3.815 \(\int \frac{(d+e x)^5}{\sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=182 \[ -\frac{21 d^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{9 d (d+e x)^3 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{(d+e x)^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{63 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}-\frac{63 d^4 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{21 d^3 (d+e x) \sqrt{d^2-e^2 x^2}}{8 e} \]

[Out]

(-63*d^4*Sqrt[d^2 - e^2*x^2])/(8*e) - (21*d^3*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(8*
e) - (21*d^2*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(20*e) - (9*d*(d + e*x)^3*Sqrt[d^2
- e^2*x^2])/(20*e) - ((d + e*x)^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (63*d^5*ArcTan[(
e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi [A]  time = 0.252128, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{21 d^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{9 d (d+e x)^3 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{(d+e x)^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{63 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}-\frac{63 d^4 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{21 d^3 (d+e x) \sqrt{d^2-e^2 x^2}}{8 e} \]

Antiderivative was successfully verified.

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

[Out]

(-63*d^4*Sqrt[d^2 - e^2*x^2])/(8*e) - (21*d^3*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(8*
e) - (21*d^2*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(20*e) - (9*d*(d + e*x)^3*Sqrt[d^2
- e^2*x^2])/(20*e) - ((d + e*x)^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (63*d^5*ArcTan[(
e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

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Rubi in Sympy [A]  time = 33.5986, size = 155, normalized size = 0.85 \[ \frac{63 d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e} - \frac{63 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{8 e} - \frac{21 d^{3} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{8 e} - \frac{21 d^{2} \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}}{20 e} - \frac{9 d \left (d + e x\right )^{3} \sqrt{d^{2} - e^{2} x^{2}}}{20 e} - \frac{\left (d + e x\right )^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)

[Out]

63*d**5*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e) - 63*d**4*sqrt(d**2 - e**2*x**2)/
(8*e) - 21*d**3*(d + e*x)*sqrt(d**2 - e**2*x**2)/(8*e) - 21*d**2*(d + e*x)**2*sq
rt(d**2 - e**2*x**2)/(20*e) - 9*d*(d + e*x)**3*sqrt(d**2 - e**2*x**2)/(20*e) - (
d + e*x)**4*sqrt(d**2 - e**2*x**2)/(5*e)

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Mathematica [A]  time = 0.110441, size = 92, normalized size = 0.51 \[ \frac{315 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (488 d^4+275 d^3 e x+144 d^2 e^2 x^2+50 d e^3 x^3+8 e^4 x^4\right )}{40 e} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(488*d^4 + 275*d^3*e*x + 144*d^2*e^2*x^2 + 50*d*e^3*x^3 +
8*e^4*x^4)) + 315*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(40*e)

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Maple [A]  time = 0.031, size = 144, normalized size = 0.8 \[{\frac{63\,{d}^{5}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{3}{x}^{4}}{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{18\,e{d}^{2}{x}^{2}}{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{61\,{d}^{4}}{5\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{5\,d{e}^{2}{x}^{3}}{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{55\,{d}^{3}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x)

[Out]

63/8*d^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/5*e^3*x^4*(-e^
2*x^2+d^2)^(1/2)-18/5*e*d^2*x^2*(-e^2*x^2+d^2)^(1/2)-61/5*d^4*(-e^2*x^2+d^2)^(1/
2)/e-5/4*d*e^2*x^3*(-e^2*x^2+d^2)^(1/2)-55/8*d^3*x*(-e^2*x^2+d^2)^(1/2)

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Maxima [A]  time = 0.777208, size = 184, normalized size = 1.01 \[ -\frac{1}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} e^{3} x^{4} - \frac{5}{4} \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{2} x^{3} - \frac{18}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e x^{2} + \frac{63 \, d^{5} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}}} - \frac{55}{8} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x - \frac{61 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*sqrt(-e^2*x^2 + d^2)*e^3*x^4 - 5/4*sqrt(-e^2*x^2 + d^2)*d*e^2*x^3 - 18/5*sq
rt(-e^2*x^2 + d^2)*d^2*e*x^2 + 63/8*d^5*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) -
55/8*sqrt(-e^2*x^2 + d^2)*d^3*x - 61/5*sqrt(-e^2*x^2 + d^2)*d^4/e

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Fricas [A]  time = 0.226991, size = 489, normalized size = 2.69 \[ -\frac{8 \, e^{10} x^{10} + 50 \, d e^{9} x^{9} + 40 \, d^{2} e^{8} x^{8} - 375 \, d^{3} e^{7} x^{7} - 1160 \, d^{4} e^{6} x^{6} - 2175 \, d^{5} e^{5} x^{5} + 6900 \, d^{7} e^{3} x^{3} + 1600 \, d^{8} e^{2} x^{2} - 4400 \, d^{9} e x + 630 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (8 \, d e^{8} x^{8} + 50 \, d^{2} e^{7} x^{7} + 112 \, d^{3} e^{6} x^{6} + 75 \, d^{4} e^{5} x^{5} - 160 \, d^{5} e^{4} x^{4} - 940 \, d^{6} e^{3} x^{3} - 320 \, d^{7} e^{2} x^{2} + 880 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40 \,{\left (5 \, d e^{5} x^{4} - 20 \, d^{3} e^{3} x^{2} + 16 \, d^{5} e -{\left (e^{5} x^{4} - 12 \, d^{2} e^{3} x^{2} + 16 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(8*e^10*x^10 + 50*d*e^9*x^9 + 40*d^2*e^8*x^8 - 375*d^3*e^7*x^7 - 1160*d^4*
e^6*x^6 - 2175*d^5*e^5*x^5 + 6900*d^7*e^3*x^3 + 1600*d^8*e^2*x^2 - 4400*d^9*e*x
+ 630*(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*(8
*d*e^8*x^8 + 50*d^2*e^7*x^7 + 112*d^3*e^6*x^6 + 75*d^4*e^5*x^5 - 160*d^5*e^4*x^4
- 940*d^6*e^3*x^3 - 320*d^7*e^2*x^2 + 880*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 [A]  time = 26.8653, size = 646, normalized size = 3.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**5*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))) + 5*d**4*e*Piecewise((x**2/(2*sqrt(d**2)), Eq(e**2, 0)), (-s
qrt(d**2 - e**2*x**2)/e**2, True)) + 10*d**3*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)) + 10*d**2*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)) + 5*d*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*sqrt(1 - e**2*x**2/d**2)) + x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) +
e**5*Piecewise((-8*d**4*sqrt(d**2 - e**2*x**2)/(15*e**6) - 4*d**2*x**2*sqrt(d**
2 - e**2*x**2)/(15*e**4) - x**4*sqrt(d**2 - e**2*x**2)/(5*e**2), Ne(e, 0)), (x**
6/(6*sqrt(d**2)), True))

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GIAC/XCAS [A]  time = 0.22879, size = 99, normalized size = 0.54 \[ \frac{63}{8} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{40} \,{\left (488 \, d^{4} e^{\left (-1\right )} +{\left (275 \, d^{3} + 2 \,{\left (72 \, d^{2} e +{\left (4 \, x e^{3} + 25 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

63/8*d^5*arcsin(x*e/d)*e^(-1)*sign(d) - 1/40*(488*d^4*e^(-1) + (275*d^3 + 2*(72*
d^2*e + (4*x*e^3 + 25*d*e^2)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)