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

Optimal. Leaf size=147 \[ \frac{x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{4 e^2}+\frac{4 d^2 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^5}+\frac{d^3 (64 d-45 e x) \sqrt{d^2-e^2 x^2}}{120 e^5} \]

[Out]

(4*d^2*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) - (d*x^3*Sqrt[d^2 - e^2*x^2])/(4*e^2) +
(x^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (d^3*(64*d - 45*e*x)*Sqrt[d^2 - e^2*x^2])/(12
0*e^5) + (3*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^5)

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Rubi [A]  time = 0.472769, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{4 e^2}+\frac{4 d^2 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^5}+\frac{d^3 (64 d-45 e x) \sqrt{d^2-e^2 x^2}}{120 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(4*d^2*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) - (d*x^3*Sqrt[d^2 - e^2*x^2])/(4*e^2) +
(x^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (d^3*(64*d - 45*e*x)*Sqrt[d^2 - e^2*x^2])/(12
0*e^5) + (3*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^5)

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Rubi in Sympy [A]  time = 52.0239, size = 129, normalized size = 0.88 \[ \frac{3 d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e^{5}} + \frac{d^{3} \left (64 d - 45 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{120 e^{5}} + \frac{4 d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{3}} - \frac{d x^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*d**5*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e**5) + d**3*(64*d - 45*e*x)*sqrt(d**
2 - e**2*x**2)/(120*e**5) + 4*d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**3) - d*x**
3*sqrt(d**2 - e**2*x**2)/(4*e**2) + x**4*sqrt(d**2 - e**2*x**2)/(5*e)

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Mathematica [A]  time = 0.100923, size = 91, normalized size = 0.62 \[ \frac{45 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (64 d^4-45 d^3 e x+32 d^2 e^2 x^2-30 d e^3 x^3+24 e^4 x^4\right )}{120 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(64*d^4 - 45*d^3*e*x + 32*d^2*e^2*x^2 - 30*d*e^3*x^3 + 24*e
^4*x^4) + 45*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(120*e^5)

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Maple [A]  time = 0.028, size = 208, normalized size = 1.4 \[ -{\frac{{x}^{2}}{5\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{d}^{2}}{15\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{4}}{{e}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{d}^{5}}{{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{5\,{d}^{3}x}{8\,{e}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{5\,{d}^{5}}{8\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{dx}{4\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/e^3*x^2*(-e^2*x^2+d^2)^(3/2)-7/15*d^2/e^5*(-e^2*x^2+d^2)^(3/2)+d^4/e^5*(-(x
+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+d^5/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))-5/8*d^3/e^4*x*(-e^2*x^2+d^2)^(1/2)-5/8*d^5/e^4/(
e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/4*d/e^4*x*(-e^2*x^2+d^2)
^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292162, size = 450, normalized size = 3.06 \[ \frac{24 \, e^{10} x^{10} - 30 \, d e^{9} x^{9} - 280 \, d^{2} e^{8} x^{8} + 345 \, d^{3} e^{7} x^{7} + 320 \, d^{4} e^{6} x^{6} - 255 \, d^{5} e^{5} x^{5} - 780 \, d^{7} e^{3} x^{3} + 720 \, d^{9} e x - 90 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (24 \, d e^{8} x^{8} - 30 \, d^{2} e^{7} x^{7} - 64 \, d^{3} e^{6} x^{6} + 75 \, d^{4} e^{5} x^{5} + 84 \, d^{6} e^{3} x^{3} - 144 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \,{\left (5 \, d e^{9} x^{4} - 20 \, d^{3} e^{7} x^{2} + 16 \, d^{5} e^{5} -{\left (e^{9} x^{4} - 12 \, d^{2} e^{7} x^{2} + 16 \, d^{4} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(24*e^10*x^10 - 30*d*e^9*x^9 - 280*d^2*e^8*x^8 + 345*d^3*e^7*x^7 + 320*d^4
*e^6*x^6 - 255*d^5*e^5*x^5 - 780*d^7*e^3*x^3 + 720*d^9*e*x - 90*(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*(24*d*e^8*x^8 - 30*d^2*e
^7*x^7 - 64*d^3*e^6*x^6 + 75*d^4*e^5*x^5 + 84*d^6*e^3*x^3 - 144*d^8*e*x)*sqrt(-e
^2*x^2 + d^2))/(5*d*e^9*x^4 - 20*d^3*e^7*x^2 + 16*d^5*e^5 - (e^9*x^4 - 12*d^2*e^
7*x^2 + 16*d^4*e^5)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x), x)

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GIAC/XCAS [A]  time = 0.297028, size = 104, normalized size = 0.71 \[ \frac{3}{8} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-5\right )}{\rm sign}\left (d\right ) + \frac{1}{120} \,{\left (64 \, d^{4} e^{\left (-5\right )} -{\left (45 \, d^{3} e^{\left (-4\right )} - 2 \,{\left (16 \, d^{2} e^{\left (-3\right )} + 3 \,{\left (4 \, x e^{\left (-1\right )} - 5 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*d^5*arcsin(x*e/d)*e^(-5)*sign(d) + 1/120*(64*d^4*e^(-5) - (45*d^3*e^(-4) - 2
*(16*d^2*e^(-3) + 3*(4*x*e^(-1) - 5*d*e^(-2))*x)*x)*x)*sqrt(-x^2*e^2 + d^2)