\left (2 \, x e + 9 \, d\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*d^3*arcsin(x*e/d)*e^(-1)*sign(d) - 1/6*sqrt(-x^2*e^2 + d^2)*(22*d^2*e^(-1) +
(2*x*e + 9*d)*x)