3.2 \(\int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=201 \[ -\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5}+\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \]

[Out]

(3*d^7*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4)
- (4*d^2*x^2*(d^2 - e^2*x^2)^(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8
*e^2) - (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) - (d^3*(128*d + 315*e*x)*(d^2 - e^2*x^
2)^(5/2))/(5040*e^5) + (3*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

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Rubi [A]  time = 0.443924, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5}+\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]

[Out]

(3*d^7*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4)
- (4*d^2*x^2*(d^2 - e^2*x^2)^(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8
*e^2) - (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) - (d^3*(128*d + 315*e*x)*(d^2 - e^2*x^
2)^(5/2))/(5040*e^5) + (3*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

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Rubi in Sympy [A]  time = 57.0033, size = 178, normalized size = 0.89 \[ \frac{3 d^{9} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{128 e^{5}} + \frac{3 d^{7} x \sqrt{d^{2} - e^{2} x^{2}}}{128 e^{4}} + \frac{d^{5} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{64 e^{4}} - \frac{d^{3} \left (384 d + 945 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{15120 e^{5}} - \frac{4 d^{2} x^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{63 e^{3}} - \frac{d x^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{8 e^{2}} - \frac{x^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{9 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)*(-e**2*x**2+d**2)**(3/2),x)

[Out]

3*d**9*atan(e*x/sqrt(d**2 - e**2*x**2))/(128*e**5) + 3*d**7*x*sqrt(d**2 - e**2*x
**2)/(128*e**4) + d**5*x*(d**2 - e**2*x**2)**(3/2)/(64*e**4) - d**3*(384*d + 945
*e*x)*(d**2 - e**2*x**2)**(5/2)/(15120*e**5) - 4*d**2*x**2*(d**2 - e**2*x**2)**(
5/2)/(63*e**3) - d*x**3*(d**2 - e**2*x**2)**(5/2)/(8*e**2) - x**4*(d**2 - e**2*x
**2)**(5/2)/(9*e)

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Mathematica [A]  time = 0.132318, size = 136, normalized size = 0.68 \[ \frac{945 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (1024 d^8+945 d^7 e x+512 d^6 e^2 x^2+630 d^5 e^3 x^3+384 d^4 e^4 x^4-7560 d^3 e^5 x^5-6400 d^2 e^6 x^6+5040 d e^7 x^7+4480 e^8 x^8\right )}{40320 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(1024*d^8 + 945*d^7*e*x + 512*d^6*e^2*x^2 + 630*d^5*e^3*x
^3 + 384*d^4*e^4*x^4 - 7560*d^3*e^5*x^5 - 6400*d^2*e^6*x^6 + 5040*d*e^7*x^7 + 44
80*e^8*x^8)) + 945*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(40320*e^5)

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Maple [A]  time = 0.036, size = 198, normalized size = 1. \[ -{\frac{d{x}^{3}}{8\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{3}x}{16\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{5}x}{64\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{7}x}{128\,{e}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{3\,{d}^{9}}{128\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{x}^{4}}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{d}^{2}{x}^{2}}{63\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{8\,{d}^{4}}{315\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x)

[Out]

-1/8*d*x^3*(-e^2*x^2+d^2)^(5/2)/e^2-1/16*d^3/e^4*x*(-e^2*x^2+d^2)^(5/2)+1/64*d^5
*x*(-e^2*x^2+d^2)^(3/2)/e^4+3/128*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^4+3/128*d^9/e^4/(
e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/9*x^4*(-e^2*x^2+d^2)^(5/
2)/e-4/63*d^2*x^2*(-e^2*x^2+d^2)^(5/2)/e^3-8/315*d^4/e^5*(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.801193, size = 257, normalized size = 1.28 \[ -\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} x^{4}}{9 \, e} + \frac{3 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}} e^{4}} + \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x^{3}}{8 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x}{64 \, e^{4}} - \frac{4 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{2} x^{2}}{63 \, e^{3}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x}{16 \, e^{4}} - \frac{8 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{4}}{315 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x^4,x, algorithm="maxima")

[Out]

-1/9*(-e^2*x^2 + d^2)^(5/2)*x^4/e + 3/128*d^9*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(
e^2)*e^4) + 3/128*sqrt(-e^2*x^2 + d^2)*d^7*x/e^4 - 1/8*(-e^2*x^2 + d^2)^(5/2)*d*
x^3/e^2 + 1/64*(-e^2*x^2 + d^2)^(3/2)*d^5*x/e^4 - 4/63*(-e^2*x^2 + d^2)^(5/2)*d^
2*x^2/e^3 - 1/16*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^4 - 8/315*(-e^2*x^2 + d^2)^(5/2)
*d^4/e^5

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Fricas [A]  time = 0.282974, size = 806, normalized size = 4.01 \[ -\frac{4480 \, e^{18} x^{18} + 5040 \, d e^{17} x^{17} - 190080 \, d^{2} e^{16} x^{16} - 214200 \, d^{3} e^{15} x^{15} + 1517184 \, d^{4} e^{14} x^{14} + 1721790 \, d^{5} e^{13} x^{13} - 4889472 \, d^{6} e^{12} x^{12} - 5609205 \, d^{7} e^{11} x^{11} + 7644672 \, d^{8} e^{10} x^{10} + 8887095 \, d^{9} e^{9} x^{9} - 5806080 \, d^{10} e^{8} x^{8} - 6781320 \, d^{11} e^{7} x^{7} + 1720320 \, d^{12} e^{6} x^{6} + 1728720 \, d^{13} e^{5} x^{5} + 504000 \, d^{15} e^{3} x^{3} - 241920 \, d^{17} e x + 1890 \,{\left (9 \, d^{10} e^{8} x^{8} - 120 \, d^{12} e^{6} x^{6} + 432 \, d^{14} e^{4} x^{4} - 576 \, d^{16} e^{2} x^{2} + 256 \, d^{18} -{\left (d^{9} e^{8} x^{8} - 40 \, d^{11} e^{6} x^{6} + 240 \, d^{13} e^{4} x^{4} - 448 \, d^{15} e^{2} x^{2} + 256 \, d^{17}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (13440 \, d e^{16} x^{16} + 15120 \, d^{2} e^{15} x^{15} - 198400 \, d^{3} e^{14} x^{14} - 224280 \, d^{4} e^{13} x^{13} + 902272 \, d^{5} e^{12} x^{12} + 1030050 \, d^{6} e^{11} x^{11} - 1795584 \, d^{7} e^{10} x^{10} - 2078685 \, d^{8} e^{9} x^{9} + 1648640 \, d^{9} e^{8} x^{8} + 1934520 \, d^{10} e^{7} x^{7} - 573440 \, d^{11} e^{6} x^{6} - 630000 \, d^{12} e^{5} x^{5} - 127680 \, d^{14} e^{3} x^{3} + 80640 \, d^{16} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40320 \,{\left (9 \, d e^{13} x^{8} - 120 \, d^{3} e^{11} x^{6} + 432 \, d^{5} e^{9} x^{4} - 576 \, d^{7} e^{7} x^{2} + 256 \, d^{9} e^{5} -{\left (e^{13} x^{8} - 40 \, d^{2} e^{11} x^{6} + 240 \, d^{4} e^{9} x^{4} - 448 \, d^{6} e^{7} x^{2} + 256 \, d^{8} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x^4,x, algorithm="fricas")

[Out]

