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

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 = 0.512918, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5}+\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x),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 = 63.692, 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**2*x**2+d**2)**(5/2)/(e*x+d),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 = 0.139971, size = 135, normalized size = 0.67 \[ \frac{945 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (1024 d^8-945 d^7 e x+512 d^6 e^2 x^2-630 d^5 e^3 x^3+384 d^4 e^4 x^4+7560 d^3 e^5 x^5-6400 d^2 e^6 x^6-5040 d e^7 x^7+4480 e^8 x^8\right )}{40320 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x),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 + 4480
*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 = 0.019, size = 330, normalized size = 1.6 \[ -{\frac{{x}^{2}}{9\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{d}^{2}}{63\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{4}}{5\,{e}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{5}x}{4\,{e}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{7}x}{8\,{e}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{3\,{d}^{9}}{8\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{3\,{d}^{3}x}{16\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{d}^{5}x}{64\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{45\,{d}^{7}x}{128\,{e}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{45\,{d}^{9}}{128\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{dx}{8\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9/e^3*x^2*(-e^2*x^2+d^2)^(7/2)-11/63*d^2/e^5*(-e^2*x^2+d^2)^(7/2)+1/5*d^4/e^5
*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+1/4*d^5/e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))
^(3/2)*x+3/8*d^7/e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x+3/8*d^9/e^4/(e^2)^(1
/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-3/16*d^3/e^4*x*(-
e^2*x^2+d^2)^(5/2)-15/64*d^5*x*(-e^2*x^2+d^2)^(3/2)/e^4-45/128*d^7*x*(-e^2*x^2+d
^2)^(1/2)/e^4-45/128*d^9/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/
2))+1/8*d/e^4*x*(-e^2*x^2+d^2)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293448, 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)^(5/2)*x^4/(e*x + d),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^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*(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))*a
rctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 3*(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 - 103
0050*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 + 1276
80*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 = 62.5952, 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**2*x**2+d**2)**(5/2)/(e*x+d),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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError