3.839 \(\int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(d + e*x)^5/(5*d*e*(d^2 - e^2*x^2)^(5/2))

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Rubi [A]  time = 0.0412624, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d + e*x)^5/(5*d*e*(d^2 - e^2*x^2)^(5/2))

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Rubi in Sympy [A]  time = 5.31678, size = 24, normalized size = 0.73 \[ \frac{\left (d + e x\right )^{5}}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

(d + e*x)**5/(5*d*e*(d**2 - e**2*x**2)**(5/2))

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Mathematica [A]  time = 0.0446283, size = 41, normalized size = 1.24 \[ \frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.009, size = 36, normalized size = 1.1 \[{\frac{ \left ( ex+d \right ) ^{6} \left ( -ex+d \right ) }{5\,de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/5*(e*x+d)^6*(-e*x+d)/d/e/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.722244, size = 200, normalized size = 6.06 \[ \frac{e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, d e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, d^{3} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d^{4}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 5/2*d*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) + 2*d^2*e*
x^2/(-e^2*x^2 + d^2)^(5/2) + 7/10*d^3*x/(-e^2*x^2 + d^2)^(5/2) + 1/5*d^4/((-e^2*
x^2 + d^2)^(5/2)*e) + 1/10*d*x/(-e^2*x^2 + d^2)^(3/2) + 1/5*x/(sqrt(-e^2*x^2 + d
^2)*d)

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Fricas [A]  time = 0.221507, size = 203, normalized size = 6.15 \[ \frac{2 \,{\left (e^{4} x^{5} + 5 \, d^{2} e^{2} x^{3} - 10 \, d^{4} x + 10 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x\right )}}{5 \,{\left (d e^{5} x^{5} - 5 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} + 5 \, d^{4} e^{2} x^{2} - 10 \, d^{5} e x + 4 \, d^{6} +{\left (d e^{4} x^{4} - 7 \, d^{3} e^{2} x^{2} + 10 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

2/5*(e^4*x^5 + 5*d^2*e^2*x^3 - 10*d^4*x + 10*sqrt(-e^2*x^2 + d^2)*d^3*x)/(d*e^5*
x^5 - 5*d^2*e^4*x^4 + 5*d^3*e^3*x^3 + 5*d^4*e^2*x^2 - 10*d^5*e*x + 4*d^6 + (d*e^
4*x^4 - 7*d^3*e^2*x^2 + 10*d^4*e*x - 4*d^5)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**5/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.230482, size = 103, normalized size = 3.12 \[ -\frac{{\left (d^{4} e^{\left (-1\right )} +{\left (5 \, d^{3} +{\left (10 \, d^{2} e +{\left (x{\left (\frac{x e^{4}}{d} + 5 \, e^{3}\right )} + 10 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-1/5*(d^4*e^(-1) + (5*d^3 + (10*d^2*e + (x*(x*e^4/d + 5*e^3) + 10*d*e^2)*x)*x)*x
)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3