-1/40320*(4480*e^18*x^18 + 5040*d*e^17*x^17 - 190080*d^2*e^16*x^16 - 214200*d^3*
e^15*x^15 + 1517184*d^4*e^14*x^14 + 1721790*d^5*e^13*x^13 - 4889472*d^6*e^12*x^1
2 - 5609205*d^7*e^11*x^11 + 7644672*d^8*e^10*x^10 + 8887095*d^9*e^9*x^9 - 580608
0*d^10*e^8*x^8 - 6781320*d^11*e^7*x^7 + 1720320*d^12*e^6*x^6 + 1728720*d^13*e^5*
x^5 + 504000*d^15*e^3*x^3 - 241920*d^17*e*x + 1890*(9*d^10*e^8*x^8 - 120*d^12*e^
6*x^6 + 432*d^14*e^4*x^4 - 576*d^16*e^2*x^2 + 256*d^18 - (d^9*e^8*x^8 - 40*d^11*
e^6*x^6 + 240*d^13*e^4*x^4 - 448*d^15*e^2*x^2 + 256*d^17)*sqrt(-e^2*x^2 + d^2))*
arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 3*(13440*d*e^16*x^16 + 15120*d^2*e^1
5*x^15 - 198400*d^3*e^14*x^14 - 224280*d^4*e^13*x^13 + 902272*d^5*e^12*x^12 + 10
30050*d^6*e^11*x^11 - 1795584*d^7*e^10*x^10 - 2078685*d^8*e^9*x^9 + 1648640*d^9*
e^8*x^8 + 1934520*d^10*e^7*x^7 - 573440*d^11*e^6*x^6 - 630000*d^12*e^5*x^5 - 127
680*d^14*e^3*x^3 + 80640*d^16*e*x)*sqrt(-e^2*x^2 + d^2))/(9*d*e^13*x^8 - 120*d^3
*e^11*x^6 + 432*d^5*e^9*x^4 - 576*d^7*e^7*x^2 + 256*d^9*e^5 - (e^13*x^8 - 40*d^2
*e^11*x^6 + 240*d^4*e^9*x^4 - 448*d^6*e^7*x^2 + 256*d^8*e^5)*sqrt(-e^2*x^2 + d^2
))

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Sympy [A]  time = 55.0427, size = 830, normalized size = 4.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x+d)*(-e**2*x**2+d**2)**(3/2),x)

[Out]

d**3*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**
2*x**2/d**2)) - I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24
*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e
**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2
*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(
1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2)), True)) + d**2*e
*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**2 -
e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d
**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True)) - d*e**2*Piecewise((-5
*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(128*e**6*sqrt(-1 + e**2*x**2/d**2)
) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/(192*e**2*s
qrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x
**9/(8*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (5*d**8*asin(e*x/
d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2/d**2)) + 5*d**5*x**3/(384*
e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2)) +
7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2
)), True)) - e**3*Piecewise((-16*d**8*sqrt(d**2 - e**2*x**2)/(315*e**8) - 8*d**6
*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d**4*x**4*sqrt(d**2 - e**2*x**2)/(10
5*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2 - e**2*x**
2)/9, Ne(e, 0)), (x**8*sqrt(d**2)/8, True))

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GIAC/XCAS [A]  time = 0.315756, size = 158, normalized size = 0.79 \[ \frac{3}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-5\right )}{\rm sign}\left (d\right ) - \frac{1}{40320} \,{\left (1024 \, d^{8} e^{\left (-5\right )} +{\left (945 \, d^{7} e^{\left (-4\right )} + 2 \,{\left (256 \, d^{6} e^{\left (-3\right )} +{\left (315 \, d^{5} e^{\left (-2\right )} + 4 \,{\left (48 \, d^{4} e^{\left (-1\right )} - 5 \,{\left (189 \, d^{3} + 2 \,{\left (80 \, d^{2} e - 7 \,{\left (8 \, x e^{3} + 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x^4,x, algorithm="giac")

[Out]

3/128*d^9*arcsin(x*e/d)*e^(-5)*sign(d) - 1/40320*(1024*d^8*e^(-5) + (945*d^7*e^(
-4) + 2*(256*d^6*e^(-3) + (315*d^5*e^(-2) + 4*(48*d^4*e^(-1) - 5*(189*d^3 + 2*(8
0*d^2*e - 7*(8*x*e^3 + 9*d*e^2)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)