3.816 \(\int \frac{(d+e x)^4}{\sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e} \]

[Out]

(-35*d^3*Sqrt[d^2 - e^2*x^2])/(8*e) - (35*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(24
*e) - (7*d*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(12*e) - ((d + e*x)^3*Sqrt[d^2 - e^2
*x^2])/(4*e) + (35*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi [A]  time = 0.185948, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e} \]

Antiderivative was successfully verified.

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

[Out]

(-35*d^3*Sqrt[d^2 - e^2*x^2])/(8*e) - (35*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(24
*e) - (7*d*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(12*e) - ((d + e*x)^3*Sqrt[d^2 - e^2
*x^2])/(4*e) + (35*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi in Sympy [A]  time = 27.2196, size = 126, normalized size = 0.85 \[ \frac{35 d^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e} - \frac{35 d^{3} \sqrt{d^{2} - e^{2} x^{2}}}{8 e} - \frac{35 d^{2} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{24 e} - \frac{7 d \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}}{12 e} - \frac{\left (d + e x\right )^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

35*d**4*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e) - 35*d**3*sqrt(d**2 - e**2*x**2)/
(8*e) - 35*d**2*(d + e*x)*sqrt(d**2 - e**2*x**2)/(24*e) - 7*d*(d + e*x)**2*sqrt(
d**2 - e**2*x**2)/(12*e) - (d + e*x)**3*sqrt(d**2 - e**2*x**2)/(4*e)

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Mathematica [A]  time = 0.086829, size = 81, normalized size = 0.54 \[ \frac{105 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (160 d^3+81 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )}{24 e} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(160*d^3 + 81*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 105*
d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e)

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Maple [A]  time = 0.011, size = 119, normalized size = 0.8 \[{\frac{35\,{d}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{2}{x}^{3}}{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{27\,{d}^{2}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{20\,{d}^{3}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{4\,de{x}^{2}}{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x)

[Out]

35/8*d^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/4*e^2*x^3*(-e^
2*x^2+d^2)^(1/2)-27/8*d^2*x*(-e^2*x^2+d^2)^(1/2)-20/3*d^3*(-e^2*x^2+d^2)^(1/2)/e
-4/3*e*d*x^2*(-e^2*x^2+d^2)^(1/2)

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Maxima [A]  time = 0.781342, size = 150, normalized size = 1.01 \[ -\frac{1}{4} \, \sqrt{-e^{2} x^{2} + d^{2}} e^{2} x^{3} - \frac{4}{3} \, \sqrt{-e^{2} x^{2} + d^{2}} d e x^{2} + \frac{35 \, d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}}} - \frac{27}{8} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x - \frac{20 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(-e^2*x^2 + d^2)*e^2*x^3 - 4/3*sqrt(-e^2*x^2 + d^2)*d*e*x^2 + 35/8*d^4*
arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) - 27/8*sqrt(-e^2*x^2 + d^2)*d^2*x - 20/3*s
qrt(-e^2*x^2 + d^2)*d^3/e

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Fricas [A]  time = 0.236044, size = 412, normalized size = 2.77 \[ \frac{24 \, d e^{7} x^{7} + 128 \, d^{2} e^{6} x^{6} + 252 \, d^{3} e^{5} x^{5} + 96 \, d^{4} e^{4} x^{4} - 924 \, d^{5} e^{3} x^{3} - 384 \, d^{6} e^{2} x^{2} + 648 \, d^{7} e x - 210 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (6 \, e^{7} x^{7} + 32 \, d e^{6} x^{6} + 33 \, d^{2} e^{5} x^{5} - 96 \, d^{3} e^{4} x^{4} - 600 \, d^{4} e^{3} x^{3} - 384 \, d^{5} e^{2} x^{2} + 648 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (e^{5} x^{4} - 8 \, d^{2} e^{3} x^{2} + 8 \, d^{4} e + 4 \,{\left (d e^{3} x^{2} - 2 \, d^{3} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(24*d*e^7*x^7 + 128*d^2*e^6*x^6 + 252*d^3*e^5*x^5 + 96*d^4*e^4*x^4 - 924*d^
5*e^3*x^3 - 384*d^6*e^2*x^2 + 648*d^7*e*x - 210*(d^4*e^4*x^4 - 8*d^6*e^2*x^2 + 8
*d^8 + 4*(d^5*e^2*x^2 - 2*d^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2
+ d^2))/(e*x)) - (6*e^7*x^7 + 32*d*e^6*x^6 + 33*d^2*e^5*x^5 - 96*d^3*e^4*x^4 - 6
00*d^4*e^3*x^3 - 384*d^5*e^2*x^2 + 648*d^6*e*x)*sqrt(-e^2*x^2 + d^2))/(e^5*x^4 -
8*d^2*e^3*x^2 + 8*d^4*e + 4*(d*e^3*x^2 - 2*d^3*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 23.653, size = 551, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**4*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))) + 4*d**3*e*Piecewise((x**2/(2*sqrt(d**2)), Eq(e**2, 0)), (-s
qrt(d**2 - e**2*x**2)/e**2, True)) + 6*d**2*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*s
qrt(1 - e**2*x**2/d**2)), True)) + 4*d*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)) + e**4*Piecewise((-3*I*d**4*acosh(e*x/d)/(8*e**5) + 3*I*d**3*x/(8
*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d*x**3/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) -
I*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (3*d**4*asin(
e*x/d)/(8*e**5) - 3*d**3*x/(8*e**4*sqrt(1 - e**2*x**2/d**2)) + d*x**3/(8*e**2*sq
rt(1 - e**2*x**2/d**2)) + x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True))

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GIAC/XCAS [A]  time = 0.227965, size = 85, normalized size = 0.57 \[ \frac{35}{8} \, d^{4} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{24} \,{\left (160 \, d^{3} e^{\left (-1\right )} +{\left (81 \, d^{2} + 2 \,{\left (3 \, x e^{2} + 16 \, d e\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35/8*d^4*arcsin(x*e/d)*e^(-1)*sign(d) - 1/24*(160*d^3*e^(-1) + (81*d^2 + 2*(3*x*
e^2 + 16*d*e)*x)*x)*sqrt(-x^2*e^2 + d^2)