3.838 \(\int \frac{(d+e x)^6}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=112 \[ \frac{2 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

(2*(d + e*x)^5)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (2*(d + e*x)^3)/(3*e*(d^2 - e^2*x^
2)^(3/2)) + (2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]]/e

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Rubi [A]  time = 0.117253, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^5)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (2*(d + e*x)^3)/(3*e*(d^2 - e^2*x^
2)^(3/2)) + (2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]]/e

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Rubi in Sympy [A]  time = 19.9273, size = 94, normalized size = 0.84 \[ \frac{2 \left (d + e x\right )^{5}}{5 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2 \left (d + e x\right )^{3}}{3 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (d + e x\right )}{e \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(-e**2*x**2+d**2)**(7/2),x)

[Out]

2*(d + e*x)**5/(5*e*(d**2 - e**2*x**2)**(5/2)) - 2*(d + e*x)**3/(3*e*(d**2 - e**
2*x**2)**(3/2)) + 2*(d + e*x)/(e*sqrt(d**2 - e**2*x**2)) - atan(e*x/sqrt(d**2 -
e**2*x**2))/e

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Mathematica [A]  time = 0.0995029, size = 84, normalized size = 0.75 \[ \frac{15 (d-e x)^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-2 \sqrt{d^2-e^2 x^2} \left (13 d^2-24 d e x+23 e^2 x^2\right )}{15 e (e x-d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(-2*Sqrt[d^2 - e^2*x^2]*(13*d^2 - 24*d*e*x + 23*e^2*x^2) + 15*(d - e*x)^3*ArcTan
[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e*(-d + e*x)^3)

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Maple [B]  time = 0.013, size = 225, normalized size = 2. \[ -{\frac{13\,{d}^{4}x}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{23\,{d}^{2}x}{30} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{38\,x}{15}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{{e}^{4}{x}^{5}}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{1\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+6\,{\frac{{e}^{3}d{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{4\,{d}^{3}e{x}^{2}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{26\,{d}^{5}}{15\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{15\,{d}^{2}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-13/10*d^4*x/(-e^2*x^2+d^2)^(5/2)+23/30*d^2*x/(-e^2*x^2+d^2)^(3/2)+38/15*x/(-e^2
*x^2+d^2)^(1/2)+1/5*e^4*x^5/(-e^2*x^2+d^2)^(5/2)-1/3*e^2*x^3/(-e^2*x^2+d^2)^(3/2
)-1/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+6*d*e^3*x^4/(-e^2*x^2
+d^2)^(5/2)-4/3*d^3*e*x^2/(-e^2*x^2+d^2)^(5/2)+26/15*d^5/e/(-e^2*x^2+d^2)^(5/2)+
15/2*e^2*d^2*x^3/(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.80603, size = 405, normalized size = 3.62 \[ \frac{1}{15} \, e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{1}{3} \, e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{6 \, d e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{15 \, d^{2} e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{4 \, d^{3} e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, d^{4} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{26 \, d^{5}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{31 \, d^{2} x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{16 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*e^6*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(
5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 1/3*e^4*x*(3*x^2/((-e^2*x^2 +
d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 6*d*e^3*x^4/(-e^2*x^2 +
d^2)^(5/2) + 15/2*d^2*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) - 4/3*d^3*e*x^2/(-e^2*x^2 +
d^2)^(5/2) - 13/10*d^4*x/(-e^2*x^2 + d^2)^(5/2) + 26/15*d^5/((-e^2*x^2 + d^2)^(
5/2)*e) + 31/30*d^2*x/(-e^2*x^2 + d^2)^(3/2) + 16/15*x/sqrt(-e^2*x^2 + d^2) - ar
csin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2)

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Fricas [A]  time = 0.23393, size = 447, normalized size = 3.99 \[ \frac{2 \,{\left (36 \, e^{5} x^{5} - 20 \, d e^{4} x^{4} - 40 \, d^{2} e^{3} x^{3} + 60 \, d^{3} e^{2} x^{2} - 60 \, d^{4} e x + 15 \,{\left (e^{5} x^{5} - 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} + 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x + 4 \, d^{5} +{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 10 \,{\left (e^{4} x^{4} - 7 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} - 6 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}}{15 \,{\left (e^{6} x^{5} - 5 \, d e^{5} x^{4} + 5 \, d^{2} e^{4} x^{3} + 5 \, d^{3} e^{3} x^{2} - 10 \, d^{4} e^{2} x + 4 \, d^{5} e +{\left (e^{5} x^{4} - 7 \, d^{2} e^{3} x^{2} + 10 \, d^{3} e^{2} x - 4 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

2/15*(36*e^5*x^5 - 20*d*e^4*x^4 - 40*d^2*e^3*x^3 + 60*d^3*e^2*x^2 - 60*d^4*e*x +
15*(e^5*x^5 - 5*d*e^4*x^4 + 5*d^2*e^3*x^3 + 5*d^3*e^2*x^2 - 10*d^4*e*x + 4*d^5
+ (e^4*x^4 - 7*d^2*e^2*x^2 + 10*d^3*e*x - 4*d^4)*sqrt(-e^2*x^2 + d^2))*arctan(-(
d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 10*(e^4*x^4 - 7*d*e^3*x^3 + 6*d^2*e^2*x^2 - 6
*d^3*e*x)*sqrt(-e^2*x^2 + d^2))/(e^6*x^5 - 5*d*e^5*x^4 + 5*d^2*e^4*x^3 + 5*d^3*e
^3*x^2 - 10*d^4*e^2*x + 4*d^5*e + (e^5*x^4 - 7*d^2*e^3*x^2 + 10*d^3*e^2*x - 4*d^
4*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**6/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.232609, size = 128, normalized size = 1.14 \[ -\arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{2 \,{\left (13 \, d^{5} e^{\left (-1\right )} +{\left (15 \, d^{4} -{\left (10 \, d^{3} e -{\left (10 \, d^{2} e^{2} +{\left (23 \, x e^{4} + 45 \, d e^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-arcsin(x*e/d)*e^(-1)*sign(d) - 2/15*(13*d^5*e^(-1) + (15*d^4 - (10*d^3*e - (10*
d^2*e^2 + (23*x*e^4 + 45*d*e^3)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)
^3